From Tealeaves Require Export
  Classes.Kleisli.DecoratedTraversableMonad
  Classes.Kleisli.Theory.DecoratedTraversableFunctor
  Classes.Kleisli.Theory.DecoratedContainerMonad.

Import Monoid.Notations.
Import Subset.Notations.
Import List.ListNotations.
Import ContainerFunctor.Notations.
Import DecoratedContainerFunctor.Notations.

#[local] Generalizable Variable W M T U G A B C.


(** * Theory of Decorated Traversable Monads *)
(**********************************************************************)
Section composition.

  Context
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{Monoid_inst: ! Monoid W}.

  Context
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd W T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt 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}.

  Context
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd W U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt 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}.

  Context
    `{ToSubset_T_inst: ToSubset T}
    `{ToSubset_U_inst: ToSubset U}
    `{! Compat_ToSubset_Traverse T}
    `{! Compat_ToSubset_Traverse U}.

  Context
    `{ToCtxset_T_inst: ToCtxset W T}
    `{ToCtxset_U_inst: ToCtxset W U}
    `{! Compat_ToCtxset_Mapdt W T}
    `{! Compat_ToCtxset_Mapdt W U}.

  Context
    `{ToCtxlist_T_inst: ToCtxlist W T}
    `{ToCtxlist_U_inst: ToCtxlist W U}
    `{! Compat_ToCtxlist_Mapdt W T}
    `{! Compat_ToCtxlist_Mapdt W U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad W T}
    `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.

  (** ** Composition in the Applicative Functor *)
  (********************************************************************)

  (** *** Composition with a Monoid *)
  (********************************************************************)
  Lemma binddt_app_const_r:
    forall {G: Type -> Type}
      `{Monoid M} {A B: Type}
      `{Applicative G} (f: W * A -> G M),
      @binddt W T U _ (G ∘ const M)
        (Map_compose G (const M))
        (Pure_compose G (const M))
        (Mult_compose G (const M)) A B f =
        binddt (U := U) (G := const (G M)) (B := B) f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (M : Type)
  (op0 : Monoid_op M) (unit0 : Monoid_unit M),
Monoid M ->
forall (A B : Type) (Map_G : Map G) (Pure_G : Pure G)
  (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G M, binddt f = binddt f
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (G : Type -> Type) (M : Type)
  (op0 : Monoid_op M) (unit0 : Monoid_unit M),
Monoid M ->
forall (A B : Type) (Map_G : Map G) (Pure_G : Pure G)
  (Mult_G : Mult G),
Applicative G ->
forall f : W * A -> G M, binddt f = binddt f
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
binddt f = binddt f fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
Map_compose G (const M) = Map_const
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
Mult_compose G (const M) = Mult_const
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
Map_compose G (const M) = Map_const ext X Y h x.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
X, Y: Type
h: X -> Y
x: (G ∘ const M) X
Map_compose G (const M) X Y h x = Map_const X Y h x
      unfold_ops @Map_compose @Map_const.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
X, Y: Type
h: X -> Y
x: (G ∘ const M) X
map (fun t : const M X => t) x = x
      now rewrite (fun_map_id (Functor := app_functor)).
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
Mult_compose G (const M) = Mult_const ext X Y [x y].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
X, Y: Type
x: (G ∘ const M) X
y: (G ∘ const M) Y
Mult_compose G (const M) X Y (x, y) =
Mult_const X Y (x, y)
      unfold_ops @Mult_compose @Mult_const.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
X, Y: Type
x: (G ∘ const M) X
y: (G ∘ const M) Y
map (fun '(x, y) => x ● y)
  (mult (fst (x, y), snd (x, y))) = x ● y
      unfold_ops @Monoid_op_applicative.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
f: W * A -> G M
X, Y: Type
x: (G ∘ const M) X
y: (G ∘ const M) Y
map (fun '(x, y) => x ● y)
  (mult (fst (x, y), snd (x, y))) =
map (uncurry monoid_op) (mult (x, y))
      reflexivity.
  Qed.

  (** ** Rewriting Lemmas for Particular Applicative Functors *)
  (********************************************************************)

  (** *** <<binddt>> withcConstant applicative functors *)
  (********************************************************************)
  Section constant_applicative.

    Context `{Monoid M}.

    Lemma binddt_constant_applicative1 {A B: Type}
      `(f: W * A -> const M (T B)):
      binddt (T := T) (U := U) f =
        binddt (T := T) (U := U) (B := False) f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
      change_right (map (F := const M) (A := U False) (B := U B)
                      (map (F := U) (A := False) (B := B) exfalso)
                      ∘ binddt (T := T) (U := U) (B := False) f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: W * A -> const M (T B)
binddt f = map (map exfalso) ∘ binddt f
      rewrite (map_binddt (G1 := const M)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: W * A -> const M (T B)
binddt f = binddt (map (map exfalso) ∘ f)
      reflexivity.
    Qed.

    Lemma binddt_constant_applicative2 (fake1 fake2: Type)
      `(f: W * A -> const M (T B)):
      binddt (T := T) (U := U) (B := fake1) f =
        binddt (T := T) (U := U) (B := fake2) f.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
fake1, fake2, A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
fake1, fake2, A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
fake1, fake2, A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
      rewrite (binddt_constant_applicative1 (B := fake1)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
fake1, fake2, A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
      rewrite (binddt_constant_applicative1 (B := fake2)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
fake1, fake2, A, B: Type
f: W * A -> const M (T B)
binddt f = binddt f
      reflexivity.
    Qed.

  End constant_applicative.

  (** ** Properties of <<mapdReduce>> *)
  (********************************************************************)

  (** *** Composition with monad operations *)
  (********************************************************************)
  Theorem mapdReduce_ret `{Monoid M}: forall `(f: W * A -> M),
      mapdReduce (T := T) f ∘ ret = f ∘ ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (A : Type) (f : W * A -> M),
mapdReduce f ∘ ret = f ∘ ret
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (A : Type) (f : W * A -> M),
mapdReduce f ∘ ret = f ∘ ret
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: W * A -> M
mapdReduce f ∘ ret = f ∘ ret
    rewrite mapdReduce_to_mapdt1. (* todo get rid of this arg *)
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: W * A -> M
mapdt f ∘ ret = f ∘ ret
    rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: W * A -> M
binddt (map ret ∘ f) ∘ ret = f ∘ ret
    rewrite (kdtm_binddt0 (G := const M) (A := A) (B := False)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: W * A -> M
map ret ∘ f ∘ ret = f ∘ ret
    reflexivity.
  Qed.

  Theorem mapdReduce_binddt `{Applicative G} `{Monoid M}:
    forall `(g: W * B -> M) `(f: W * A -> G (T B)),
      map (mapdReduce g) ∘ binddt f =
        mapdReduce (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> G (T B)),
map (mapdReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> G (T B)),
map (mapdReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
map (mapdReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    rewrite mapdReduce_to_mapdt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
map (mapdt g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
map (binddt (map ret ∘ g)) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    rewrite (kdtm_binddt2 (G2 := const M) (G1 := G)). (* TODO args *)
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
binddt (kc7 G (const M) (map ret ∘ g) f) =
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    rewrite mapdReduce_to_mapdt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
binddt (kc7 G (const M) (map ret ∘ g) f) =
mapdt
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a)))
    rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
binddt (kc7 G (const M) (map ret ∘ g) f) =
binddt
  (map ret
   ∘ (fun '(w, a) =>
      map (mapdReduce (g ⦿ w)) (f (w, a))))
    rewrite binddt_app_const_r.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
binddt (kc7 G (const M) (map ret ∘ g) f) =
binddt
  (map ret
   ∘ (fun '(w, a) =>
      map (mapdReduce (g ⦿ w)) (f (w, a))))
    unfold mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
binddt (kc7 G (const M) (map ret ∘ g) f) =
binddt
  (map ret
   ∘ (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a))))
    (* TODO Make mapdt_to_binddt work immediately here *)
    fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
kc7 G (const M) (map ret ∘ g) f =
map ret
∘ (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a))) ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
kc7 G (const M) (map ret ∘ g) f (w, a) =
(map ret
 ∘ (fun '(w, a) => map (mapdt (g ⦿ w)) (f (w, a))))
  (w, a)
    unfold compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
kc7 G (const M) (map ret ○ g) f (w, a) =
map ret (map (mapdt (g ⦿ w)) (f (w, a))) cbn.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
map (binddt ((map ret ○ g) ⦿ w)) (f (w, a)) =
map ret (map (mapdt (g ⦿ w)) (f (w, a)))
    rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
map (binddt ((map ret ○ g) ⦿ w)) (f (w, a)) =
map ret (map (binddt (map ret ∘ g ⦿ w)) (f (w, a)))
    reflexivity.
  Qed.

  Corollary mapdReduce_binddt_I `{Monoid M}:
    forall `(g: W * B -> M) `(f: W * A -> T B),
      mapdReduce g ∘ binddt (U := U) (G := fun A => A) f =
        mapdReduce (T := U) (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> T B),
mapdReduce g ∘ binddt f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> T B),
mapdReduce g ∘ binddt f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
mapdReduce g ∘ binddt f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    change (mapdReduce g) with
      (map (F := fun A => A) (mapdReduce (T := U) g)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
map (mapdReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    rewrite (mapdReduce_binddt (G := fun A => A)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => map (mapdReduce (g ⦿ w)) (f (w, a))) =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    reflexivity.
  Qed.

  Corollary mapdReduce_bindd `{Monoid M}:
    forall `(g: W * B -> M) `(f: W * A -> T B),
      mapdReduce g ∘ bindd f =
        mapdReduce (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> T B),
mapdReduce g ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : W * B -> M) (A : Type)
  (f : W * A -> T B),
mapdReduce g ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
mapdReduce g ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
mapdReduce g ∘ binddt f =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    rewrite mapdReduce_binddt_I.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: W * B -> M
A: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a))) =
mapdReduce
  (fun '(w, a) => mapdReduce (g ⦿ w) (f (w, a)))
    reflexivity.
  Qed.

  (** ** Properties of <<mapReduce>> *)
  (********************************************************************)
  Corollary mapReduce_binddt `{Applicative G} `{Monoid M}:
    forall `(g: B -> M) `(f: W * A -> G (T B)),
      map (mapReduce g) ∘ binddt f =
        mapdReduce (fun '(w, a) => map (mapReduce g) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (B : Type) (g : B -> M) (A : Type)
  (f : W * A -> G (T B)),
map (mapReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapReduce g) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (B : Type) (g : B -> M) (A : Type)
  (f : W * A -> G (T B)),
map (mapReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapReduce g) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
map (mapReduce g) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapReduce g) (f (w, a)))
    rewrite mapReduce_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
map (mapdReduce (g ∘ extract)) ∘ binddt f =
mapdReduce
  (fun '(w, a) => map (mapReduce g) (f (w, a)))
    rewrite mapReduce_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
map (mapdReduce (g ∘ extract)) ∘ binddt f =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (g ∘ extract)) (f (w, a)))
    rewrite mapdReduce_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce ((g ∘ extract) ⦿ w)) (f (w, a))) =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (g ∘ extract)) (f (w, a)))
    fequal; ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
map (mapdReduce ((g ∘ extract) ⦿ w)) (f (w, a)) =
map (mapdReduce (g ∘ extract)) (f (w, a))
    rewrite extract_preincr2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H: Applicative G
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> G (T B)
w: W
a: A
map (mapdReduce (g ∘ extract)) (f (w, a)) =
map (mapdReduce (g ∘ extract)) (f (w, a))
    reflexivity.
  Qed.

  Corollary mapReduce_bindd `{Monoid M}:
    forall `(g: B -> M) `(f: W * A -> T B),
      mapReduce g ∘ bindd f =
        mapdReduce (fun '(w, a) => mapReduce g (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : B -> M) (A : Type)
  (f : W * A -> T B),
mapReduce g ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : B -> M) (A : Type)
  (f : W * A -> T B),
mapReduce g ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
mapReduce g ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
    rewrite bindd_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
mapReduce g ∘ binddt f =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
    rewrite mapReduce_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
mapdReduce (g ∘ extract) ∘ binddt f =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
    rewrite mapdReduce_binddt_I.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) =>
   mapdReduce ((g ∘ extract) ⦿ w) (f (w, a))) =
mapdReduce (fun '(w, a) => mapReduce g (f (w, a)))
    fequal; ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
w: W
a: A
mapdReduce ((g ∘ extract) ⦿ w) (f (w, a)) =
mapReduce g (f (w, a))
    rewrite extract_preincr2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
w: W
a: A
mapdReduce (g ∘ extract) (f (w, a)) =
mapReduce g (f (w, a))
    rewrite mapReduce_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
M: Type
op0: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: B -> M
A: Type
f: W * A -> T B
w: W
a: A
mapdReduce (g ∘ extract) (f (w, a)) =
mapdReduce (g ∘ extract) (f (w, a))
    reflexivity.
  Qed.

  (** ** Properties of <<tolistd>> *)
  (********************************************************************)

  (** *** Composition with <<binddt>> / <<ret>> *)
  (********************************************************************)
  Lemma toctxlist_ret:
    forall (A: Type) (a: A),
      toctxlist (ret (T := T) a) =
        [ (Ƶ, a) ].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (a : A),
toctxlist (ret a) = [(Ƶ, a)]
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (a : A),
toctxlist (ret a) = [(Ƶ, a)]
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
toctxlist (ret a) = [(Ƶ, a)]
    rewrite toctxlist_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
mapdReduce ret (ret a) = [(Ƶ, a)]
    compose near a on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
(mapdReduce ret ∘ ret) a = [(Ƶ, a)]
    rewrite mapdReduce_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
(ret ∘ ret) a = [(Ƶ, a)]
    reflexivity.
  Qed.

  Lemma toctxlist_binddt:
    forall `{Applicative G} `(f: W * A -> G (T B)),
      map (F := G) toctxlist ∘ binddt (G := G) f =
        mapdReduce (T := U)
          (fun '(w, a) =>
             map (mapdReduce (T := T) (ret (T := list) ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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 : W * A -> G (T B)),
map toctxlist ∘ binddt f =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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 : W * A -> G (T B)),
map toctxlist ∘ binddt f =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
map toctxlist ∘ binddt f =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a)))
    rewrite toctxlist_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
map (mapdReduce ret) ∘ binddt f =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a)))
    rewrite mapdReduce_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a))) =
mapdReduce
  (fun '(w, a) =>
   map (mapdReduce (ret ⦿ w)) (f (w, a)))
    reflexivity.
  Qed.

  (** *** Composition with <<bindd>> *)
  (********************************************************************)
  Corollary toctxlist_bindd: forall `(f: W * A -> T B),
      toctxlist ∘ bindd f =
        mapdReduce (T := U)
          (fun '(w, a) => (mapdReduce (ret (T := list) ⦿ w)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxlist ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxlist ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
toctxlist ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a)))
    rewrite toctxlist_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce ret ∘ bindd f =
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a)))
    rewrite mapdReduce_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a)))
    reflexivity.
  Qed.

  (** ** Properties of <<tolist>> *)
  (********************************************************************)

  (** *** Composition with <<binddt>> *)
  (********************************************************************)
  Corollary tolist_binddt: forall `{Applicative G} `(f: W * A -> G (T B)),
      map tolist ∘ binddt f = mapdReduce (T := U) (map tolist ∘ f).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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 : W * A -> G (T B)),
map tolist ∘ binddt f = mapdReduce (map tolist ∘ f)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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 : W * A -> G (T B)),
map tolist ∘ binddt f = mapdReduce (map tolist ∘ f)
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
map tolist ∘ binddt f = mapdReduce (map tolist ∘ f)
    rewrite tolist_to_mapReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
map (mapReduce ret) ∘ binddt f =
mapdReduce (map tolist ∘ f)
    rewrite mapReduce_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
mapdReduce
  (fun '(w, a) => map (mapReduce ret) (f (w, a))) =
mapdReduce (map tolist ∘ f)
    (* todo why isn't reflexivity enough... b.c. destructing the pair? *)
    fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
(fun '(w, a) => map (mapReduce ret) (f (w, a))) =
map tolist ∘ f ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
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: W * A -> G (T B)
w: W
a: A
map (mapReduce ret) (f (w, a)) =
(map tolist ∘ f) (w, a)
    reflexivity.
  Qed.

  (** *** Composition with <<bindd>> *)
  (********************************************************************)
  Corollary tolist_bindd: forall `(f: W * A -> T B),
      tolist ∘ bindd f =
        mapdReduce (fun '(w, a) => mapReduce (ret (T := list)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
tolist ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
tolist ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
tolist ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    rewrite tolist_to_mapReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapReduce ret ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    rewrite mapReduce_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a))) =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    reflexivity.
  Qed.

  Corollary tolist_to_binddt: forall (A: Type),
      tolist = binddt (G := const (list A))
                 (B := False) (ret (T := list) ∘ extract).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, tolist = binddt (ret ∘ extract)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, tolist = binddt (ret ∘ extract)
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
tolist = binddt (ret ∘ extract)
    rewrite tolist_to_traverse1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
traverse ret = binddt (ret ∘ extract)
    rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ ret ∘ extract) =
binddt (ret ∘ extract)
    reflexivity.
  Qed.

  (** ** Properties of <<toctxset>>  *)
  (********************************************************************)
  Corollary toctxset_ret: forall (A: Type) (a: A),
      toctxset (ret (T := T) a) = {{ (Ƶ, a) }}.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (a : A),
toctxset (ret a) = {{(Ƶ, a)}}
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (a : A),
toctxset (ret a) = {{(Ƶ, a)}}
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
toctxset (ret a) = {{(Ƶ, a)}}
    rewrite toctxset_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
mapdReduce ret (ret a) = {{(Ƶ, a)}}
    compose near a on left.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
(mapdReduce ret ∘ ret) a = {{(Ƶ, a)}}
    rewrite mapdReduce_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
(ret ∘ ret) a = {{(Ƶ, a)}}
    reflexivity.
  Qed.

  Import DecoratedTraversableMonad.DerivedInstances.

  Lemma toctxset_bindd:
    forall `(f: W * A -> T B),
      toctxset ∘ bindd (T := T) (U := U) f =
        bindd (U := ctxset W) (toctxset (F := T) ∘ f) ∘ toctxset (F := U).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
    rewrite toctxset_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce ret ∘ bindd f =
bindd (toctxset ∘ f) ∘ toctxset
    rewrite mapdReduce_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
bindd (toctxset ∘ f) ∘ toctxset
    rewrite toctxset_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
bindd (mapdReduce ret ∘ f) ∘ toctxset
    rewrite toctxset_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
bindd (mapdReduce ret ∘ f) ∘ mapdReduce ret
    rewrite mapdReduce_morphism.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce
  (fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
mapdReduce (bindd (mapdReduce ret ∘ f) ∘ ret)
    fequal.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
(fun '(w, a) => mapdReduce (ret ⦿ w) (f (w, a))) =
bindd (mapdReduce ret ∘ f) ∘ ret
    ext [w a].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
(bindd (mapdReduce ret ∘ f) ∘ ret) (w, a)
    change_right
      (bindd (T := ctxset W) (mapdReduce (ret (T := subset)) ∘ f) {{(w, a)}}).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
bindd (mapdReduce ret ∘ f) {{(w, a)}}
    rewrite bindd_ctxset_one.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
DecoratedMonad.shift subset
  (w, (mapdReduce ret ∘ f) (w, a))
    unfold compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
DecoratedMonad.shift subset
  (w, mapdReduce ret (f (w, a)))
    rewrite (DecoratedMonad.shift_spec subset
               (W := W) (op := op) (A := B)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
map (map_fst (fun m : W => w ● m))
  (mapdReduce ret (f (w, a)))
    compose near (f (w, a)) on right.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
(map (map_fst (fun m : W => w ● m)) ∘ mapdReduce ret)
  (f (w, a))
    rewrite mapdReduce_morphism.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
mapdReduce (map (map_fst (fun m : W => w ● m)) ∘ ret)
  (f (w, a))
    rewrite (natural (ϕ := @ret subset _)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
w: W
a: A
mapdReduce (ret ⦿ w) (f (w, a)) =
mapdReduce (ret ∘ map (map_fst (fun m : W => w ● m)))
  (f (w, a))
    reflexivity.
  Qed.

  Corollary toctxset_to_binddt: forall (A: Type),
      toctxset (F := U) = binddt (G := const (subset (W * A)))
                            (B := False) (ret (T := subset)).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset = binddt ret
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset = binddt ret
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
toctxset = binddt ret
    rewrite toctxset_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapdReduce ret = binddt ret
    rewrite mapdReduce_to_mapdt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapdt ret = binddt ret
    rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ ret) = binddt ret
    reflexivity.
  Qed.

  (** ** Properties of <<tosubset>>  *)
  (********************************************************************)
  Lemma tosubset_bindd: forall `(f: W * A -> T B),
      tosubset ∘ bindd f =
        mapdReduce (fun '(w, a) => mapReduce (ret (T := subset)) (f (w, a))).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
tosubset ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
tosubset ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
tosubset ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    rewrite tosubset_to_mapReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapReduce ret ∘ bindd f =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    rewrite mapReduce_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a))) =
mapdReduce (fun '(w, a) => mapReduce ret (f (w, a)))
    reflexivity.
  Qed.

  Corollary tosubset_to_binddt: forall (A: Type),
      tosubset = binddt (G := const (subset A))
                   (B := False) (ret (T := subset) ∘ extract).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, tosubset = binddt (ret ∘ extract)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, tosubset = binddt (ret ∘ extract)
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
tosubset = binddt (ret ∘ extract)
    rewrite tosubset_to_mapReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
mapReduce ret = binddt (ret ∘ extract)
    rewrite mapReduce_to_traverse1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
traverse ret = binddt (ret ∘ extract)
    rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
binddt (map ret ∘ ret ∘ extract) =
binddt (ret ∘ extract)
    reflexivity.
  Qed.

  (** ** Properties of <<∈d>> *)
  (********************************************************************)
  Section properties_element_ctx_of.

    Lemma element_ctx_of_ret: forall {A: Type} (w: W) (a1 a2: A),
        (w, a1) ∈d ret (T := T) a2 <-> w = Ƶ /\ a1 = a2.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (w : W) (a1 a2 : A),
(w, a1) ∈d ret a2 <-> w = Ƶ /\ a1 = a2
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (w : W) (a1 a2 : A),
(w, a1) ∈d ret a2 <-> w = Ƶ /\ a1 = a2
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
w: W
a1, a2: A
(w, a1) ∈d ret a2 <-> w = Ƶ /\ a1 = a2
      unfold element_ctx_of.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
w: W
a1, a2: A
toctxset (ret a2) (w, a1) <-> w = Ƶ /\ a1 = a2
      rewrite toctxset_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
w: W
a1, a2: A
{{(Ƶ, a2)}} (w, a1) <-> w = Ƶ /\ a1 = a2
      unfold subset_one.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
w: W
a1, a2: A
(Ƶ, a2) = (w, a1) <-> w = Ƶ /\ a1 = a2
      rewrite pair_equal_spec.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
w: W
a1, a2: A
Ƶ = w /\ a2 = a1 <-> w = Ƶ /\ a1 = a2
      easy.
    Qed.

    Corollary element_ctx_of_to_binddt: forall (A: Type) (t: U A) (w: W) (a: A),
        (w, a) ∈d t = binddt (G := const Prop)
                        (Pure_G := @Pure_const Prop Monoid_unit_false)
                        (Mult_G := @Mult_const Prop Monoid_op_or)
                        (B := False) (eq (w, a)) t.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (t : U A) (w : W) (a : A),
(w, a) ∈d t = binddt (eq (w, a)) t
    Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (t : U A) (w : W) (a : A),
(w, a) ∈d t = binddt (eq (w, a)) t
      intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
w: W
a: A
(w, a) ∈d t = binddt (eq (w, a)) t
      rewrite element_ctx_of_to_mapdReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
w: W
a: A
mapdReduce {{(w, a)}} t = binddt (eq (w, a)) t
      rewrite mapdReduce_to_mapdt1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
w: W
a: A
mapdt {{(w, a)}} t = binddt (eq (w, a)) t
      rewrite mapdt_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
w: W
a: A
binddt (map ret ∘ {{(w, a)}}) t = binddt (eq (w, a)) t
      reflexivity.
    Qed.

  End properties_element_ctx_of.
  (** ** Properties of <<∈>> *)
  (********************************************************************)
  Import TraversableMonad.DerivedInstances.

  Corollary element_of_to_binddt: forall (A: Type) (t: U A) (a: A),
      a ∈ t = binddt (G := const Prop)
                (Pure_G := @Pure_const Prop Monoid_unit_false)
                (Mult_G := @Mult_const Prop Monoid_op_or)
                (B := False) (eq a ∘ extract) t.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (t : U A) (a : A),
a ∈ t = binddt (eq a ∘ extract) t
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A : Type) (t : U A) (a : A),
a ∈ t = binddt (eq a ∘ extract) t
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
a: A
a ∈ t = binddt (eq a ∘ extract) t
    rewrite element_of_to_mapReduce.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
a: A
mapReduce {{a}} t = binddt (eq a ∘ extract) t
    rewrite mapReduce_to_traverse1.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
a: A
traverse {{a}} t = binddt (eq a ∘ extract) t
    rewrite traverse_to_binddt.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
t: U A
a: A
binddt (map ret ∘ {{a}} ∘ extract) t =
binddt (eq a ∘ extract) t
    reflexivity.
  Qed.

End composition.

(** * Monad Homomorphism Instances *)
(**********************************************************************)
Section instances.


  Context
    `{op: Monoid_op W}
    `{unit: Monoid_unit W}
    `{Monoid_inst: ! Monoid W}.

  Context
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd W T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt 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}.

  Context
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd W U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt 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}.

  Context
    `{ToSubset_T_inst: ToSubset T}
    `{ToSubset_U_inst: ToSubset U}
    `{! Compat_ToSubset_Traverse T}
    `{! Compat_ToSubset_Traverse U}.

  Context
    `{ToCtxset_T_inst: ToCtxset W T}
    `{ToCtxset_U_inst: ToCtxset W U}
    `{! Compat_ToCtxset_Mapdt W T}
    `{! Compat_ToCtxset_Mapdt W U}.

  Context
    `{ToCtxlist_T_inst: ToCtxlist W T}
    `{ToCtxlist_U_inst: ToCtxlist W U}
    `{! Compat_ToCtxlist_Mapdt W T}
    `{! Compat_ToCtxlist_Mapdt W U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad W T}
    `{Module_inst: ! DecoratedTraversableRightPreModule W T U}.

  #[export] Instance:
    DecoratedMonadHom W T (ctxset W) (@toctxset W T _).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedMonadHom W T (ctxset W)
  (@toctxset W T ToCtxset_T_inst)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedMonadHom W T (ctxset W)
  (@toctxset W T ToCtxset_T_inst)
    constructor.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset ∘ ret = ret
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
      rewrite toctxset_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
bindd (toctxset ∘ f) ∘ toctxset =
bindd (toctxset ∘ f) ∘ toctxset
      reflexivity.
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset ∘ ret = ret intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
toctxset ∘ ret = ret
      ext a [w a'].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
(toctxset ∘ ret) a (w, a') = ret a (w, a') unfold compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
toctxset (ret a) (w, a') = ret a (w, a')
      rewrite toctxset_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
{{(Ƶ, a)}} (w, a') = ret a (w, a')
      cbv.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
((unit, a) = (w, a')) = (a = a' /\ w = unit)
      apply propositional_extensionality.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
(unit, a) = (w, a') <-> a = a' /\ w = unit
      rewrite pair_equal_spec.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
unit = w /\ a = a' <-> a = a' /\ w = unit
      easy.
  Qed.

  #[export] Instance DTM_ctxset_DecoratedMonadHom:
    DecoratedMonadHom W T (ctxset W) (@toctxset W T _).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedMonadHom W T (ctxset W)
  (@toctxset W T ToCtxset_T_inst)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
DecoratedMonadHom W T (ctxset W)
  (@toctxset W T ToCtxset_T_inst)
    constructor.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset ∘ ret = ret
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
      rewrite toctxset_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
bindd (toctxset ∘ f) ∘ toctxset =
bindd (toctxset ∘ f) ∘ toctxset
      reflexivity.
    -
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall A : Type, toctxset ∘ ret = ret intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
toctxset ∘ ret = ret
      ext a [w a'].
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
(toctxset ∘ ret) a (w, a') = ret a (w, a') unfold compose.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
toctxset (ret a) (w, a') = ret a (w, a')
      rewrite toctxset_ret.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
{{(Ƶ, a)}} (w, a') = ret a (w, a')
      cbv.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
((unit, a) = (w, a')) = (a = a' /\ w = unit)
      apply propositional_extensionality.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
(unit, a) = (w, a') <-> a = a' /\ w = unit
      rewrite pair_equal_spec.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A: Type
a: A
w: W
a': A
unit = w /\ a = a' <-> a = a' /\ w = unit
      easy.
  Qed.

  #[export] Instance DTM_ctxset_DecoratedModuleHom:
    ParallelDecoratedRightModuleHom
      T (ctxset W) U (ctxset W)
      (@toctxset W T _) (@toctxset W U _).
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
ParallelDecoratedRightModuleHom T (ctxset W) U
  (ctxset W) (@toctxset W T ToCtxset_T_inst)
  (@toctxset W U ToCtxset_U_inst)
  Proof.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
ParallelDecoratedRightModuleHom T (ctxset W) U
  (ctxset W) (@toctxset W T ToCtxset_T_inst)
  (@toctxset W U ToCtxset_U_inst)
    constructor.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
forall (A B : Type) (f : W * A -> T B),
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
    intros.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
toctxset ∘ bindd f = bindd (toctxset ∘ f) ∘ toctxset
    rewrite toctxset_bindd.
W: Type
op: Monoid_op W
unit: Monoid_unit W
Monoid_inst: Monoid W
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd W T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt W T
Bindd_T_inst: Bindd W T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt W T T
Compat_Map_Binddt0: Compat_Map_Binddt W T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt W T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt W T T
Compat_Bind_Binddt0: Compat_Bind_Binddt W T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt W T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt W T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt W T T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd W U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt W U
Bindd_U_inst: Bindd W T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt W T U
Compat_Map_Binddt1: Compat_Map_Binddt W T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt W T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt W T U
Compat_Bind_Binddt1: Compat_Bind_Binddt W T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt W T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt W T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt W T U
ToSubset_T_inst: ToSubset T
ToSubset_U_inst: ToSubset U
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
Compat_ToSubset_Traverse1: Compat_ToSubset_Traverse U
ToCtxset_T_inst: ToCtxset W T
ToCtxset_U_inst: ToCtxset W U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt W T
Compat_ToCtxset_Mapdt1: Compat_ToCtxset_Mapdt W U
ToCtxlist_T_inst: ToCtxlist W T
ToCtxlist_U_inst: ToCtxlist W U
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt W T
Compat_ToCtxlist_Mapdt1: Compat_ToCtxlist_Mapdt W U
Monad_inst: DecoratedTraversableMonad W T
Module_inst: DecoratedTraversableRightPreModule W T U
A, B: Type
f: W * A -> T B
bindd (toctxset ∘ f) ∘ toctxset =
bindd (toctxset ∘ f) ∘ toctxset
    reflexivity.
  Qed.

End instances.