From Tealeaves Require Export
  Classes.Kleisli.DecoratedFunctor
  Classes.Kleisli.DecoratedMonad
  Classes.Kleisli.TraversableFunctor
  Classes.Kleisli.TraversableMonad
  Classes.Kleisli.DecoratedTraversableFunctor
  Functors.Early.Writer.

Import Monoid.Notations.
Import Product.Notations.
Import Monad.Notations.
Import Comonad.Notations.
Import DecoratedMonad.Notations.
Import TraversableFunctor.Notations.
Import TraversableMonad.Notations.
Import DecoratedTraversableFunctor.Notations.

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

(** * Decorated Traversable Monads *)
(**********************************************************************)

(** ** Operation <<binddt>>  *)
(**********************************************************************)
Class Binddt
  (W: Type)
  (T: Type -> Type)
  (U: Type -> Type)
  := binddt:
    forall (G: Type -> Type)
      `{Map_G: Map G} `{Pure_G: Pure G} `{Mult_G: Mult G}
      (A B: Type),
      (W * A -> G (T B)) -> U A -> G (U B).

#[global] Arguments binddt {W}%type_scope {T} {U}%function_scope {Binddt}
  {G}%function_scope {Map_G Pure_G Mult_G}
  {A B}%type_scope _%function_scope _.

(** ** Kleisli composition *)
(**********************************************************************)
Definition kc7
  {W: Type}
  {T: Type -> Type}
  `{Return_T: Return T}
  `{Binddt_WTT: Binddt W T T}
  `{op: Monoid_op W} `{unit: Monoid_unit W}
  {A B C: Type}
  (G1 G2: Type -> Type)
  `{Map G1} `{Pure G1} `{Mult G1}
  `{Map G2} `{Pure G2} `{Mult G2}:
  (W * B -> G2 (T C)) ->
  (W * A -> G1 (T B)) ->
  (W * A -> G1 (G2 (T C))) :=
  fun g f '(w, a) =>
    map (F := G1) (A := T B) (B := G2 (T C))
      (binddt (g ⦿ w)) (f (w, a)).

#[local] Infix "⋆7" := (kc7 _ _) (at level 60): tealeaves_scope.

(** ** Typeclass *)
(**********************************************************************)
Class DecoratedTraversableRightPreModule
  (W: Type)
  (T U: Type -> Type)
  `{op: Monoid_op W}
  `{unit: Monoid_unit W}
  `{Return_T: Return T}
  `{Bindd_T: Binddt W T T}
  `{Bindd_U: Binddt W T U} :=
  { kdtm_binddt1: forall (A: Type),
      binddt (G := fun A => A)
        (ret (T := T) ∘ extract (W := (W ×))) = @id (U A);
    kdtm_binddt2:
    forall `{Applicative G1} `{Applicative G2}
      `(g: W * B -> G2 (T C))
      `(f: W * A -> G1 (T B)),
      map (F := G1) (binddt g) ∘ binddt f =
        binddt (U := U) (G := G1 ∘ G2) (g ⋆7 f);
    kdtm_morph:
    forall (G1 G2: Type -> Type) `{morph: ApplicativeMorphism G1 G2 ϕ}
      `(f: W * A -> G1 (T B)),
      ϕ (U B) ∘ binddt f = binddt (ϕ (T B) ∘ f);
  }.

Class DecoratedTraversableMonad
  (W: Type)
  (T: Type -> Type)
  `{op: Monoid_op W}
  `{unit: Monoid_unit W}
  `{Return_inst: Return T}
  `{Bindd_T_inst: Binddt W T T} :=
  { kdtm_monoid :> Monoid W;
    kdtm_binddt0: forall `{Applicative G} `(f: W * A -> G (T B)),
      binddt f ∘ ret = f ∘ ret (T := (W ×));
    kdtm_premod :> DecoratedTraversableRightPreModule W T T;
  }.

Class DecoratedTraversableRightModule
  (W: Type)
  (T U: Type -> Type)
  `{op: Monoid_op W}
  `{unit: Monoid_unit W}
  `{Return_inst: Return T}
  `{Bindd_T_inst: Binddt W T T}
  `{Bindd_U_inst: Binddt W T U}
  :=
  { kdtmod_monad :> DecoratedTraversableMonad W T;
    kdtmod_premod :> DecoratedTraversableRightPreModule W T U;
  }.

(** ** Homomorphisms *)
(**********************************************************************)
Class DecoratedTraversableMonadHom
  (T U: Type -> Type)
  `{Return T} `{Binddt W T T}
  `{Return U} `{Binddt W U U}
  (ϕ: forall (A: Type), T A -> U A) :=
  { kmon_hom_ret: forall (A: Type),
      ϕ A ∘ @ret T _ A = @ret U _ A;
    kmon_hom_bind:
    forall `{Applicative G} `(f: W * A -> G (T B)),
      map (F := G) (ϕ B) ∘ @binddt W T T _ G _ _ _ A B f =
        @binddt W U U _ G _ _ _ A B (map (F := G) (ϕ B) ∘ f) ∘ ϕ A;
  }.

(** ** Kleisi Category Laws *)
(**********************************************************************)
Section decorated_monad_kleisli_composition.

  (*
    #[local] Set Typeclasses Iterative Deepening.
   *)

  Context
    `{Return_T: Return T}
    `{Binddt_WTT: Binddt W T T}
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{! Monoid W}.

  (** *** Interaction with [incr], [preincr] *)
  (********************************************************************)
  Section incr.

    Context
      `{Applicative G1}
      `{Applicative G2}
      {A B C: Type}.

    Lemma kc7_incr:
      forall `(g: W * B -> G2 (T C)) `(f: W * A -> G1 (T B)) (w: W),
        (g ∘ incr w) ⋆7 (f ∘ incr w) = (g ⋆7 f) ∘ incr w.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 (T B))
  (w : W),
g ∘ incr w ⋆7 f ∘ incr w = (g ⋆7 f) ∘ incr w
    Proof.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 (T B))
  (w : W),
g ∘ incr w ⋆7 f ∘ incr w = (g ⋆7 f) ∘ incr w
      intros.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
g ∘ incr w ⋆7 f ∘ incr w = (g ⋆7 f) ∘ incr w unfold kc7.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
(fun '(w0, a) =>
 map (binddt ((g ∘ incr w) ⦿ w0))
   ((f ∘ incr w) (w0, a))) =
(fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
∘ incr w ext [w' a].
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w, w': W
a: A
map (binddt ((g ∘ incr w) ⦿ w'))
  ((f ∘ incr w) (w', a)) =
((fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
 ∘ incr w) (w', a)
      unfold preincr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w, w': W
a: A
map (binddt (g ∘ incr w ∘ incr w'))
  ((f ∘ incr w) (w', a)) =
((fun '(w, a) => map (binddt (g ∘ incr w)) (f (w, a)))
 ∘ incr w) (w', a) reassociate ->.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w, w': W
a: A
map (binddt (g ∘ (incr w ∘ incr w')))
  ((f ∘ incr w) (w', a)) =
((fun '(w, a) => map (binddt (g ∘ incr w)) (f (w, a)))
 ∘ incr w) (w', a)
      rewrite incr_incr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w, w': W
a: A
map (binddt (g ∘ incr (w ● w')))
  ((f ∘ incr w) (w', a)) =
((fun '(w, a) => map (binddt (g ∘ incr w)) (f (w, a)))
 ∘ incr w) (w', a)
      reflexivity.
    Qed.

    Lemma kc7_preincr:
      forall `(g: W * B -> G2 (T C)) `(f: W * A -> G1 (T B)) (w: W),
        (g ⦿ w) ⋆7 (f ⦿ w) = (g ⋆7 f) ⦿ w.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 (T B))
  (w : W), g ⦿ w ⋆7 f ⦿ w = (g ⋆7 f) ⦿ w
    Proof.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 (T B))
  (w : W), g ⦿ w ⋆7 f ⦿ w = (g ⋆7 f) ⦿ w
      intros.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
g ⦿ w ⋆7 f ⦿ w = (g ⋆7 f) ⦿ w unfold preincr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
g ∘ incr w ⋆7 f ∘ incr w = (g ⋆7 f) ∘ incr w rewrite kc7_incr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
(g ⋆7 f) ∘ incr w = (g ⋆7 f) ∘ incr w
      reflexivity.
    Qed.

  End incr.

  Context
    `{! DecoratedTraversableMonad W T}.

  Context
    `{Applicative G}
    {A B C D: Type}.

  (** *** Left identity *)
  (********************************************************************)
  Lemma kc7_id1: forall (f: W * A -> G (T B)),
      kc7 G (fun A => A) (ret (T := T) ∘ extract (W := (W ×))) f = f.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
forall f : W * A -> G (T B), ret ∘ extract ⋆7 f = f
  Proof.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
forall f : W * A -> G (T B), ret ∘ extract ⋆7 f = f
    intros.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
ret ∘ extract ⋆7 f = f
    unfold kc7.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
(fun '(w, a) =>
 map (binddt ((ret ∘ extract) ⦿ w)) (f (w, a))) = f
    ext [w a].
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
w: W
a: A
map (binddt ((ret ∘ extract) ⦿ w)) (f (w, a)) =
f (w, a)
    rewrite preincr_assoc.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
w: W
a: A
map (binddt (ret ∘ extract ⦿ w)) (f (w, a)) = f (w, a)
    rewrite extract_preincr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
w: W
a: A
map (binddt (ret ∘ extract)) (f (w, a)) = f (w, a)
    rewrite kdtm_binddt1.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
w: W
a: A
map id (f (w, a)) = f (w, a)
    rewrite (fun_map_id (F := G)).
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
f: W * A -> G (T B)
w: W
a: A
id (f (w, a)) = f (w, a)
    reflexivity.
  Qed.

  (** *** Right identity *)
  (********************************************************************)
  Lemma kc7_id2: forall (g: W * A -> G (T B)),
      kc7 (fun A => A) G g (ret (T := T) ∘ extract (W := (W ×))) = g.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
forall g : W * A -> G (T B), g ⋆7 ret ∘ extract = g
  Proof.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
forall g : W * A -> G (T B), g ⋆7 ret ∘ extract = g
    intros.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
g ⋆7 ret ∘ extract = g
    unfold kc7.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
(fun '(w, a) =>
 map (binddt (g ⦿ w)) ((ret ∘ extract) (w, a))) = g
    ext [w a].
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
map (binddt (g ⦿ w)) ((ret ∘ extract) (w, a)) =
g (w, a)
    unfold_ops @Map_I.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
binddt (g ⦿ w) ((ret ∘ extract) (w, a)) = g (w, a)
    change_left ((binddt (T := T) (g ⦿ w) ∘ ret (T := T)) a).
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
(binddt (g ⦿ w) ∘ ret) a = g (w, a)
    rewrite kdtm_binddt0.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
(g ⦿ w ∘ ret) a = g (w, a)
    change (g (w ● Ƶ, a) = g (w, a)).
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
g (w ● Ƶ, a) = g (w, a)
    simpl_monoid.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
g: W * A -> G (T B)
w: W
a: A
g (w, a) = g (w, a)
    reflexivity.
  Qed.

  (** *** Associativity *)
  (********************************************************************)
  Lemma kc7_assoc
    `{Applicative G3}
    `{Applicative G2}
    `{Applicative G1}:
    forall (h: W * C -> G3 (T D))
      (g: W * B -> G2 (T C))
      (f: W * A -> G1 (T B)),
      kc7 (G1 ∘ G2) G3 h (g ⋆7 f) = kc7 G1 (G2 ∘ G3) (h ⋆7 g) f.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
forall (h : W * C -> G3 (T D)) (g : W * B -> G2 (T C))
  (f : W * A -> G1 (T B)),
h ⋆7 (g ⋆7 f) = (h ⋆7 g) ⋆7 f
  Proof.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
forall (h : W * C -> G3 (T D)) (g : W * B -> G2 (T C))
  (f : W * A -> G1 (T B)),
h ⋆7 (g ⋆7 f) = (h ⋆7 g) ⋆7 f
    intros.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
h ⋆7 (g ⋆7 f) = (h ⋆7 g) ⋆7 f
    unfold kc7 at 1 2.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
(fun '(w, a) =>
 map (binddt (h ⦿ w))
   (map (binddt (g ⦿ w)) (f (w, a)))) = (h ⋆7 g) ⋆7 f
    ext [w a].
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (binddt (h ⦿ w)) (map (binddt (g ⦿ w)) (f (w, a))) =
((h ⋆7 g) ⋆7 f) (w, a)
    unfold_ops @Map_compose.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (map (binddt (h ⦿ w)))
  (map (binddt (g ⦿ w)) (f (w, a))) =
((h ⋆7 g) ⋆7 f) (w, a)
    compose near (f (w, a)) on left.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
(map (map (binddt (h ⦿ w))) ∘ map (binddt (g ⦿ w)))
  (f (w, a)) = ((h ⋆7 g) ⋆7 f) (w, a)
    rewrite (fun_map_map (F := G1)).
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (map (binddt (h ⦿ w)) ∘ binddt (g ⦿ w)) (f (w, a)) =
((h ⋆7 g) ⋆7 f) (w, a)
    rewrite kdtm_binddt2.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (binddt (h ⦿ w ⋆7 g ⦿ w)) (f (w, a)) =
((h ⋆7 g) ⋆7 f) (w, a)
    rewrite kc7_preincr.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (binddt ((h ⋆7 g) ⦿ w)) (f (w, a)) =
((h ⋆7 g) ⋆7 f) (w, a)
    unfold kc7 at 2.
T: Type -> Type
Return_T: Return T
W: Type
Binddt_WTT: Binddt W T T
op: Monoid_op W
unit: Monoid_unit W
Monoid0: Monoid W
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B, C, D: Type
G3: Type -> Type
Map_G0: Map G3
Pure_G0: Pure G3
Mult_G0: Mult G3
H0: Applicative G3
G2: Type -> Type
Map_G1: Map G2
Pure_G1: Pure G2
Mult_G1: Mult G2
H1: Applicative G2
G1: Type -> Type
Map_G2: Map G1
Pure_G2: Pure G1
Mult_G2: Mult G1
H2: Applicative G1
h: W * C -> G3 (T D)
g: W * B -> G2 (T C)
f: W * A -> G1 (T B)
w: W
a: A
map (binddt ((h ⋆7 g) ⦿ w)) (f (w, a)) =
map (binddt ((h ⋆7 g) ⦿ w)) (f (w, a))
    reflexivity.
  Qed.

End decorated_monad_kleisli_composition.

(** * Derived Structures *)
(**********************************************************************)

(** ** Derived Operations *)
(**********************************************************************)
Module DerivedOperations.
  Section operations.

    Context
      (W: Type)
      (T: Type -> Type)
      (U: Type -> Type)
      `{Return_T: Return T}
      `{Bindd_WTU: Binddt W T U}.

    #[export] Instance Map_Binddt: Map U :=
      fun (A B: Type) (f: A -> B) =>
        binddt (G := fun A => A)
          (ret (T := T) ∘ f ∘ extract (W := (W ×))).

    #[export] Instance Mapdt_Binddt: Mapdt W U
      := fun G _ _ _ A B f =>
           binddt (map (F := G) (ret (T := T)) ∘ f).

    #[export] Instance Bindd_Binddt: Bindd W T U
      := fun A B f => binddt (G := fun A => A) f.

    #[export] Instance Bindt_Binddt: Bindt T U
      := fun G _ _ _ A B f =>
           binddt (f ∘ extract (W := (W ×))).

    #[export] Instance Bind_Binddt: Bind T U
      := fun A B f =>
           binddt (G := fun A => A)
             (f ∘ extract (W := (W ×))).

    #[export] Instance Mapd_Binddt: Mapd W U
      := fun A B f =>
           binddt (G := fun A => A) (ret (T := T) ∘ f).

    #[export] Instance Traverse_Binddt: Traverse U
      := fun G _ _ _ A B f =>
           binddt (map (F := G) ret ∘ f ∘ extract).

  End operations.
End DerivedOperations.

(** ** Compatibility Typeclasses *)
(**********************************************************************)
Section decorated_traversable_monad_compat.

  Context
    (W: Type)
    (T: Type -> Type)
    (U: Type -> Type)
    `{Return_T: Return T}
    `{Binddt_inst: Binddt W T U}
    `{Map_inst: Map U}
    `{Mapd_inst: Mapd W U}
    `{Traverse_inst: Traverse U}
    `{Bind_inst: Bind T U}
    `{Mapdt_inst: Mapdt W U}
    `{Bindd_inst: Bindd W T U}
    `{Bindt_inst: Bindt T U}.

  Class Compat_Map_Binddt: Prop :=
    compat_map_binddt:
      Map_inst =
        @DerivedOperations.Map_Binddt W T U Return_T Binddt_inst.

  Class Compat_Mapd_Binddt: Prop :=
    compat_mapd_binddt:
      Mapd_inst =
        @DerivedOperations.Mapd_Binddt W T U Return_T Binddt_inst.

  Class Compat_Traverse_Binddt: Prop :=
    compat_traverse_binddt:
      Traverse_inst =
        @DerivedOperations.Traverse_Binddt W T U Return_T Binddt_inst.

  Class Compat_Bind_Binddt: Prop :=
    compat_bind_binddt:
      Bind_inst =
        @DerivedOperations.Bind_Binddt W T U Binddt_inst.

  Class Compat_Bindd_Binddt: Prop :=
    compat_bindd_binddt:
      Bindd_inst =
        @DerivedOperations.Bindd_Binddt W T U Binddt_inst.

  Class Compat_Bindt_Binddt: Prop :=
    compat_bindt_binddt:
      Bindt_inst =
        @DerivedOperations.Bindt_Binddt W T U Binddt_inst.

  Class Compat_Mapdt_Binddt: Prop :=
    compat_mapdt_binddt:
      Mapdt_inst =
        @DerivedOperations.Mapdt_Binddt W T U Return_T Binddt_inst.

  Class Compat_Full_Binddt: Prop :=
    { compat_map_binddt_full :> Compat_Map_Binddt;
      compat_mapd_binddt_full :> Compat_Mapd_Binddt;
      compat_traverse_binddt_full :> Compat_Traverse_Binddt;
      compat_bind_binddt_full :> Compat_Bind_Binddt;
      compat_bindd_binddt_full :> Compat_Bindd_Binddt;
      compat_bindt_binddt_full :> Compat_Bindt_Binddt;
      compat_mapdt_binddt_full :> Compat_Mapdt_Binddt;
    }.

End decorated_traversable_monad_compat.

Section decorated_traversable_monad_compat_self.

  Context
    (W: Type)
    (T: Type -> Type)
    (U: Type -> Type)
    `{Return_T: Return T}
    `{Binddt_WTU: Binddt W T U}.

  #[export] Instance Compat_Map_Binddt_Self:
    Compat_Map_Binddt W T U
      (Map_inst := DerivedOperations.Map_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Map_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Map_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Mapd_Binddt_Self:
    Compat_Mapd_Binddt W T U
      (Mapd_inst := DerivedOperations.Mapd_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Mapd_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Mapd_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Traverse_Binddt_Self:
    Compat_Traverse_Binddt W T U
      (Traverse_inst := DerivedOperations.Traverse_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Traverse_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Traverse_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Bind_Binddt_Self:
    Compat_Bind_Binddt W T U
      (Bind_inst := DerivedOperations.Bind_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bind_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bind_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Bindd_Binddt_Self:
    Compat_Bindd_Binddt W T U
      (Bindd_inst := DerivedOperations.Bindd_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bindd_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bindd_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Bindt_Binddt_Self:
    Compat_Bindt_Binddt W T U
      (Bindt_inst := DerivedOperations.Bindt_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bindt_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Bindt_Binddt W T U
    reflexivity.
  Qed.

  #[export] Instance Compat_Mapdt_Binddt_Self:
    Compat_Mapdt_Binddt W T U
      (Mapdt_inst := DerivedOperations.Mapdt_Binddt W T U).
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt W T U
  Proof.
W: Type
T, U: Type -> Type
Return_T: Return T
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt W T U
    reflexivity.
  Qed.

  Ltac normalize_binddt :=
    repeat (rewrite (compat_map_binddt W T U) ||
              rewrite (compat_mapd_binddt W T U) ||
                rewrite (compat_traverse_binddt W T U) ||
                  rewrite (compat_mapdt_binddt W T U) ||
                    rewrite (compat_bind_binddt W T U) ||
                      rewrite (compat_bindd_binddt W T U) ||
                        rewrite (compat_bindt_binddt W T U)).

  Ltac solve_compat :=
    hnf;
    normalize_binddt;
    unfold_ops @DerivedOperations.Map_Bind;
    unfold_ops @DerivedOperations.Map_Traverse;
    unfold_ops @DerivedOperations.Map_Mapd;
    unfold_ops @DerivedOperations.Map_Mapdt;
    unfold_ops @DerivedOperations.Mapd_Mapdt;
    unfold_ops @DerivedOperations.Traverse_Mapdt;
    unfold_ops @DerivedOperations.Map_Bindd;
    unfold_ops @DerivedOperations.Mapd_Bindd;
    unfold_ops @DerivedOperations.Bind_Bindd;
    unfold_ops @DerivedOperations.Map_Bindt;
    unfold_ops @DerivedOperations.Traverse_Bindt;
    unfold_ops @DerivedOperations.Bind_Bindt;
    normalize_binddt;
    reflexivity.

  Context
    `{Map_U: Map U}
    `{Mapd_WU: Mapd W U}
    `{Traverse_U: Traverse U}
    `{Mapdt_WU: Mapdt W U}
    `{Bind_TU: Bind T U}
    `{Bindd_WTU: Bindd W T U}
    `{Bindt_TU: Bindt T U}
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Mapd_Binddt W T U}
    `{! Compat_Traverse_Binddt W T U}
    `{! Compat_Mapdt_Binddt W T U}
    `{! Compat_Bind_Binddt W T U}
    `{! Compat_Bindd_Binddt W T U}
    `{! Compat_Bindt_Binddt W T U}.

  (** *** Compatibility with <<traverse>>, <<mapd>>, and <<bind>>.*)
  #[export] Instance Compat_Map_Traverse_Binddt:
    Compat_Map_Traverse U := ltac:(solve_compat).

  #[export] Instance Compat_Map_Mapd_Binddt:
    Compat_Map_Mapd W U := ltac:(solve_compat).

  #[export] Instance Compat_Map_Bind_Binddt:
    Compat_Map_Bind T U := ltac:(solve_compat).

  (** *** Compatibility with <<bindd>> *)
  #[export] Instance Compat_Map_Bindd_Binddt:
    Compat_Map_Bindd W T U := ltac:(solve_compat).

  #[export] Instance Compat_Mapd_Bindd_Binddt:
    Compat_Mapd_Bindd W T U := ltac:(solve_compat).

  #[export] Instance Compat_Bind_Bindd_Binddt:
    Compat_Bind_Bindd W T U := ltac:(solve_compat).

  (** *** Compatibility with <<bindt>> *)
  #[export] Instance Compat_Map_Bindt_Bindtt:
    Compat_Map_Bindt T U := ltac:(solve_compat).

  #[export] Instance Compat_Traverse_Bindt_Bindtt:
    Compat_Traverse_Bindt T U := ltac:(solve_compat).

  #[export] Instance Compat_Bind_Bindt_Bindtt:
    Compat_Bind_Bindt T U := ltac:(solve_compat).

  (** *** Compatibility with <<mapdt>> *)
  #[export] Instance Compat_Map_Mapdt_Binddt:
    Compat_Map_Mapdt W U := ltac:(solve_compat).

  #[export] Instance Compat_Mapd_Mapdt_Binddt:
    Compat_Mapd_Mapdt W U := ltac:(solve_compat).

  #[export] Instance Compat_Traverse_Mapdt_Binddt:
    Compat_Traverse_Mapdt W U := ltac:(solve_compat).

End decorated_traversable_monad_compat_self.

(** ** Rewriting <<X>> to <<binddt>> Lemmas *)
(**********************************************************************)
Section rewriting.

  Context
    `{Return_T: Return T}
    `{Map_U: Map U}
    `{Mapd_WU: Mapd W U}
    `{Traverse_U: Traverse U}
    `{Bind_TU: Bind T U}
    `{Mapdt_WU: Mapdt W U}
    `{Bindd_WTU: Bindd W T U}
    `{Bindt_TU: Bindt T U}
    `{Binddt_WTU: Binddt W T U}.

  Lemma map_to_binddt `{! Compat_Map_Binddt W T U}:
    forall `(f: A -> B),
      map (F := U) f =
        binddt (G := fun A => A) (ret (T := T) ∘ f ∘ extract).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = binddt (ret ∘ f ∘ extract)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = binddt (ret ∘ f ∘ extract)
    rewrite (compat_map_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
forall (A B : Type) (f : A -> B),
DerivedOperations.Map_Binddt W T U A B f =
binddt (ret ∘ f ∘ extract)
    reflexivity.
  Qed.

  Lemma mapd_to_binddt `{! Compat_Mapd_Binddt W T U}:
    forall `(f: W * A -> B),
      mapd (T := U) f =
        binddt (G := fun A => A) (ret (T := T) ∘ f).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T U
forall (A B : Type) (f : W * A -> B),
mapd f = binddt (ret ∘ f)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T U
forall (A B : Type) (f : W * A -> B),
mapd f = binddt (ret ∘ f)
    rewrite (compat_mapd_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T U
forall (A B : Type) (f : W * A -> B),
DerivedOperations.Mapd_Binddt W T U A B f =
binddt (ret ∘ f)
    reflexivity.
  Qed.

  Lemma traverse_to_binddt `{! Compat_Traverse_Binddt W T U}
    `{Applicative G}:
    forall `(f: A -> G B),
      traverse (G := G) (T := U) f =
        binddt (U := U) (map (F := G) (ret (T := T)) ∘ f ∘ extract).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
forall (A B : Type) (f : A -> G B),
traverse f = binddt (map ret ∘ f ∘ extract)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
forall (A B : Type) (f : A -> G B),
traverse f = binddt (map ret ∘ f ∘ extract)
    rewrite (compat_traverse_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
forall (A B : Type) (f : A -> G B),
DerivedOperations.Traverse_Binddt W T U G Map_G Pure_G
  Mult_G A B f = binddt (map ret ∘ f ∘ extract)
    reflexivity.
  Qed.

  Lemma bind_to_binddt `{! Compat_Bind_Binddt W T U}:
    forall `(f: A -> T B),
      bind (U := U) f = binddt (U := U) (f ∘ extract).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> T B),
bind f = binddt (f ∘ extract)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> T B),
bind f = binddt (f ∘ extract)
    rewrite (compat_bind_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> T B),
DerivedOperations.Bind_Binddt W T U A B f =
binddt (f ∘ extract)
    reflexivity.
  Qed.

  Lemma mapdt_to_binddt `{! Compat_Mapdt_Binddt W T U}:
    forall `{Applicative G} `(f: W * A -> G B),
      mapdt (G := G) f = binddt (map (ret (T := T)) ∘ f).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : W * A -> G B),
mapdt f = binddt (map ret ∘ f)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : W * A -> G B),
mapdt f = binddt (map ret ∘ f)
    rewrite (compat_mapdt_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : W * A -> G B),
DerivedOperations.Mapdt_Binddt W T U G Map_G Pure_G
  Mult_G A B f = binddt (map ret ∘ f)
    reflexivity.
  Qed.

  Lemma mapdt_to_binddt' `{! Compat_Mapdt_Binddt W T U}:
    forall `{Applicative G} (A B: Type),
      mapdt (T := U) (G := G) (A := A) (B := B) =
        binddt ∘ compose (map ret).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall A B : Type, mapdt = binddt ∘ compose (map ret)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall A B : Type, mapdt = binddt ∘ compose (map ret)
    intros.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
mapdt = binddt ∘ compose (map ret)
    ext f.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: W * A -> G B
mapdt f = (binddt ∘ compose (map ret)) f
    rewrite mapdt_to_binddt.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: W * A -> G B
binddt (map ret ∘ f) = (binddt ∘ compose (map ret)) f
    reflexivity.
  Qed.

  Lemma bindd_to_binddt `{! Compat_Bindd_Binddt W T U}:
    forall (A B: Type),
      bindd (T := T) (U := U) (A := A) (B := B) =
        binddt (G := fun A => A).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
forall A B : Type, bindd = binddt
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
forall A B : Type, bindd = binddt
    rewrite (compat_bindd_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
forall A B : Type,
DerivedOperations.Bindd_Binddt W T U A B = binddt
    reflexivity.
  Qed.

  Lemma bindt_to_binddt `{! Compat_Bindt_Binddt W T U}
    `{Applicative G} `(f: A -> G (T B)):
    bindt (G := G) f = binddt (U := U) (f ∘ extract).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
bindt f = binddt (f ∘ extract)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
bindt f = binddt (f ∘ extract)
    rewrite (compat_bindt_binddt W T U).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
DerivedOperations.Bindt_Binddt W T U G Map_G Pure_G
  Mult_G A B f = binddt (f ∘ extract)
    reflexivity.
  Qed.

  Lemma map_to_bind
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Bind_Binddt W T U}
   : forall (A B: Type) (f: A -> B),
      map f = bind (ret ∘ f).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = bind (ret ∘ f)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = bind (ret ∘ f)
    rewrite (compat_map_bind
               (Compat_Map_Bind :=
                  Compat_Map_Bind_Binddt W T U)).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bind_Binddt0: Compat_Bind_Binddt W T U
forall (A B : Type) (f : A -> B),
DerivedOperations.Map_Bind T U A B f = bind (ret ∘ f)
    reflexivity.
  Qed.

  Lemma map_to_bindd
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Bindd_Binddt W T U}
   : forall (A B: Type) (f: A -> B),
      map (F := U) f = bindd (ret ∘ f ∘ extract).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = bindd (ret ∘ f ∘ extract)
  Proof.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
forall (A B : Type) (f : A -> B),
map f = bindd (ret ∘ f ∘ extract)
    intros.
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
A, B: Type
f: A -> B
map f = bindd (ret ∘ f ∘ extract)
    rewrite (compat_map_bindd W T U
               (Compat_Map_Bindd :=
                  Compat_Map_Bindd_Binddt W T U)).
T: Type -> Type
Return_T: Return T
U: Type -> Type
Map_U: Map U
W: Type
Mapd_WU: Mapd W U
Traverse_U: Traverse U
Bind_TU: Bind T U
Mapdt_WU: Mapdt W U
Bindd_WTU: Bindd W T U
Bindt_TU: Bindt T U
Binddt_WTU: Binddt W T U
Compat_Map_Binddt0: Compat_Map_Binddt W T U
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T U
A, B: Type
f: A -> B
DerivedOperations.Map_Bindd W T U A B f =
bindd (ret ∘ f ∘ extract)
    reflexivity.
  Qed.

End rewriting.

(** ** Composition with the Identity Applicative Functor *)
(**********************************************************************)
Section properties.

  Context
    `{DecoratedTraversableRightModule W T U}.

  Lemma binddt_app_id_l:
    forall {G: Type -> Type} {A B: Type}
           `{Applicative G} (f: W * A -> G (T B)),
      @binddt W T U _ ((fun A => A) ∘ G)
        (Map_compose (fun A => A) G)
        (Pure_compose (fun A => A) G)
        (Mult_compose (fun A => A) G) A B f =
        binddt (T := T) f.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
forall (G : Type -> Type) (A B : Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G (T B), binddt f = binddt f
  Proof.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
forall (G : Type -> Type) (A B : Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G (T B), binddt f = binddt f
    intros.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
G: Type -> Type
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G (T B)
binddt f = binddt f fequal.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
G: Type -> Type
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G (T B)
Mult_compose (fun A : Type => A) G = Mult_G now rewrite (Mult_compose_identity2 G).
  Qed.

  Lemma binddt_app_id_r:
    forall {G: Type -> Type} {A B: Type}
           `{Applicative G} (f: W * A -> G (T B)),
      @binddt W T U _ (G ∘ (fun A => A))
        (Map_compose G (fun A => A))
        (Pure_compose G (fun A => A))
        (Mult_compose G (fun A => A)) A B f =
        binddt f.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
forall (G : Type -> Type) (A B : Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G (T B), binddt f = binddt f
  Proof.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
forall (G : Type -> Type) (A B : Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G (T B), binddt f = binddt f
    intros.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
G: Type -> Type
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G (T B)
binddt f = binddt f fequal.
W: Type
T, U: Type -> Type
op: Monoid_op W
unit0: Monoid_unit W
Return_inst: Return T
Bindd_T_inst: Binddt W T T
Bindd_U_inst: Binddt W T U
H: DecoratedTraversableRightModule W T U
G: Type -> Type
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G (T B)
Mult_compose G (fun A : Type => A) = Mult_G now rewrite (Mult_compose_identity1 G).
  Qed.

End properties.

(** ** Derived Kleisli Composition Laws *)
(**********************************************************************)
Section derived_instances.

  (** *** Section Context *)
  (********************************************************************)
  Context
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd W T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt W T}
    `{Bindd_T_inst: Bindd W T T}
    `{Bindt_T_inst: Bindt T T}
    `{Binddt_T_inst: Binddt W T T}
    `{! Compat_Map_Binddt W T T}
    `{! Compat_Mapd_Binddt W T T}
    `{! Compat_Traverse_Binddt W T T}
    `{! Compat_Bind_Binddt W T T}
    `{! Compat_Mapdt_Binddt W T T}
    `{! Compat_Bindd_Binddt W T T}
    `{! Compat_Bindt_Binddt W T T}
    `{Monad_inst: ! DecoratedTraversableMonad W T}
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd W U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt W U}
    `{Bindd_U_inst: Bindd W T U}
    `{Bindt_U_inst: Bindt T U}
    `{Binddt_U_inst: Binddt W T U}
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Mapd_Binddt W T U}
    `{! Compat_Traverse_Binddt W T U}
    `{! Compat_Bind_Binddt W T U}
    `{! Compat_Mapdt_Binddt W T U}
    `{! Compat_Bindd_Binddt W T U}
    `{! Compat_Bindt_Binddt W T U}
    `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.

  (** *** Tactical support *)
  (********************************************************************)
  #[local] Ltac unfold_map_id :=
    repeat (change (map (F := fun A => A) ?f) with f).

  #[local] Hint Unfold kc7 kc6 kc5 kc kc3 kc2 kc1: tealeaves_kc_unfold.

  #[local] Ltac setup :=
    intros;
    autounfold with tealeaves_kc_unfold;
    ext [w a];
    unfold_map_id.

  Ltac kdtmf_normalize_T :=
    repeat
      ( rewrite (map_to_binddt (U := T) (T := T)) ||
          rewrite (bind_to_binddt (U := T) (T := T)) ||
            rewrite (traverse_to_binddt (U := T) (T := T)) ||
              rewrite (mapd_to_binddt (U := T) (T := T)) ||
                rewrite (mapdt_to_binddt (U := T) (T := T)) ||
                  rewrite (bindd_to_binddt (U := T) (T := T)) ||
                    rewrite (bindt_to_binddt (U := T) (T := T))).

  Ltac solve_kc7 :=
    setup;
    kdtmf_normalize_T;
    try (reflexivity ||
           rewrite preincr_assoc;
         rewrite extract_preincr;
         reflexivity).

  (** *** Homogeneous Kleisli composition laws *)
  (********************************************************************)
  Section composition.

    Context
      `{Applicative G1}
      `{Applicative G2}.

    Variables (A B C: Type).

    (** **** Lemmas <<kc7_xx>> *)
    (******************************************************************)
    Lemma kc7_33:
      forall (g: W * B -> G2 C) (f: W * A -> G1 B),
        (map (F := G2) (ret (T := T)) ∘ g) ⋆7
          (map (F := G1) (ret (T := T)) ∘ f) =
          map (F := G1 ∘ G2) (ret (T := T)) ∘ (g ⋆3 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 B),
map ret ∘ g ⋆7 map ret ∘ f = map ret ∘ (g ⋆3 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 B),
map ret ∘ g ⋆7 map ret ∘ f = map ret ∘ (g ⋆3 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
map ret ∘ g ⋆7 map ret ∘ f = map ret ∘ (g ⋆3 f)
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
(fun '(w, a) =>
 map (binddt ((map ret ∘ g) ⦿ w))
   ((map ret ∘ f) (w, a))) = map ret ∘ (g ⋆3 f)
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (binddt ((map ret ∘ g) ⦿ w))
  ((map ret ∘ f) (w, a)) = (map ret ∘ (g ⋆3 f)) (w, a)
      (* LHS *)
      unfold compose at 2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (binddt ((map ret ∘ g) ⦿ w)) (map ret (f (w, a))) =
(map ret ∘ (g ⋆3 f)) (w, a)
      compose near (f (w, a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
(map (binddt ((map ret ∘ g) ⦿ w)) ∘ map ret)
  (f (w, a)) = (map ret ∘ (g ⋆3 f)) (w, a)
      rewrite (fun_map_map (F := G1)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (binddt ((map ret ∘ g) ⦿ w) ∘ ret) (f (w, a)) =
(map ret ∘ (g ⋆3 f)) (w, a)
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map ((map ret ∘ g) ⦿ w ∘ ret) (f (w, a)) =
(map ret ∘ (g ⋆3 f)) (w, a)
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map ret ∘ (g ⋆3 f)) (w, a)
      (* RHS *)
      unfold kc3.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map ret ∘ (map g ∘ strength ∘ cobind f)) (w, a)
      unfold_ops @Map_compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map (map ret) ∘ (map g ∘ strength ∘ cobind f)) (w, a)
      do 2 reassociate <- on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map (map ret) ∘ map g ∘ strength ∘ cobind f) (w, a)
      unfold_compose_in_compose; rewrite (fun_map_map (F := G1)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map (map ret ∘ g) ∘ strength ∘ cobind f) (w, a)
      unfold_ops @Cobind_reader; unfold strength.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map (map ret ∘ g) ∘ (fun '(a, t) => map (pair a) t)
 ∘ (fun '(e, a) => (e, f (e, a)))) (w, a)
      unfold compose at 3 4.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
map (map ret ∘ g) (map (pair w) (f (w, a)))
      compose near (f (w, a)) on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
(map (map ret ∘ g) ∘ map (pair w)) (f (w, a))
      rewrite (fun_map_map (F := G1)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
w: W
a: A
map (map ret ∘ g ∘ pair w) (f (w, a)) =
map (map ret ∘ g ∘ pair w) (f (w, a))
      reflexivity.
    Qed.

    Lemma kc7_55:
      forall `(g: W * B -> T C) `(f: W * A -> T B),
        kc7 (fun A => A) (fun A => A) g f = g ⋆5 f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> T B),
g ⋆7 f = g ⋆5 f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> T B),
g ⋆7 f = g ⋆5 f
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
g ⋆7 f = g ⋆5 f
      unfold kc5.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
g ⋆7 f = (fun '(w, a) => bindd (g ⦿ w) (f (w, a)))
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
w: W
a: A
(g ⋆7 f) (w, a) = bindd (g ⦿ w) (f (w, a))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
w: W
a: A
(g ⋆7 f) (w, a) = binddt (g ⦿ w) (f (w, a))
      reflexivity.
    Qed.

    Lemma kc7_44:
      forall `(g: W * B -> C) `(f: W * A -> B),
        kc7 (fun A => A) (fun A => A)
          (ret (T := T) ∘ g) (ret (T := T) ∘ f) =
          ret (T := T) ∘ (g ⋆1 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> B),
ret ∘ g ⋆7 ret ∘ f = ret ∘ (g ⋆1 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> B),
ret ∘ g ⋆7 ret ∘ f = ret ∘ (g ⋆1 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
ret ∘ g ⋆7 ret ∘ f = ret ∘ (g ⋆1 f)
      rewrite kc7_55.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
ret ∘ g ⋆5 ret ∘ f = ret ∘ (g ⋆1 f)
      unfold kc5.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
(fun '(w, a) =>
 bindd ((ret ∘ g) ⦿ w) ((ret ∘ f) (w, a))) =
ret ∘ (g ⋆1 f)
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
bindd ((ret ∘ g) ⦿ w) ((ret ∘ f) (w, a)) =
(ret ∘ (g ⋆1 f)) (w, a)
      unfold compose at 2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
bindd ((ret ∘ g) ⦿ w) (ret (f (w, a))) =
(ret ∘ (g ⋆1 f)) (w, a)
      compose near (f (w, a)) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
(bindd ((ret ∘ g) ⦿ w) ∘ ret) (f (w, a)) =
(ret ∘ (g ⋆1 f)) (w, a)
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
(binddt ((ret ∘ g) ⦿ w) ∘ ret) (f (w, a)) =
(ret ∘ (g ⋆1 f)) (w, a)
      rewrite (kdtm_binddt0 (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
((ret ∘ g) ⦿ w ∘ ret) (f (w, a)) =
(ret ∘ (g ⋆1 f)) (w, a)
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
w: W
a: A
(ret ∘ g ∘ pair w) (f (w, a)) =
(ret ∘ (g ⋆1 f)) (w, a)
      reflexivity.
    Qed.

    Lemma kc7_66:
      forall (g: B -> G2 (T C)) (f: A -> G1 (T B)),
        kc7 G1 G2
          (g ∘ extract (W := (W ×)))
          (f ∘ extract (W := (W ×))) =
          (g ⋆6 f) ∘ extract (W := (W ×)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : A -> G1 (T B)),
g ∘ extract ⋆7 f ∘ extract = (g ⋆6 f) ∘ extract
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : A -> G1 (T B)),
g ∘ extract ⋆7 f ∘ extract = (g ⋆6 f) ∘ extract
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
g ∘ extract ⋆7 f ∘ extract = (g ⋆6 f) ∘ extract
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
(fun '(w, a) =>
 map (binddt ((g ∘ extract) ⦿ w))
   ((f ∘ extract) (w, a))) = (g ⋆6 f) ∘ extract
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
w: W
a: A
map (binddt ((g ∘ extract) ⦿ w))
  ((f ∘ extract) (w, a)) = ((g ⋆6 f) ∘ extract) (w, a)
      rewrite preincr_assoc.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
w: W
a: A
map (binddt (g ∘ extract ⦿ w)) ((f ∘ extract) (w, a)) =
((g ⋆6 f) ∘ extract) (w, a)
      rewrite extract_preincr.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
w: W
a: A
map (binddt (g ∘ extract)) ((f ∘ extract) (w, a)) =
((g ⋆6 f) ∘ extract) (w, a)
      unfold kc6.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
w: W
a: A
map (binddt (g ∘ extract)) ((f ∘ extract) (w, a)) =
(map (bindt g) ∘ f ∘ extract) (w, a)
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
w: W
a: A
map (binddt (g ∘ extract)) ((f ∘ extract) (w, a)) =
(map (binddt (g ∘ extract)) ∘ f ∘ extract) (w, a)
      reflexivity.
    Qed.

    Lemma kc7_22:
      forall (g: B -> G2 C) (f: A -> G1 B),
        kc7 G1 G2
          (map (F := G2) (ret (T := T)) ∘ g ∘ extract (W := (W ×)))
          (map (F := G1) (ret (T := T)) ∘ f ∘ extract (W := (W ×))) =
          kc2 (G1 := G1 ∘ G2) (ret (T := T))
            ((map (F := G1) g ∘ f)) ∘ extract (W := (W ×)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : A -> G1 B),
map ret ∘ g ∘ extract ⋆7 map ret ∘ f ∘ extract =
(ret ⋆2 map g ∘ f) ∘ extract
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : A -> G1 B),
map ret ∘ g ∘ extract ⋆7 map ret ∘ f ∘ extract =
(ret ⋆2 map g ∘ f) ∘ extract
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
map ret ∘ g ∘ extract ⋆7 map ret ∘ f ∘ extract =
(ret ⋆2 map g ∘ f) ∘ extract
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
(fun '(w, a) =>
 map (binddt ((map ret ∘ g ∘ extract) ⦿ w))
   ((map ret ∘ f ∘ extract) (w, a))) =
(ret ⋆2 map g ∘ f) ∘ extract
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (binddt ((map ret ∘ g ∘ extract) ⦿ w))
  ((map ret ∘ f ∘ extract) (w, a)) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      rewrite preincr_assoc.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (binddt (map ret ∘ g ∘ extract ⦿ w))
  ((map ret ∘ f ∘ extract) (w, a)) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      do 2 reassociate ->.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (binddt (map ret ∘ (g ∘ extract ⦿ w)))
  ((map ret ∘ (f ∘ extract)) (w, a)) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      rewrite extract_preincr.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (binddt (map ret ∘ (g ∘ extract)))
  ((map ret ∘ (f ∘ extract)) (w, a)) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      unfold compose at 1 2 3 4; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map
  (binddt (fun a : W * B => map ret (g (extract a))))
  (map ret (f a)) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      compose near (f a) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
(map
   (binddt (fun a : W * B => map ret (g (extract a))))
 ∘ map ret) (f a) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map
  (binddt (fun a : W * B => map ret (g (extract a)))
   ∘ ret) (f a) =
((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map ((fun a : W * B => map ret (g (extract a))) ∘ ret)
  (f a) = ((ret ⋆2 map g ∘ f) ∘ extract) (w, a)
      unfold kc2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map ((fun a : W * B => map ret (g (extract a))) ∘ ret)
  (f a) = (map ret ∘ (map g ∘ f) ∘ extract) (w, a)
      unfold_ops @Map_compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map ((fun a : W * B => map ret (g (extract a))) ∘ ret)
  (f a) =
(map (map ret) ∘ (map g ∘ f) ∘ extract) (w, a)
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (map ret ○ g) (f a) = map (map ret) (map g (f a))
      compose near (f a) on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (map ret ○ g) (f a) =
(map (map ret) ∘ map g) (f a)
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
w: W
a: A
map (map ret ○ g) (f a) = map (map ret ∘ g) (f a)
      reflexivity.
    Qed.

    Lemma kc7_11:
      forall (g: B -> T C) (f: A -> T B),
        kc7 (fun A => A) (fun A => A)
          (g ∘ extract (W := (W ×)))
          (f ∘ extract (W := (W ×))) =
          (g ⋆ f) ∘ extract (W := (W ×)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : A -> T B),
g ∘ extract ⋆7 f ∘ extract = (g ⋆ f) ∘ extract
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : A -> T B),
g ∘ extract ⋆7 f ∘ extract = (g ⋆ f) ∘ extract
      setup.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
w: W
a: A
binddt ((g ∘ extract) ⦿ w) ((f ∘ extract) (w, a)) =
(bind g ∘ f ∘ extract) (w, a)
      rewrite (bind_to_binddt (T := T)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
w: W
a: A
binddt ((g ∘ extract) ⦿ w) ((f ∘ extract) (w, a)) =
(binddt (g ∘ extract) ∘ f ∘ extract) (w, a)
      rewrite preincr_assoc.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
w: W
a: A
binddt (g ∘ extract ⦿ w) ((f ∘ extract) (w, a)) =
(binddt (g ∘ extract) ∘ f ∘ extract) (w, a)
      rewrite extract_preincr.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
w: W
a: A
binddt (g ∘ extract) ((f ∘ extract) (w, a)) =
(binddt (g ∘ extract) ∘ f ∘ extract) (w, a)
      reflexivity.
    Qed.

    Lemma kc7_00:
      forall (g: B -> C) (f: A -> B),
        kc7 (fun A => A) (fun A => A)
          (ret (T := T) ∘ g ∘ extract (W := (W ×)))
          (ret (T := T) ∘ f ∘ extract (W := (W ×))) =
          (ret (T := T) ∘ g ∘ f ∘ extract (W := (W ×))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> C) (f : A -> B),
ret ∘ g ∘ extract ⋆7 ret ∘ f ∘ extract =
ret ∘ g ∘ f ∘ extract
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> C) (f : A -> B),
ret ∘ g ∘ extract ⋆7 ret ∘ f ∘ extract =
ret ∘ g ∘ f ∘ extract
      setup; unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> C
f: A -> B
w: W
a: A
binddt ((fun a : W * B => ret (g (extract a))) ⦿ w)
  (ret (f a)) = ret (g (f a))
      compose near (f a) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> C
f: A -> B
w: W
a: A
(binddt ((fun a : W * B => ret (g (extract a))) ⦿ w)
 ∘ ret) (f a) = ret (g (f a))
      unfold_ops @Map_I.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> C
f: A -> B
w: W
a: A
(binddt ((fun a : W * B => ret (g (extract a))) ⦿ w)
 ∘ ret) (f a) = ret (g (f a))
      rewrite (kdtm_binddt0 (T := T) (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> C
f: A -> B
w: W
a: A
((fun a : W * B => ret (g (extract a))) ⦿ w ∘ ret)
  (f a) = ret (g (f a))
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> C
f: A -> B
w: W
a: A
((fun a : W * B => ret (g (extract a))) ∘ pair w)
  (f a) = ret (g (f a))
      reflexivity.
    Qed.

  End composition.

  (** ** Rewriting rules for special cases of <<kc7>> *)
  (********************************************************************)
  Section kc7_special_cases.

    Context
      `{Applicative G1}
      `{Applicative G2}
      {A B C D: Type}.

    (** *** Lemmas <<kc7_x7>> *)
    (******************************************************************)
    Lemma kc7_07:
      forall (g: B -> C) (f: W * A -> G1 (T B)),
        ret (T := T) ∘ g ∘ extract (W := (W ×)) ⋆7 f =
          map (F := G1) (map (F := T) g) ∘ f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> C) (f : W * A -> G1 (T B)),
ret ∘ g ∘ extract ⋆7 f = map (map g) ∘ f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> C) (f : W * A -> G1 (T B)),
ret ∘ g ∘ extract ⋆7 f = map (map g) ∘ f
      solve_kc7.
    Qed.

    Lemma kc7_17:
      forall (g: B -> T C) (f: W * A -> G1 (T B)),
        g ∘ extract (W := (W ×)) ⋆7 f =
          map (F := G1) (bind (U := T) (T := T) g) ∘ f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> T C) (f : W * A -> G1 (T B)),
g ∘ extract ⋆7 f = map (bind g) ∘ f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> T C) (f : W * A -> G1 (T B)),
g ∘ extract ⋆7 f = map (bind g) ∘ f
      solve_kc7.
    Qed.

    Lemma kc7_27:
      forall (g: B -> G2 C) (f: W * A -> G1 (T B)),
        map (F := G2) (ret (T := T)) ∘ g ∘ extract (W := (W ×)) ⋆7 f =
          map (F := G1) (traverse (T := T) (G := G2) g) ∘ f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 C) (f : W * A -> G1 (T B)),
map ret ∘ g ∘ extract ⋆7 f = map (traverse g) ∘ f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 C) (f : W * A -> G1 (T B)),
map ret ∘ g ∘ extract ⋆7 f = map (traverse g) ∘ f
      solve_kc7.
    Qed.

    Lemma kc7_67:
      forall (g: B -> G2 (T C)) (f: W * A -> G1 (T B)),
        (g ∘ extract (W := (W ×))) ⋆7 f = g ⋆6 f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 (T B)),
g ∘ extract ⋆7 f = g ⋆6 f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 (T B)),
g ∘ extract ⋆7 f = g ⋆6 f
      solve_kc7.
    Qed.

    Lemma kc7_47:
      forall (g: W * B -> C) (f: W * A -> G1 (T B)),
        kc7 G1 (fun A => A) (ret (T := T) ∘ g) f =
          (fun '(w, a) => map (F := G1) (mapd (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> C) (f : W * A -> G1 (T B)),
ret ∘ g ⋆7 f =
(fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> C) (f : W * A -> G1 (T B)),
ret ∘ g ⋆7 f =
(fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      solve_kc7.
    Qed.


    Lemma kc7_57:
      forall (g: W * B -> T C) (f: W * A -> G1 (T B)),
        kc7 G1 (fun A => A) g f = g ⋆7 f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> T C) (f : W * A -> G1 (T B)),
g ⋆7 f = g ⋆7 f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> T C) (f : W * A -> G1 (T B)),
g ⋆7 f = g ⋆7 f
      reflexivity.
    Qed.

    Lemma kc7_37:
      forall (g: W * B -> G2 C) (f: W * A -> G1 (T B)),
        (map (F := G2) (ret (T := T)) ∘ g) ⋆7 f =
          (fun '(w, a) => map (F := G1) (mapdt (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 (T B)),
map ret ∘ g ⋆7 f =
(fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 (T B)),
map ret ∘ g ⋆7 f =
(fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      solve_kc7.
    Qed.

    (** *** Lemmas <<kc7_7x>> *)
    (******************************************************************)
    Lemma kc7_73:
      forall (g: W * B -> G2 (T C)) (f: W * A -> G1 B),
        g ⋆7 (map (F := G1) ret ∘ f) = g ⋆3 f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 B),
g ⋆7 map ret ∘ f = g ⋆3 f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 B),
g ⋆7 map ret ∘ f = g ⋆3 f
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
g ⋆7 map ret ∘ f = g ⋆3 f unfold kc7, kc3.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
(fun '(w, a) =>
 map (binddt (g ⦿ w)) ((map ret ∘ f) (w, a))) =
map g ∘ strength ∘ cobind f
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) ((map ret ∘ f) (w, a)) =
(map g ∘ strength ∘ cobind f) (w, a)
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) (map ret (f (w, a))) =
map g (map (pair w) (f (w, a)))
      compose near (f (w, a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
(map (binddt (g ⦿ w)) ∘ map ret) (f (w, a)) =
(map g ∘ map (pair w)) (f (w, a))
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w) ∘ ret) (f (w, a)) =
(map g ∘ map (pair w)) (f (w, a))
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w) ∘ ret) (f (w, a)) =
map (g ∘ pair w) (f (w, a))
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (g ⦿ w ∘ ret) (f (w, a)) =
map (g ∘ pair w) (f (w, a))
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (g ∘ pair w) (f (w, a)) =
map (g ∘ pair w) (f (w, a))
      reflexivity.
    Qed.

    Lemma kc7_75:
      forall (g: W * B -> G2 (T C)) (f: W * A -> T B),
        kc7 (fun A => A) G2 g f =
          fun '(w, a) => binddt (g ⦿ w) (f (w, a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> T B),
g ⋆7 f = (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> T B),
g ⋆7 f = (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      reflexivity.
    Qed.

    Lemma kc7_74:
      forall (g: W * B -> G2 (T C)) (f: W * A -> B),
        kc7 (fun A => A) G2 g (ret (T := T) ∘ f) = g ⋆1 f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> B),
g ⋆7 ret ∘ f = g ⋆1 f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> B),
g ⋆7 ret ∘ f = g ⋆1 f
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
g ⋆7 ret ∘ f = g ⋆1 f unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
(fun '(w, a) =>
 map (binddt (g ⦿ w)) ((ret ∘ f) (w, a))) = g ⋆1 f
      ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
map (binddt (g ⦿ w)) ((ret ∘ f) (w, a)) =
(g ⋆1 f) (w, a)
      unfold compose at 1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
map (binddt (g ⦿ w)) (ret (f (w, a))) =
(g ⋆1 f) (w, a)
      compose near (f (w, a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
(map (binddt (g ⦿ w)) ∘ ret) (f (w, a)) =
(g ⋆1 f) (w, a)
      change (map (F := fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
(binddt (g ⦿ w) ∘ ret) (f (w, a)) = (g ⋆1 f) (w, a)
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
(g ⦿ w ∘ ret) (f (w, a)) = (g ⋆1 f) (w, a)
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: W * A -> B
w: W
a: A
(g ∘ pair w) (f (w, a)) = (g ⋆1 f) (w, a)
      reflexivity.
    Qed.

    Lemma kc7_76:
      forall (g: W * B -> G2 (T C)) (f: A -> G1 (T B)),
        g ⋆7 (f ∘ extract (W := (W ×))) =
          fun '(w, a) => map (F := G1) (binddt (g ⦿ w)) (f a).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 (T B)),
g ⋆7 f ∘ extract =
(fun '(w, a) => map (binddt (g ⦿ w)) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 (T B)),
g ⋆7 f ∘ extract =
(fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      solve_kc7.
    Qed.

    Lemma kc7_72:
      forall (g: W * B -> G2 (T C)) (f: A -> G1 B),
        g ⋆7 (map (F := G1) ret ∘ f ∘ extract (W := (W ×))) =
          fun '(w, a) => map (F := G1) (g ∘ pair w) (f a).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 B),
g ⋆7 map ret ∘ f ∘ extract =
(fun '(w, a) => map (g ∘ pair w) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 B),
g ⋆7 map ret ∘ f ∘ extract =
(fun '(w, a) => map (g ∘ pair w) (f a))
      setup.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) ((map ret ∘ f ∘ extract) (w, a)) =
map (g ∘ pair w) (f a)
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) (map ret (f a)) =
map (g ○ pair w) (f a)
      compose near (f a) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
(map (binddt (g ⦿ w)) ∘ map ret) (f a) =
map (g ○ pair w) (f a)
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
map (binddt (g ⦿ w) ∘ ret) (f a) =
map (g ○ pair w) (f a)
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
map (g ⦿ w ∘ ret) (f a) = map (g ○ pair w) (f a)
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> G1 B
w: W
a: A
map (g ∘ pair w) (f a) = map (g ○ pair w) (f a)
      reflexivity.
    Qed.

    Lemma kc7_71:
      forall (g: W * B -> G2 (T C)) (f: A -> T B),
        kc7 (fun A => A) G2 g (f ∘ extract (W := (W ×))) =
          fun '(w, a) => binddt (g ⦿ w) (f a).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> T B),
g ⋆7 f ∘ extract =
(fun '(w, a) => binddt (g ⦿ w) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> T B),
g ⋆7 f ∘ extract =
(fun '(w, a) => binddt (g ⦿ w) (f a))
      solve_kc7.
    Qed.

    Lemma kc7_70:
      forall (g: W * B -> G2 (T C)) (f: A -> B),
        kc7 (fun A => A) G2 g (ret (T := T) ∘ f ∘ extract (W := (W ×))) =
          g ∘ map (F := (W ×)) f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> B),
g ⋆7 ret ∘ f ∘ extract = g ∘ map f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> G2 (T C)) (f : A -> B),
g ⋆7 ret ∘ f ∘ extract = g ∘ map f
      setup.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
binddt (g ⦿ w) ((ret ∘ f ∘ extract) (w, a)) =
(g ∘ map f) (w, a)
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
binddt (g ⦿ w) (ret (f a)) = g (id w, f a)
      compose near (f a) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
(binddt (g ⦿ w) ∘ ret) (f a) = g (id w, f a)
      change (map (F := fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
(binddt (g ⦿ w) ∘ ret) (f a) = g (id w, f a)
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
(g ⦿ w ∘ ret) (f a) = g (id w, f a)
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> G2 (T C)
f: A -> B
w: W
a: A
(g ∘ pair w) (f a) = g (id w, f a)
      reflexivity.
    Qed.

    (** *** Other lemmas *)
    (******************************************************************)
    Lemma kc7_56:
      forall (g: W * B -> T C) (f: W * A -> G1 B),
        g ⋆7 map (F := G1)(ret (T := T)) ∘ f =
          (fun '(w, a) => map (F := G1) (g ∘ pair w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> T C) (f : W * A -> G1 B),
g ⋆7 map ret ∘ f =
(fun '(w, a) => map (g ∘ pair w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : W * B -> T C) (f : W * A -> G1 B),
g ⋆7 map ret ∘ f =
(fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      setup.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) ((map ret ∘ f) (w, a)) =
map (g ∘ pair w) (f (w, a))
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w)) (map ret (f (w, a))) =
map (g ○ pair w) (f (w, a))
      compose near (f (w, a)) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
(map (binddt (g ⦿ w)) ∘ map ret) (f (w, a)) =
map (g ○ pair w) (f (w, a))
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
map (binddt (g ⦿ w) ∘ ret) (f (w, a)) =
map (g ○ pair w) (f (w, a))
      rewrite (kdtm_binddt0 (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
map (g ⦿ w ∘ ret) (f (w, a)) =
map (g ○ pair w) (f (w, a))
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: W * B -> T C
f: W * A -> G1 B
w: W
a: A
map (g ∘ pair w) (f (w, a)) =
map (g ○ pair w) (f (w, a))
      reflexivity.
    Qed.

    Lemma kc7_63:
      forall (g: B -> G2 (T C)) (f: W * A -> G1 B),
        g ∘ extract ⋆7 map ret ∘ f = map g ∘ f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 B),
g ∘ extract ⋆7 map ret ∘ f = map g ∘ f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 B),
g ∘ extract ⋆7 map ret ∘ f = map g ∘ f
      setup.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt ((g ∘ extract) ⦿ w))
  ((map ret ∘ f) (w, a)) = (map g ∘ f) (w, a)
      unfold compose; cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt ((g ○ extract) ⦿ w)) (map ret (f (w, a))) =
map g (f (w, a))
      compose near (f (w, a)) on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
(map (binddt ((g ○ extract) ⦿ w)) ∘ map ret)
  (f (w, a)) = map g (f (w, a))
      rewrite fun_map_map.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (binddt ((g ○ extract) ⦿ w) ∘ ret) (f (w, a)) =
map g (f (w, a))
      rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map ((g ○ extract) ⦿ w ∘ ret) (f (w, a)) =
map g (f (w, a))
      rewrite preincr_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C, D: Type
g: B -> G2 (T C)
f: W * A -> G1 B
w: W
a: A
map (g ○ extract ∘ pair w) (f (w, a)) =
map g (f (w, a))
      reflexivity.
    Qed.

  End kc7_special_cases.

End derived_instances.

(** ** Derived Composition Laws *)
(**********************************************************************)
Section other_composition_laws.

  Context
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd W T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt W T}
    `{Bindd_T_inst: Bindd W T T}
    `{Bindt_T_inst: Bindt T T}
    `{Binddt_T_inst: Binddt W T T}
    `{! Compat_Map_Binddt W T T}
    `{! Compat_Mapd_Binddt W T T}
    `{! Compat_Traverse_Binddt W T T}
    `{! Compat_Bind_Binddt W T T}
    `{! Compat_Mapdt_Binddt W T T}
    `{! Compat_Bindd_Binddt W T T}
    `{! Compat_Bindt_Binddt W T T}
    `{Monad_inst: ! DecoratedTraversableMonad W T}
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd W U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt W U}
    `{Bindd_U_inst: Bindd W T U}
    `{Bindt_U_inst: Bindt T U}
    `{Binddt_U_inst: Binddt W T U}
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Mapd_Binddt W T U}
    `{! Compat_Traverse_Binddt W T U}
    `{! Compat_Bind_Binddt W T U}
    `{! Compat_Mapdt_Binddt W T U}
    `{! Compat_Bindd_Binddt W T U}
    `{! Compat_Bindt_Binddt W T U}
    `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.


  (*
    Context
    {W: Type}
    {T: Type -> Type}
    `{Return_T: Return T}
    `{Map_T: Map T}
    `{Mapd_T: Mapd W T}
    `{Traverse_T: Traverse T}
    `{Bind_T: Bind T T}
    `{Mapdt_T: Mapdt W T}
    `{Bindd_T: Bindd W T T}
    `{Bindt_T: Bindt T T}
    `{Binddt_T: Binddt W T T}
    `{Monoid_unit_W: Monoid_unit W}
    `{Monoid_op_W: Monoid_op W}
    `{! DecoratedTraversableMonad W T}
    `{! Compat_Full_Binddt W T T}.

    Context
    {U: Type -> Type}
    `{Map_U: Map U}
    `{Mapd_U: Mapd W U}
    `{Traverse_U: Traverse U}
    `{Bind_U: Bind T U}
    `{Mapdt_U: Mapdt W U}
    `{Bindd_U: Bindd W T U}
    `{Bindt_U: Bindt T U}
    `{Binddt_U: Binddt W T U}
    `{! DecoratedTraversableRightModule W T U}
    `{! Compat_Full_Binddt W T U}.
    *)

  (** *** Homogeneous composition laws *)
  (********************************************************************)
  Section composition.

    Context
      {G1: Type -> Type}
      `{Map G1} `{Pure G1} `{Mult G1} `{! Applicative G1}
      {G2: Type -> Type}
      `{Map G2} `{Pure G2} `{Mult G2} `{! Applicative G2}.

    Variables (A B C: Type).

    (* composition_33 *)
    Lemma mapdt_mapdt: forall
        (g: W * B -> G2 C)
        (f: W * A -> G1 B),
        map (F := G1) (A := U B) (B := G2 (U C))
          (mapdt (T := U) g) ∘ mapdt (T := U) f =
          mapdt (T := U) (G := G1 ∘ G2) (g ⋆3 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 B),
map (mapdt g) ∘ mapdt f = mapdt (g ⋆3 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 B),
map (mapdt g) ∘ mapdt f = mapdt (g ⋆3 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
map (mapdt g) ∘ mapdt f = mapdt (g ⋆3 f)
      do 2 rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
map (binddt (map ret ∘ g)) ∘ binddt (map ret ∘ f) =
mapdt (g ⋆3 f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
binddt (map ret ∘ g ⋆7 map ret ∘ f) = mapdt (g ⋆3 f)
      rewrite kc7_33.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
binddt (map ret ∘ (g ⋆3 f)) = mapdt (g ⋆3 f)
      rewrite <- mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
mapdt (g ⋆3 f) = mapdt (g ⋆3 f)
      reflexivity.
    Qed.

    (* composition_55 *)
    Lemma bindd_bindd: forall
        (g: W * B -> T C)
        (f: W * A -> T B),
        bindd g ∘ bindd (U := U) f = bindd (U := U) (g ⋆5 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> T B),
bindd g ∘ bindd f = bindd (g ⋆5 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> T B),
bindd g ∘ bindd f = bindd (g ⋆5 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
bindd g ∘ bindd f = bindd (g ⋆5 f)
      do 2 rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
binddt g ∘ binddt f = bindd (g ⋆5 f)
      change (binddt g) with (map (F := fun A => A) (binddt g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
map (binddt g) ∘ binddt f = bindd (g ⋆5 f)
      change ((fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
map (binddt g) ∘ binddt f = bindd (g ⋆5 f)
      rewrite (kdtm_binddt2 (G1 := fun A => A) (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
binddt (g ⋆7 f) = bindd (g ⋆5 f)
      rewrite (binddt_app_id_l (T := T) (U := U)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
binddt (g ⋆7 f) = bindd (g ⋆5 f)
      rewrite kc7_55.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
binddt (g ⋆5 f) = bindd (g ⋆5 f)
      rewrite <- bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
bindd (g ⋆5 f) = bindd (g ⋆5 f)
      reflexivity.
    Qed.

    (* composition_44 *)
    Lemma mapd_mapd: forall (g: W * B -> C) (f: W * A -> B),
        mapd g ∘ mapd f = mapd (g ∘ cobind (W := (W ×)) f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> B),
mapd g ∘ mapd f = mapd (g ∘ cobind f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> B),
mapd g ∘ mapd f = mapd (g ∘ cobind f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
mapd g ∘ mapd f = mapd (g ∘ cobind f)
      do 2 rewrite mapd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
binddt (ret ∘ g) ∘ binddt (ret ∘ f) =
mapd (g ∘ cobind f)
      change (binddt (ret (T := T) ∘ g)) with
        (map (F := fun A => A) (binddt (ret (T := T) ∘ g))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
map (binddt (ret ∘ g)) ∘ binddt (ret ∘ f) =
mapd (g ∘ cobind f)
      change ((fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
map (binddt (ret ∘ g)) ∘ binddt (ret ∘ f) =
mapd (g ∘ cobind f)
      rewrite (kdtm_binddt2 (G1 := fun A => A) (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
binddt (ret ∘ g ⋆7 ret ∘ f) = mapd (g ∘ cobind f)
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
binddt (ret ∘ g ⋆7 ret ∘ f) = mapd (g ∘ cobind f)
      rewrite kc7_44.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
binddt (ret ∘ (g ⋆1 f)) = mapd (g ∘ cobind f)
      rewrite <- mapd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> B
mapd (g ⋆1 f) = mapd (g ∘ cobind f)
      reflexivity.
    Qed.

    (* composition_66 *)
    Lemma bindt_bindt:
      forall (g: B -> G2 (T C)) (f: A -> G1 (T B)),
        map (F := G1) (bindt g) ∘ bindt f =
          bindt (U := U) (G := G1 ∘ G2) (g ⋆6 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : A -> G1 (T B)),
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : A -> G1 (T B)),
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
      do 2 rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (binddt (g ∘ extract)) ∘ binddt (f ∘ extract) =
bindt (g ⋆6 f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
binddt (g ∘ extract ⋆7 f ∘ extract) = bindt (g ⋆6 f)
      rewrite kc7_66.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
binddt ((g ⋆6 f) ∘ extract) = bindt (g ⋆6 f)
      rewrite <- (bindt_to_binddt (A := A) (B := C)
                    (U := U) (T := T) (G := G1 ○ G2)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
bindt (g ⋆6 f) = bindt (g ⋆6 f)
      reflexivity.
    Qed.

    (* composition_22 *)
    Lemma traverse_traverse: forall (g: B -> G2 C) (f: A -> G1 B),
        map (F := G1) (traverse (G := G2) g) ∘ traverse (G := G1) f =
          traverse (G := G1 ∘ G2) (map g ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : A -> G1 B),
map (traverse g) ∘ traverse f = traverse (map g ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : A -> G1 B),
map (traverse g) ∘ traverse f = traverse (map g ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
map (traverse g) ∘ traverse f = traverse (map g ∘ f)
      do 2 rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
map (binddt (map ret ∘ g ∘ extract))
∘ binddt (map ret ∘ f ∘ extract) =
traverse (map g ∘ f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
binddt
  (map ret ∘ g ∘ extract ⋆7 map ret ∘ f ∘ extract) =
traverse (map g ∘ f)
      rewrite kc7_22.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
binddt ((ret ⋆2 map g ∘ f) ∘ extract) =
traverse (map g ∘ f)
      unfold kc2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
binddt (map ret ∘ (map g ∘ f) ∘ extract) =
traverse (map g ∘ f)
      rewrite <- (traverse_to_binddt
                    (T := T)
                    (G := G1 ∘ G2)
                    (@map G1 H B (G2 C) g ∘ f)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
traverse (map g ∘ f) = traverse (map g ∘ f)
      reflexivity.
    Qed.

    (* composition_11 *)
    Lemma bind_bind: forall (g: B -> T C) (f: A -> T B),
        bind g ∘ bind f = bind (U := U) (g ⋆ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : A -> T B),
bind g ∘ bind f = bind (g ⋆ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : A -> T B),
bind g ∘ bind f = bind (g ⋆ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
bind g ∘ bind f = bind (g ⋆ f)
      do 2 rewrite bind_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
binddt (g ∘ extract) ∘ binddt (f ∘ extract) =
bind (g ⋆ f)
      change (binddt (g ∘ extract)) with
        (map (F := fun A => A) (binddt (g ∘ extract))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
map (binddt (g ∘ extract)) ∘ binddt (f ∘ extract) =
bind (g ⋆ f)
      change ((fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
map (binddt (g ∘ extract)) ∘ binddt (f ∘ extract) =
bind (g ⋆ f)
      rewrite (kdtm_binddt2 (G1 := fun A => A) (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
binddt (g ∘ extract ⋆7 f ∘ extract) = bind (g ⋆ f)
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
binddt (g ∘ extract ⋆7 f ∘ extract) = bind (g ⋆ f)
      rewrite kc7_11.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
binddt ((g ⋆ f) ∘ extract) = bind (g ⋆ f)
      rewrite <- bind_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: A -> T B
bind (g ⋆ f) = bind (g ⋆ f)
      reflexivity.
    Qed.

    (* composition_00 *)
    Lemma map_map: forall (f: A -> B) (g: B -> C),
        map g ∘ map f = map (F := U) (g ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (f : A -> B) (g : B -> C),
map g ∘ map f = map (g ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (f : A -> B) (g : B -> C),
map g ∘ map f = map (g ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
map g ∘ map f = map (g ∘ f)
      do 2 rewrite map_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
binddt (ret ∘ g ∘ extract)
∘ binddt (ret ∘ f ∘ extract) = map (g ∘ f)
      change (binddt (?ret ∘ g ∘ ?extract)) with
        (map (F := fun A => A) (binddt (ret ∘ g ∘ extract))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
map (binddt (ret ∘ g ∘ extract))
∘ binddt (ret ∘ f ∘ extract) = map (g ∘ f)
      change ((fun A => A) ?f) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
map (binddt (ret ∘ g ∘ extract))
∘ binddt (ret ∘ f ∘ extract) = map (g ∘ f)
      rewrite (kdtm_binddt2 (G1 := fun A => A) (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
binddt (ret ∘ g ∘ extract ⋆7 ret ∘ f ∘ extract) =
map (g ∘ f)
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
binddt (ret ∘ g ∘ extract ⋆7 ret ∘ f ∘ extract) =
map (g ∘ f)
      rewrite kc7_00.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
binddt (ret ∘ g ∘ f ∘ extract) = map (g ∘ f)
      change (?ret ∘ g ∘ f ∘ ?extract) with
        (ret ∘ (g ∘ f) ∘ extract).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
binddt (ret ∘ (g ∘ f) ∘ extract) = map (g ∘ f)
      rewrite <- map_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
f: A -> B
g: B -> C
map (g ∘ f) = map (g ∘ f)
      reflexivity.
    Qed.

    (*
      End composition.

      Section composition_special_cases_top.

      Context
      `{Applicative G1}
      `{Applicative G2}
      {A B C: Type}.

     *)

    (** *** <<binddt>> on the right *)
    (******************************************************************)
    (* composition_67 *)
    Lemma mapdt_binddt:
      forall (g: W * B -> G2 C)
             (f: W * A -> G1 (T B)),
        map (F := G1) (mapdt g) ∘ binddt f =
          binddt (G := G1 ∘ G2)
            (fun '(w, a) => map (F := G1) (mapdt (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 (T B)),
map (mapdt g) ∘ binddt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> G1 (T B)),
map (mapdt g) ∘ binddt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 (T B)
map (mapdt g) ∘ binddt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 (T B)
map (binddt (map ret ∘ g)) ∘ binddt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 (T B)
binddt (map ret ∘ g ⋆7 f) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      rewrite kc7_37.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 (T B)
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a))) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_57 *)
    Lemma bindd_binddt:
      forall (g: W * B -> T C)
             (f: W * A -> G1 (T B)),
        map (F := G1) (bindd g) ∘ binddt f =
          binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> G1 (T B)),
map (bindd g) ∘ binddt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> G1 (T B)),
map (bindd g) ∘ binddt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
map (bindd g) ∘ binddt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f (w, a)))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
map (binddt g) ∘ binddt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f (w, a)))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
map (binddt g) ∘ binddt f =
binddt
  (fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
      rewrite (kdtm_binddt2 (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
binddt (g ⋆7 f) =
binddt
  (fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
binddt (g ⋆7 f) =
binddt
  (fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 (T B)
binddt
  (fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a))) =
binddt
  (fun '(w, a) => map (binddt (g ⦿ w)) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_47 *)
    Lemma mapd_binddt: forall
        (g: W * B -> C)
        (f: W * A -> G1 (T B)),
        map (F := G1) (mapd g) ∘ binddt f =
          binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> G1 (T B)),
map (mapd g) ∘ binddt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : W * A -> G1 (T B)),
map (mapd g) ∘ binddt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> G1 (T B)
map (mapd g) ∘ binddt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      rewrite mapd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> G1 (T B)
map (binddt (ret ∘ g)) ∘ binddt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      rewrite (kdtm_binddt2 (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> G1 (T B)
binddt (ret ∘ g ⋆7 f) =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> G1 (T B)
binddt (ret ∘ g ⋆7 f) =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      rewrite kc7_47.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> C
f: W * A -> G1 (T B)
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a))) =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_67 *)
    Lemma bindt_binddt:
      forall (g: B -> G2 (T C))
             (f: W * A -> G1 (T B)),
        map (F := G1) (bindt g) ∘ binddt f =
          binddt (G := G1 ∘ G2) (map (F := G1) (bindt g) ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 (T B)),
map (bindt g) ∘ binddt f = binddt (map (bindt g) ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 (T B)),
map (bindt g) ∘ binddt f = binddt (map (bindt g) ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 (T B)
map (bindt g) ∘ binddt f = binddt (map (bindt g) ∘ f)
      rewrite bindt_to_binddt at 1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 (T B)
map (binddt (g ∘ extract)) ∘ binddt f =
binddt (map (bindt g) ∘ f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 (T B)
binddt (g ∘ extract ⋆7 f) = binddt (map (bindt g) ∘ f)
      rewrite kc7_67.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 (T B)
binddt (g ⋆6 f) = binddt (map (bindt g) ∘ f)
      reflexivity.
    Qed.

    (* composition_27 *)
    Lemma traverse_binddt: forall
        (g: B -> G2 C)
        (f: W * A -> G1 (T B)),
        map (F := G1) (traverse (T := T) (G := G2) g) ∘ binddt f =
          binddt (T := T) (G := G1 ∘ G2)
            (map (F := G1) (traverse (T := T) (G := G2)  g) ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : W * A -> G1 (T B)),
map (traverse g) ∘ binddt f =
binddt (map (traverse g) ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : W * A -> G1 (T B)),
map (traverse g) ∘ binddt f =
binddt (map (traverse g) ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> G1 (T B)
map (traverse g) ∘ binddt f =
binddt (map (traverse g) ∘ f)
      rewrite traverse_to_binddt at 1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> G1 (T B)
map (binddt (map ret ∘ g ∘ extract)) ∘ binddt f =
binddt (map (traverse g) ∘ f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> G1 (T B)
binddt (map ret ∘ g ∘ extract ⋆7 f) =
binddt (map (traverse g) ∘ f)
      rewrite kc7_27.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> G1 (T B)
binddt (map (traverse g) ∘ f) =
binddt (map (traverse g) ∘ f)
      reflexivity.
    Qed.

    (* composition_17 *)
    Lemma bind_binddt: forall
        (g: B -> T C)
        (f: W * A -> G1 (T B)),
        map (F := G1) (bind g) ∘ binddt f =
          binddt (map (F := G1) (bind g) ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : W * A -> G1 (T B)),
map (bind g) ∘ binddt f = binddt (map (bind g) ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> T C) (f : W * A -> G1 (T B)),
map (bind g) ∘ binddt f = binddt (map (bind g) ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: W * A -> G1 (T B)
map (bind g) ∘ binddt f = binddt (map (bind g) ∘ f)
      rewrite bind_to_binddt at 1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: W * A -> G1 (T B)
map (binddt (g ∘ extract)) ∘ binddt f =
binddt (map (bind g) ∘ f)
      rewrite (kdtm_binddt2 (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: W * A -> G1 (T B)
binddt (g ∘ extract ⋆7 f) = binddt (map (bind g) ∘ f)
      rewrite kc7_17.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: W * A -> G1 (T B)
binddt (map (bind g) ∘ f) = binddt (map (bind g) ∘ f)
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> T C
f: W * A -> G1 (T B)
binddt (map (bind g) ∘ f) = binddt (map (bind g) ∘ f)
      reflexivity.
    Qed.

    (* composition_07 *)
    Lemma map_binddt:
      forall (g: B -> C)
             (f: W * A -> G1 (T B)),
        map (F := G1) (map (F := U) g) ∘ binddt f =
          binddt (map (F := G1) (map (F := T) g) ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> C) (f : W * A -> G1 (T B)),
map (map g) ∘ binddt f = binddt (map (map g) ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : B -> C) (f : W * A -> G1 (T B)),
map (map g) ∘ binddt f = binddt (map (map g) ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> C
f: W * A -> G1 (T B)
map (map g) ∘ binddt f = binddt (map (map g) ∘ f)
      rewrite (map_to_binddt (U := U)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> C
f: W * A -> G1 (T B)
map (binddt (ret ∘ g ∘ extract)) ∘ binddt f =
binddt (map (map g) ∘ f)
      rewrite (kdtm_binddt2 (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> C
f: W * A -> G1 (T B)
binddt (ret ∘ g ∘ extract ⋆7 f) =
binddt (map (map g) ∘ f)
      rewrite kc7_07.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> C
f: W * A -> G1 (T B)
binddt (map (map g) ∘ f) = binddt (map (map g) ∘ f)
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: B -> C
f: W * A -> G1 (T B)
binddt (map (map g) ∘ f) = binddt (map (map g) ∘ f)
      reflexivity.
    Qed.

    (** *** <<binddt>> on the left *)
    (******************************************************************)
    (* composition_73 *)
    Lemma binddt_mapdt: forall
        (g: W * B -> G2 (T C))
        (f: W * A -> G1 B),
        map (F := G1) (binddt g) ∘ mapdt f =
          binddt (U := U) (G := G1 ∘ G2)
            (fun '(w, a) => map (fun b => g (w, b)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 B),
map (binddt g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> G1 B),
map (binddt g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
map (binddt g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
map (binddt g) ∘ binddt (map ret ∘ f) =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
binddt (g ⋆7 map ret ∘ f) =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      rewrite kc7_73.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
binddt (g ⋆3 f) =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      rewrite kc3_spec.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> G1 B
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a))) =
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_75 *)
    Lemma binddt_bindd: forall
        (g: W * B -> G2 (T C))
        (f: W * A -> T B),
        binddt g ∘ bindd f =
          binddt (U := U) (fun '(w, a) => binddt (g ⦿ w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> T B),
binddt g ∘ bindd f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> T B),
binddt g ∘ bindd f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> T B
binddt g ∘ bindd f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> T B
binddt g ∘ binddt f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      change (binddt g) with (map (F := fun A => A) (binddt g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> T B
map (binddt g) ∘ binddt f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      rewrite (kdtm_binddt2 (G1 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> T B
binddt (g ⋆7 f) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> T B
binddt (g ⋆7 f) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_74 *)
    Lemma binddt_mapd: forall
        (g: W * B -> G2 (T C))
        (f: W * A -> B),
        binddt g ∘ mapd f = binddt (g ⋆1 f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> B),
binddt g ∘ mapd f = binddt (g ⋆1 f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : W * A -> B),
binddt g ∘ mapd f = binddt (g ⋆1 f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
binddt g ∘ mapd f = binddt (g ⋆1 f)
      rewrite mapd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
binddt g ∘ binddt (ret ∘ f) = binddt (g ⋆1 f)
      change (binddt g) with (map (F := fun A => A) (binddt g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
map (binddt g) ∘ binddt (ret ∘ f) = binddt (g ⋆1 f)
      rewrite (kdtm_binddt2 (G1 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
binddt (g ⋆7 ret ∘ f) = binddt (g ⋆1 f)
      rewrite kc7_74.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
binddt (g ⋆1 f) = binddt (g ⋆1 f)
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: W * A -> B
binddt (g ⋆1 f) = binddt (g ⋆1 f)
      reflexivity.
    Qed.

    (* composition_76 *)
    Lemma binddt_bindt: forall
        (g: W * B -> G2 (T C))
        (f: A -> G1 (T B)),
        map (F := G1) (binddt g) ∘ bindt f =
          binddt (T := T) (G := G1 ∘ G2)
            (fun '(w, a) => map (F := G1) (binddt (g ⦿ w)) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 (T B)),
map (binddt g) ∘ bindt f =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 (T B)),
map (binddt g) ∘ bindt f =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 (T B)
map (binddt g) ∘ bindt f =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 (T B)
map (binddt g) ∘ binddt (f ∘ extract) =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 (T B)
binddt (g ⋆7 f ∘ extract) =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      rewrite kc7_76.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 (T B)
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a)) =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      reflexivity.
    Qed.

    (* composition_72 *)
    Lemma binddt_traverse: forall
        (g: W * B -> G2 (T C))
        (f: A -> G1 B),
        map (F := G1) (binddt g) ∘ traverse (T := U) (G := G1) f =
          binddt (G := G1 ∘ G2)
            (fun '(w, a) => map (fun b => g (w, b)) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 B),
map (binddt g) ∘ traverse f =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> G1 B),
map (binddt g) ∘ traverse f =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 B
map (binddt g) ∘ traverse f =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
      rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 B
map (binddt g) ∘ binddt (map ret ∘ f ∘ extract) =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 B
binddt (g ⋆7 map ret ∘ f ∘ extract) =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
      rewrite kc7_72.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> G1 B
binddt (fun '(w, a) => map (g ∘ pair w) (f a)) =
binddt (fun '(w, a) => map (g ○ pair w) (f a))
      reflexivity.
    Qed.

    (* composition_71 *)
    Lemma binddt_bind: forall
        (g: W * B -> G2 (T C))
        (f: A -> T B),
        binddt g ∘ bind f =
          binddt (U := U) (fun '(w, a) => binddt (g ⦿ w) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> T B),
binddt g ∘ bind f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> T B),
binddt g ∘ bind f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
binddt g ∘ bind f =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      rewrite bind_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
binddt g ∘ binddt (f ∘ extract) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      change (binddt g) with (map (F := fun A => A) (binddt g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
map (binddt g) ∘ binddt (f ∘ extract) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      rewrite (kdtm_binddt2 (G1 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
binddt (g ⋆7 f ∘ extract) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
binddt (g ⋆7 f ∘ extract) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      rewrite kc7_71.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> T B
binddt (fun '(w, a) => binddt (g ⦿ w) (f a)) =
binddt (fun '(w, a) => binddt (g ⦿ w) (f a))
      reflexivity.
    Qed.

    (* composition_70 *)
    Lemma binddt_map: forall
        (g: W * B -> G2 (T C))
        (f: A -> B),
        binddt g ∘ map (F := U) f =
          binddt (U := U) (g ∘ map (F := (W ×)) f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> B),
binddt g ∘ map f = binddt (g ∘ map f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 (T C)) (f : A -> B),
binddt g ∘ map f = binddt (g ∘ map f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
binddt g ∘ map f = binddt (g ∘ map f)
      rewrite (map_to_binddt (U := U)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
binddt g ∘ binddt (ret ∘ f ∘ extract) =
binddt (g ∘ map f)
      change (binddt g) with (map (F := fun A => A) (binddt g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
map (binddt g) ∘ binddt (ret ∘ f ∘ extract) =
binddt (g ∘ map f)
      rewrite (kdtm_binddt2 (G1 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
binddt (g ⋆7 ret ∘ f ∘ extract) = binddt (g ∘ map f)
      rewrite binddt_app_id_l.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
binddt (g ⋆7 ret ∘ f ∘ extract) = binddt (g ∘ map f)
      rewrite kc7_70.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
H: Map G1
H0: Pure G1
H1: Mult G1
Applicative0: Applicative G1
G2: Type -> Type
H2: Map G2
H3: Pure G2
H4: Mult G2
Applicative1: Applicative G2
A, B, C: Type
g: W * B -> G2 (T C)
f: A -> B
binddt (g ∘ map f) = binddt (g ∘ map f)
      reflexivity.
    Qed.

  End composition.

  Section composition_special_cases_middle.

    Context
      `{Applicative G1}
      `{Applicative G2}
      {A B C: Type}.

    (** *** <<bindd>>, <<mapdt>>, <<bindt>> *)
    (******************************************************************)
    (* composition_56 *)
    Lemma bindd_mapdt: forall
        (g: W * B -> T C)
        (f: W * A -> G1 B),
        map (F := G1) (bindd g) ∘ mapdt f =
          binddt (U := U) (fun '(w, a) => map (g ∘ pair w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> G1 B),
map (bindd g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : W * A -> G1 B),
map (bindd g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 B
map (bindd g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 B
map (binddt g) ∘ mapdt f =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      rewrite (binddt_mapdt (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 B
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a))) =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: W * A -> G1 B
binddt (fun '(w, a) => map (g ○ pair w) (f (w, a))) =
binddt (fun '(w, a) => map (g ∘ pair w) (f (w, a)))
      reflexivity.
    Qed.

    (* composition_65 *)
    Lemma mapdt_bindd: forall
        (g: W * B -> G2 C)
        (f: W * A -> T B),
        mapdt g ∘ bindd f =
          binddt (U := U) (fun '(w, a) => mapdt (g ⦿ w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> T B),
mapdt g ∘ bindd f =
binddt (fun '(w, a) => mapdt (g ⦿ w) (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : W * A -> T B),
mapdt g ∘ bindd f =
binddt (fun '(w, a) => mapdt (g ⦿ w) (f (w, a)))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
mapdt g ∘ bindd f =
binddt (fun '(w, a) => mapdt (g ⦿ w) (f (w, a)))
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
binddt (map ret ∘ g) ∘ bindd f =
binddt (fun '(w, a) => mapdt (g ⦿ w) (f (w, a)))
      rewrite binddt_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
binddt
  (fun '(w, a) =>
   binddt ((map ret ∘ g) ⦿ w) (f (w, a))) =
binddt (fun '(w, a) => mapdt (g ⦿ w) (f (w, a)))
      fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
(fun '(w, a) => binddt ((map ret ∘ g) ⦿ w) (f (w, a))) =
(fun '(w, a) => mapdt (g ⦿ w) (f (w, a))) ext [w a]. (* TODO can this be improved? *)
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
w: W
a: A
binddt ((map ret ∘ g) ⦿ w) (f (w, a)) =
mapdt (g ⦿ w) (f (w, a))
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> T B
w: W
a: A
binddt ((map ret ∘ g) ⦿ w) (f (w, a)) =
binddt (map ret ∘ g ⦿ w) (f (w, a))
      reflexivity.
    Qed.

    (* composition_65 *)
    Lemma bindt_bindd: forall
        (g: B -> G2 (T C))
        (f: W * A -> T B),
        bindt g ∘ bindd f = binddt (U := U) (bindt g ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> T B),
bindt g ∘ bindd f = binddt (bindt g ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> T B),
bindt g ∘ bindd f = binddt (bindt g ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
bindt g ∘ bindd f = binddt (bindt g ∘ f)
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
binddt (g ∘ extract) ∘ bindd f = binddt (bindt g ∘ f)
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
binddt (g ∘ extract) ∘ bindd f =
binddt (binddt (g ∘ extract) ∘ f)
      rewrite binddt_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
binddt
  (fun '(w, a) =>
   binddt ((g ∘ extract) ⦿ w) (f (w, a))) =
binddt (binddt (g ∘ extract) ∘ f)
      try rewrite kc7_65. (* TODO *)
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
binddt
  (fun '(w, a) =>
   binddt ((g ∘ extract) ⦿ w) (f (w, a))) =
binddt (binddt (g ∘ extract) ∘ f)
      fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
(fun '(w, a) => binddt ((g ∘ extract) ⦿ w) (f (w, a))) =
binddt (g ∘ extract) ∘ f ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
w: W
a: A
binddt ((g ∘ extract) ⦿ w) (f (w, a)) =
(binddt (g ∘ extract) ∘ f) (w, a)
      rewrite extract_preincr2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> T B
w: W
a: A
binddt (g ∘ extract) (f (w, a)) =
(binddt (g ∘ extract) ∘ f) (w, a)
      reflexivity.
    Qed.

    (* composition_56 *)
    Lemma bindd_bindt: forall
        (g: W * B -> T C)
        (f: A -> G1 (T B)),
        map (F := G1) (bindd g) ∘ bindt f =
          binddt (U := U) (fun '(w, a) => map (bindd (g ⦿ w)) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : A -> G1 (T B)),
map (bindd g) ∘ bindt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> T C) (f : A -> G1 (T B)),
map (bindd g) ∘ bindt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
map (bindd g) ∘ bindt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
map (binddt g) ∘ bindt f =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
map (binddt g) ∘ binddt (f ∘ extract) =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      rewrite (kdtm_binddt2 (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
binddt (g ⋆7 f ∘ extract) =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt (g ⦿ w)) ((f ∘ extract) (w, a))) =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt (g ⦿ w)) ((f ∘ extract) (w, a))) =
binddt (fun '(w, a) => map (bindd (g ⦿ w)) (f a))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> T C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt (g ⦿ w)) ((f ∘ extract) (w, a))) =
binddt (fun '(w, a) => map (binddt (g ⦿ w)) (f a))
      reflexivity.
    Qed.

    Lemma mapdt_bindt: forall
        (g: W * B -> G2 C)
        (f: A -> G1 (T B)),
        map (F := G1) (mapdt g) ∘ bindt f =
          binddt (U := U) (G := G1 ∘ G2)
            (fun '(w, a) => map(mapdt (g ⦿ w)) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : A -> G1 (T B)),
map (mapdt g) ∘ bindt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> G2 C) (f : A -> G1 (T B)),
map (mapdt g) ∘ bindt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
map (mapdt g) ∘ bindt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
map (binddt (map ret ∘ g)) ∘ bindt f =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
map (binddt (map ret ∘ g)) ∘ binddt (f ∘ extract) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
binddt (map ret ∘ g ⋆7 f ∘ extract) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      rewrite kc7_76.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt ((map ret ∘ g) ⦿ w)) (f a)) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      try rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt ((map ret ∘ g) ⦿ w)) (f a)) =
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a))
      rewrite (compat_mapdt_binddt W T T).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt ((map ret ∘ g) ⦿ w)) (f a)) =
binddt
  (fun '(w, a) =>
   map
     (DerivedOperations.Mapdt_Binddt W T T G2 Map_G0
        Pure_G0 Mult_G0 B C (g ⦿ w)) (f a))
      unfold DerivedOperations.Mapdt_Binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: A -> G1 (T B)
binddt
  (fun '(w, a) =>
   map (binddt ((map ret ∘ g) ⦿ w)) (f a)) =
binddt
  (fun '(w, a) => map (binddt (map ret ∘ g ⦿ w)) (f a))
      reflexivity.
    Qed.

    Lemma bindt_mapdt: forall
        (g: B -> G2 (T C))
        (f: W * A -> G1 B),
        map (F := G1) (bindt g) ∘ mapdt f =
          binddt (U := U) (G := G1 ∘ G2) (map (F := G1) g ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 B),
map (bindt g) ∘ mapdt f = binddt (map g ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 (T C)) (f : W * A -> G1 B),
map (bindt g) ∘ mapdt f = binddt (map g ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 B
map (bindt g) ∘ mapdt f = binddt (map g ∘ f)
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 B
map (bindt g) ∘ binddt (map ret ∘ f) =
binddt (map g ∘ f)
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 B
map (binddt (g ∘ extract)) ∘ binddt (map ret ∘ f) =
binddt (map g ∘ f)
      rewrite kdtm_binddt2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 B
binddt (g ∘ extract ⋆7 map ret ∘ f) =
binddt (map g ∘ f)
      rewrite kc7_63.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: W * A -> G1 B
binddt (map g ∘ f) = binddt (map g ∘ f)
      reflexivity.
    Qed.

  End composition_special_cases_middle.

  (* The lemmas below can cite the ones above *)
  Section composition_special_cases_bottom.

    Context
      `{Applicative G1}
      `{Applicative G2}
      {A B C: Type}.

    (** *** <<bindd>> on the right *)
    (******************************************************************)

    (* composition_25 *)
    Lemma traverse_bindd: forall
        (g: B -> G2 C)
        (f: W * A -> T B),
        traverse (T := U) (G := G2) g ∘ bindd f =
          binddt (fun '(w, a) => traverse (G := G2) g (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : W * A -> T B),
traverse g ∘ bindd f =
binddt (fun '(w, a) => traverse g (f (w, a)))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : B -> G2 C) (f : W * A -> T B),
traverse g ∘ bindd f =
binddt (fun '(w, a) => traverse g (f (w, a)))
      (* TODO Use traverse_to_bindt to solve simpler  *)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
traverse g ∘ bindd f =
binddt (fun '(w, a) => traverse g (f (w, a)))
      rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
binddt (map ret ∘ g ∘ extract) ∘ bindd f =
binddt (fun '(w, a) => traverse g (f (w, a)))
      rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
binddt (map ret ∘ g ∘ extract) ∘ binddt f =
binddt (fun '(w, a) => traverse g (f (w, a)))
      change (binddt (?mapret ∘ g ∘ ?extract)) with
        (map (F := fun A => A) (binddt (mapret ∘ g ∘ extract))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
map (binddt (map ret ∘ g ∘ extract)) ∘ binddt f =
binddt (fun '(w, a) => traverse g (f (w, a)))
      rewrite (kdtm_binddt2 (T := T) (G1 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
binddt (map ret ∘ g ∘ extract ⋆7 f) =
binddt (fun '(w, a) => traverse g (f (w, a)))
      rewrite (binddt_app_id_l).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
binddt (map ret ∘ g ∘ extract ⋆7 f) =
binddt (fun '(w, a) => traverse g (f (w, a)))
      fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
map ret ∘ g ∘ extract ⋆7 f =
(fun '(w, a) => traverse g (f (w, a))) ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
w: W
a: A
(map ret ∘ g ∘ extract ⋆7 f) (w, a) =
traverse g (f (w, a))
      unfold kc7.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
w: W
a: A
map (binddt ((map ret ∘ g ∘ extract) ⦿ w)) (f (w, a)) =
traverse g (f (w, a))
      rewrite preincr_assoc.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
w: W
a: A
map (binddt (map ret ∘ g ∘ extract ⦿ w)) (f (w, a)) =
traverse g (f (w, a))
      rewrite extract_preincr.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
w: W
a: A
map (binddt (map ret ∘ g ∘ extract)) (f (w, a)) =
traverse g (f (w, a))
      rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: W * A -> T B
w: W
a: A
map (binddt (map ret ∘ g ∘ extract)) (f (w, a)) =
binddt (map ret ∘ g ∘ extract) (f (w, a))
      reflexivity.
    Qed.

    (* composition_52 *)
    (* TODO bindd_traverse *)

    (* composition_46 *)
    Lemma mapd_bindt: forall (g: W * B -> C) (f: A -> G1 (T B)),
        map (F := G1) (mapd g) ∘ bindt (G := G1) f =
          binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : A -> G1 (T B)),
map (mapd g) ∘ bindt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
forall (g : W * B -> C) (f : A -> G1 (T B)),
map (mapd g) ∘ bindt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
      introv.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
map (mapd g) ∘ bindt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
      rewrite mapd_to_mapdt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
map (mapdt g) ∘ bindt f =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
      rewrite (mapdt_bindt (G2 := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a)) =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
      rewrite binddt_app_id_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
binddt (fun '(w, a) => map (mapdt (g ⦿ w)) (f a)) =
binddt (fun '(w, a) => map (mapd (g ⦿ w)) (f a))
      fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
(fun '(w, a) => map (mapdt (g ⦿ w)) (f a)) =
(fun '(w, a) => map (mapd (g ⦿ w)) (f a)) ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
w: W
a: A
map (mapdt (g ⦿ w)) (f a) = map (mapd (g ⦿ w)) (f a)
      rewrite mapd_to_mapdt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: W * B -> C
f: A -> G1 (T B)
w: W
a: A
map (mapdt (g ⦿ w)) (f a) = map (mapdt (g ⦿ w)) (f a)
      reflexivity.
    Qed.

    (* composition_64 *)
    (* TODO bindt_mapd *)

    (* composition_16 *)
    (* TODO bind_mapdt *)

    (* composition_61 *)
    (* TODO mapdt_bind *)

  End composition_special_cases_bottom.

End other_composition_laws.

(** ** Derived Identity/<<ret>>/Appl. Homomorphism Laws *)
(**********************************************************************)
Section derived_instances.

  Context
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd W T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt W T}
    `{Bindd_T_inst: Bindd W T T}
    `{Bindt_T_inst: Bindt T T}
    `{Binddt_T_inst: Binddt W T T}
    `{! Compat_Map_Binddt W T T}
    `{! Compat_Mapd_Binddt W T T}
    `{! Compat_Traverse_Binddt W T T}
    `{! Compat_Bind_Binddt W T T}
    `{! Compat_Mapdt_Binddt W T T}
    `{! Compat_Bindd_Binddt W T T}
    `{! Compat_Bindt_Binddt W T T}
    `{Monad_inst: ! DecoratedTraversableMonad W T}
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd W U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt W U}
    `{Bindd_U_inst: Bindd W T U}
    `{Bindt_U_inst: Bindt T U}
    `{Binddt_U_inst: Binddt W T U}
    `{! Compat_Map_Binddt W T U}
    `{! Compat_Mapd_Binddt W T U}
    `{! Compat_Traverse_Binddt W T U}
    `{! Compat_Bind_Binddt W T U}
    `{! Compat_Mapdt_Binddt W T U}
    `{! Compat_Bindd_Binddt W T U}
    `{! Compat_Bindt_Binddt W T U}
    `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.

  (** *** <<binddt>> purity *)
  (********************************************************************)
  Lemma binddt_pure:
    forall (A: Type) `{Applicative G},
      binddt (pure (F := G) ∘ ret (T := T) (A := A) ∘ extract) =
        pure (F := G).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G -> binddt (pure ∘ ret ∘ extract) = pure
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G -> binddt (pure ∘ ret ∘ extract) = pure
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
binddt (pure ∘ ret ∘ extract) = pure
    reassociate -> on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
binddt (pure ∘ (ret ∘ extract)) = pure
    rewrite <- (kdtm_morph (fun A => A) G (ϕ := @pure G _)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
pure ∘ binddt (ret ∘ extract) = pure
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
pure ∘ id = pure
    reflexivity.
  Qed.

  (** *** Identity laws *)
  (********************************************************************)
  Lemma bindd_id: forall A: Type,
      bindd (U := U) (ret ∘ extract (W := (W ×))) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bindd (ret ∘ extract) = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bindd (ret ∘ extract) = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
bindd (ret ∘ extract) = id
    rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma bindt_id: forall A: Type,
      bindt (G := fun A => A) (ret (T := T)) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bindt ret = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bindt ret = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
bindt ret = id
    rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma mapdt_id: forall A: Type,
      mapdt (G := fun A => A) (extract (W := (W ×))) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, mapdt extract = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, mapdt extract = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapdt extract = id
    rewrite (mapdt_to_binddt (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ extract) = id
    unfold_ops @Map_I.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma mapd_id: forall A: Type,
      mapd (extract (W := (W ×))) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, mapd extract = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, mapd extract = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapd extract = id
    rewrite mapd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma traverse_id: forall A: Type,
      traverse (T := U) (G := fun A => A) (@id A) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, traverse id = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, traverse id = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
traverse id = id
    rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ id ∘ extract) = id
    change (?f ∘ id) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ extract) = id
    unfold_ops @Map_I.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma bind_id: forall A: Type,
      bind (ret (T := T)) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bind ret = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, bind ret = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
bind ret = id
    rewrite bind_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  Lemma map_id: forall A: Type,
      map (F := U) (@id A) = @id (U A).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, map id = id
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, map id = id
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
map id = id
    rewrite map_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ id ∘ extract) = id
    change (?f ∘ id) with f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (ret ∘ extract) = id
    rewrite kdtm_binddt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
id = id
    reflexivity.
  Qed.

  (** *** Composition with <<ret>> *)
  (********************************************************************)
  Lemma bindd_ret:
    forall (A B: Type) (f: W * A -> T B),
      bindd (T := T) f ∘ ret (T := T) = f ∘ ret (T := (W ×)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
bindd f ∘ ret = f ∘ ret
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
bindd f ∘ ret = f ∘ ret
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
bindd f ∘ ret = f ∘ ret
    rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
binddt f ∘ ret = f ∘ ret
    rewrite (kdtm_binddt0 (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
f ∘ ret = f ∘ ret
    reflexivity.
  Qed.

  Lemma bindt_ret:
    forall (G: Type -> Type) `{Applicative G}
      (A B: Type) (f: A -> G (T B)),
      bindt f ∘ ret (T := T) = f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G (T B)),
bindt f ∘ ret = f
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G (T B)),
bindt f ∘ ret = f
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
bindt f ∘ ret = f
    rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
binddt (f ∘ extract) ∘ ret = f
    rewrite kdtm_binddt0.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
f ∘ extract ∘ ret = f
    reflexivity.
  Qed.

  Lemma traverse_ret:
    forall (G: Type -> Type) `{Applicative G}
      (A B: Type) (f: A -> G B),
      traverse f ∘ ret (T := T) = map ret ∘ f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G B),
traverse f ∘ ret = map ret ∘ f
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G B),
traverse f ∘ ret = map ret ∘ f
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G B
traverse f ∘ ret = map ret ∘ f
    rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G B
binddt (map ret ∘ f ∘ extract) ∘ ret = map ret ∘ f
    rewrite (kdtm_binddt0).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G B
map ret ∘ f ∘ extract ∘ ret = map ret ∘ f
    reflexivity.
  Qed.

  Lemma bind_ret:
    forall (A B: Type) (f: A -> T B),
      bind f ∘ ret (T := T) = f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : A -> T B), bind f ∘ ret = f
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : A -> T B), bind f ∘ ret = f
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: A -> T B
bind f ∘ ret = f
    rewrite bind_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: A -> T B
binddt (f ∘ extract) ∘ ret = f
    rewrite (kdtm_binddt0 (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: A -> T B
f ∘ extract ∘ ret = f
    reflexivity.
  Qed.

  (** *** Naturality in Applicative Morphisms *)
  (********************************************************************)
  Section applicative_morphisms.

    Context
      (G1 G2: Type -> Type)
      `{Map G1} `{Mult G1} `{Pure G1}
      `{Map G2} `{Mult G2} `{Pure G2}
      (ϕ: forall A: Type, G1 A -> G2 A)
      `{morph: ! ApplicativeMorphism G1 G2 ϕ }.

    Lemma bindt_morph:
      forall (A B: Type) (f: A -> G1 (T B)),
        ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : A -> G1 (T B)),
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : A -> G1 (T B)),
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
      inversion morph.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
      rewrite bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (f ∘ extract) = bindt (ϕ (T B) ∘ f)
      rewrite (kdtm_morph G1 G2).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (ϕ (T B) ∘ (f ∘ extract)) = bindt (ϕ (T B) ∘ f)
      reassociate <- on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (ϕ (T B) ∘ f ∘ extract) = bindt (ϕ (T B) ∘ f)
      rewrite <- bindt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
bindt (ϕ (T B) ∘ f) = bindt (ϕ (T B) ∘ f)
      reflexivity.
    Qed.

    Lemma mapdt_morph:
      forall (A B: Type) (f: W * A -> G1 B),
        ϕ (U B) ∘ mapdt (T := U) f = mapdt (ϕ B ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : W * A -> G1 B),
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : W * A -> G1 B),
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f)
      inversion morph.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f)
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) = mapdt (ϕ B ∘ f)
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) =
binddt (map ret ∘ (ϕ B ∘ f))
      reassociate <- on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) =
binddt (map ret ∘ ϕ B ∘ f)
      rewrite (natural (ϕ := ϕ)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) =
binddt (ϕ (T B) ∘ map ret ∘ f)
      reassociate -> on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) =
binddt (ϕ (T B) ∘ (map ret ∘ f))
      rewrite <- (kdtm_morph G1 G2).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f) =
ϕ (U B) ∘ binddt (map ret ∘ f)
      rewrite <- mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ mapdt f = ϕ (U B) ∘ mapdt f
      reflexivity.
    Qed.

    Lemma traverse_morph:
      forall (A B: Type) (f: A -> G1 B),
        ϕ (U B) ∘ traverse (T := U) (G := G1) f =
          traverse (T := U) (G := G2) (ϕ B ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : A -> G1 B),
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f)
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
forall (A B : Type) (f : A -> G1 B),
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f)
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f)
      inversion morph.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f)
      rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
ϕ (U B) ∘ binddt (map ret ∘ f ∘ extract) =
traverse (ϕ B ∘ f)
      rewrite (kdtm_morph G1 G2).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (ϕ (T B) ∘ (map ret ∘ f ∘ extract)) =
traverse (ϕ B ∘ f)
      do 2 reassociate <- on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (ϕ (T B) ∘ map ret ∘ f ∘ extract) =
traverse (ϕ B ∘ f)
      rewrite <- (natural (ϕ := ϕ)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (map ret ∘ ϕ B ∘ f ∘ extract) =
traverse (ϕ B ∘ f)
      reassociate -> near f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
binddt (map ret ∘ (ϕ B ∘ f) ∘ extract) =
traverse (ϕ B ∘ f)
      rewrite <- traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
Monad_inst: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morph: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
appmor_app_F: Applicative G1
appmor_app_G: Applicative G2
appmor_natural: forall (A B : Type) (f : A -> B)
  (x : G1 A),
ϕ B (map f x) = map f (ϕ A x)
appmor_pure: forall (A : Type) (a : A),
ϕ A (pure a) = pure a
appmor_mult: forall (A B : Type) (x : G1 A)
  (y : G1 B),
ϕ (A * B) (mult (x, y)) =
mult (ϕ A x, ϕ B y)
traverse (ϕ B ∘ f) = traverse (ϕ B ∘ f)
      reflexivity.
    Qed.

  End applicative_morphisms.

End derived_instances.

(** ** Derived Typeclass Instances *)
(**********************************************************************)
Module DerivedInstances.
  Section derived_instances.

    Context
      (W: Type)
      (T: Type -> Type)
      `{op: Monoid_op W}
      `{unit: Monoid_unit W}
      `{ret_inst: Return T}
      `{Map_T_inst: Map T}
      `{Mapd_T_inst: Mapd W T}
      `{Traverse_T_inst: Traverse T}
      `{Bind_T_inst: Bind T T}
      `{Mapdt_T_inst: Mapdt W T}
      `{Bindd_T_inst: Bindd W T T}
      `{Bindt_T_inst: Bindt T T}
      `{Binddt_T_inst: Binddt W T T}
      `{! Compat_Map_Binddt W T T}
      `{! Compat_Mapd_Binddt W T T}
      `{! Compat_Traverse_Binddt W T T}
      `{! Compat_Bind_Binddt W T T}
      `{! Compat_Mapdt_Binddt W T T}
      `{! Compat_Bindd_Binddt W T T}
      `{! Compat_Bindt_Binddt W T T}
      `{dtm_T: ! DecoratedTraversableMonad W T}.

    (** *** Exported instances *)
    (******************************************************************)
    #[export] Instance
      DecoratedMonad_DecoratedTraversableMonad:
      DecoratedMonad W T.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
DecoratedMonad W T
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
DecoratedMonad W T
      constructor.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
Monoid W
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
DecoratedRightPreModule W T T
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
forall (A B : Type) (f : W * A -> T B),
bindd f ∘ ret = f ∘ ret
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
Monoid W typeclasses eauto.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
DecoratedRightPreModule W T T constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A: Type
bindd (ret ∘ extract) = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
bindd g ∘ bindd f = bindd (g ⋆5 f)
        apply bindd_id.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A, B, C: Type
g: W * B -> T C
f: W * A -> T B
bindd g ∘ bindd f = bindd (g ⋆5 f)
        apply bindd_bindd.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
forall (A B : Type) (f : W * A -> T B),
bindd f ∘ ret = f ∘ ret intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A, B: Type
f: W * A -> T B
bindd f ∘ ret = f ∘ ret
        apply bindd_ret.
    Qed.

    #[export] Instance
      TraversableMonad_DecoratedTraversableMonad:
      TraversableMonad T.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
TraversableMonad T
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
TraversableMonad T
      constructor.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G (T B)),
bindt f ∘ ret = f
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
TraversableRightPreModule T T
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (f : A -> G (T B)),
bindt f ∘ ret = f intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
A, B: Type
f: A -> G (T B)
bindt f ∘ ret = f now apply bindt_ret.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
TraversableRightPreModule T T constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A: Type
bindt ret = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (T B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
        apply bindt_id.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (T B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
        apply bindt_bindt.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (T B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
        apply (bindt_morph G1 G2 ϕ).
    Qed.

    #[export] Instance
      Monad_DecoratedTraversableMonad:
      Monad T :=
      DecoratedMonad.DerivedInstances.Monad_DecoratedMonad W T.

    #[export] Instance
      Functor_DecoratedTraversableMonad:
      Functor T.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
Functor T
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
Functor T
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A: Type
map id = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A, B, C: Type
f: A -> B
g: B -> C
map g ∘ map f = map (g ∘ f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A: Type
map id = id apply map_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
A, B, C: Type
f: A -> B
g: B -> C
map g ∘ map f = map (g ∘ f) apply map_map.
    Qed.

    (** *** Local instances *)
    (******************************************************************)
    (*
    #[export] Instance
      DecoratedRightPreModule_DecoratedTraversableMonad:
      DecoratedRightPreModule W T T.
    Proof.
      constructor; intros.
      apply bindd_id.
      apply bindd_bindd.
    Qed.

    #[export] Instance
      DecoratedMonad_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      DecoratedMonad W T :=
      {| kdm_bindd0 := bindd_ret;
      |}.

    #[export] Instance
      TraversableRightPreModule_DecoratedTraversableMonad:
      TraversableRightPreModule T T.
    Proof.
      constructor; intros.
      - apply bindt_id.
      - apply bindt_bindt.
      - apply (bindt_morph G1 G2 ϕ).
    Qed.

    #[export] Instance
      TraversableMonad_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      TraversableMonad T :=
      {| ktm_bindt0 := bindt_ret;
      |}.

    #[export] Instance
      DecoratedTraversableFunctor_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      DecoratedTraversableFunctor W T.
    Proof.
      constructor; intros.
      - apply mapdt_id.
      - apply mapdt_mapdt.
      - apply (mapdt_morph G1 G2 ϕ).
    Qed.

    #[export] Instance
      TraversableFunctor_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      TraversableFunctor T.
    Proof.
      constructor.
      - apply traverse_id.
      - intros. apply traverse_traverse.
      - intros. apply traverse_morph. auto.
    Qed.


    #[export] Instance
      DecoratedFunctor_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      DecoratedFunctor W T.
    Proof.
      constructor.
      - apply mapd_id.
      - intros. apply mapd_mapd.
    Qed.

    #[export] Instance
      Functor_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      Functor T.
    Proof.
      constructor; intros.
      - apply map_id.
      - apply map_map.
    Qed.
*)

    #[local] Instance
      DecoratedTraversableRightModule_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      DecoratedTraversableRightModule W T T.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T, DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
DecoratedTraversableRightModule W T T
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T, DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
DecoratedTraversableRightModule W T T
      constructor; typeclasses eauto.
    Qed.

    (*
    #[local] Instance
      TraversableRightModule_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      TraversableRightModule T T.
    Proof.
      constructor; typeclasses eauto.
    Qed.

    #[local] Instance
      DecoratedRightModule_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      DecoratedRightModule W T T.
    Proof.
      constructor; typeclasses eauto.
    Qed.

    #[local] Instance
      RightModule_DecoratedTraversableMonad
      `{! DecoratedTraversableMonad W T}:
      RightModule T T.
    Proof.
      constructor; typeclasses eauto.
    Qed.
    *)

    Context
      (U: Type -> Type)
      `{Map_U_inst: Map U}
      `{Mapd_U_inst: Mapd W U}
      `{Traverse_U_inst: Traverse U}
      `{Bind_U_inst: Bind T U}
      `{Mapdt_U_inst: Mapdt W U}
      `{Bindd_U_inst: Bindd W T U}
      `{Bindt_U_inst: Bindt T U}
      `{Binddt_U_inst: Binddt W T U}
      `{! Compat_Map_Binddt W T U}
      `{! Compat_Mapd_Binddt W T U}
      `{! Compat_Traverse_Binddt W T U}
      `{! Compat_Bind_Binddt W T U}
      `{! Compat_Mapdt_Binddt W T U}
      `{! Compat_Bindd_Binddt W T U}
      `{! Compat_Bindt_Binddt W T U}
      `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.

    (** *** Exported instances *)
    (******************************************************************)
    #[export] Instance
      TraversableRightPreModule_DecoratedTraversableRightPreModule:
      TraversableRightPreModule T U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
TraversableRightPreModule T U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
TraversableRightPreModule T U
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
bindt ret = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (bindt g) ∘ bindt f = bindt (g ⋆6 f)
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
bindt ret = id apply bindt_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 (T C)
f: A -> G1 (T B)
map (bindt g) ∘ bindt f = bindt (g ⋆6 f) apply bindt_bindt.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 (T B)
ϕ (U B) ∘ bindt f = bindt (ϕ (T B) ∘ f) apply (bindt_morph G1 G2 ϕ).
    Qed.

    #[export] Instance
      DecoratedRightPreModule_DecoratedTraversableRightPreModule:
      DecoratedRightPreModule W T U :=
      {| kdmod_bindd1 := bindd_id;
         kdmod_bindd2 := bindd_bindd;
      |}.

    #[export] Instance
      DecoratedTraversableFunctor_DecoratedTraversableRightPreModule:
      DecoratedTraversableFunctor W U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableFunctor W U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableFunctor W U
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapdt extract = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H0: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
map (mapdt g) ∘ mapdt f = mapdt (g ⋆3 f)
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H0: Map G1
H1: Mult G1
H2: Pure G1
H3: Map G2
H4: Mult G2
H5: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapdt extract = id apply mapdt_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H0: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H1: Applicative G2
A, B, C: Type
g: W * B -> G2 C
f: W * A -> G1 B
map (mapdt g) ∘ mapdt f = mapdt (g ⋆3 f) apply mapdt_mapdt.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H0: Map G1
H1: Mult G1
H2: Pure G1
H3: Map G2
H4: Mult G2
H5: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: W * A -> G1 B
ϕ (U B) ∘ mapdt f = mapdt (ϕ B ∘ f) apply (mapdt_morph G1 G2 ϕ).
    Qed.

    #[export] Instance
      DecoratedFunctor_DecoratedTraversableRightPreModule:
      DecoratedFunctor W U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedFunctor W U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedFunctor W U
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapd extract = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B, C: Type
g: W * B -> C
f: W * A -> B
mapd g ∘ mapd f = mapd (g ⋆1 f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapd extract = id apply mapd_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B, C: Type
g: W * B -> C
f: W * A -> B
mapd g ∘ mapd f = mapd (g ⋆1 f) apply mapd_mapd.
    Qed.

    #[export] Instance
      TraversableFunctor_DecoratedTraversableRightPreModule:
      TraversableFunctor U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
TraversableFunctor U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
TraversableFunctor U
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
traverse id = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
map (traverse g) ∘ traverse f = traverse (g ⋆2 f)
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
traverse id = id apply traverse_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1: Type -> Type
Map_G: Map G1
Pure_G: Pure G1
Mult_G: Mult G1
H: Applicative G1
G2: Type -> Type
Map_G0: Map G2
Pure_G0: Pure G2
Mult_G0: Mult G2
H0: Applicative G2
A, B, C: Type
g: B -> G2 C
f: A -> G1 B
map (traverse g) ∘ traverse f = traverse (g ⋆2 f) apply traverse_traverse.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
ϕ (U B) ∘ traverse f = traverse (ϕ B ∘ f) apply traverse_morph.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
G1, G2: Type -> Type
H: Map G1
H0: Mult G1
H1: Pure G1
H2: Map G2
H3: Mult G2
H4: Pure G2
ϕ: forall A : Type, G1 A -> G2 A
morphism: ApplicativeMorphism G1 G2 ϕ
A, B: Type
f: A -> G1 B
ApplicativeMorphism G1 G2 ϕ auto.
    Qed.

    #[export] Instance
      Functor_DecoratedTraversableRightPreModule:
      Functor U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
Functor U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
Functor U
      constructor; intros.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
map id = id
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B, C: Type
f: A -> B
g: B -> C
map g ∘ map f = map (g ∘ f)
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
map id = id apply map_id.
      -
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
A, B, C: Type
f: A -> B
g: B -> C
map g ∘ map f = map (g ∘ f) apply map_map.
    Qed.

    (** *** Local instances *)
    (******************************************************************)
    #[local] Instance
      DecoratedTraversableRightModule_DecoratedTraversableRightPreModule
      `{! DecoratedTraversableMonad W T}:
      DecoratedTraversableRightModule W T U.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
DecoratedTraversableRightModule W T U
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
DecoratedTraversableRightModule W T U
      constructor; typeclasses eauto.
    Qed.

    #[local] Instance
      TraversableRightModule_DecoratedTraversableRightPreModule:
      TraversableRightModule T U :=
      {| ktmod_monad := _
      |}.

    #[local] Instance
      DecoratedRightModule_DecoratedTraversableRightPreModule
      `{! DecoratedTraversableMonad W T}:
      DecoratedRightModule W T U :=
      {| kdmod_monad := _
      |}.

    #[local] Instance
      RightModule_DecoratedTraversableRightPreModule
      `{! DecoratedTraversableMonad W T}:
      RightModule T T.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
RightModule T T
    Proof.
W: Type
T: Type -> Type
op: Monoid_op W
unit: Monoid_unit W
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
dtm_T: DecoratedTraversableMonad W T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedTraversableMonad0: DecoratedTraversableMonad
  W T
RightModule T T
      constructor; typeclasses eauto.
    Qed.

  End derived_instances.
End DerivedInstances.

(** * Notations *)
(**********************************************************************)
Module Notations.
  Infix "⋆7" := (kc7 _ _) (at level 60): tealeaves_scope.
End Notations.