From Tealeaves Require Import
  Classes.Categorical.ContainerFunctor
  Classes.Multisorted.DecoratedTraversableMonad
  Functors.List
  Functors.Subset
  Functors.Constant.

Import ContainerFunctor.Notations.
Import Subset.Notations.
Import TypeFamily.Notations.
Import Monoid.Notations.
Import List.ListNotations.
#[local] Open Scope list_scope.

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

(** * Sorted lists with context *)
(**********************************************************************)
Section list.

  Context
    `{ix: Index}
    `{Monoid W}.

  Instance: MReturn (fun k A => list (W * (K * A))) :=
    fun A (k: K) (a: A) =>
      [(Ƶ, (k, a))].

  (** This operation is a context- and tag-sensitive substitution
   operation on lists of annotated values. It is used internally to
   reason about the interaction between <<mbinddt>> and
   <<tolistmd>>. *)
  Fixpoint mbinddt_list
           `(f: forall (k: K), W * A -> list (W * (K * B)))
           (l: list (W * (K * A))): list (W * (K * B)) :=
    match l with
    | nil => nil
    | cons (w, (k, a)) rest =>
      map (F := list) (incr w) (f k (w, a)) ++ mbinddt_list f rest
    end.

  Lemma mbinddt_list_nil: forall
      `(f: forall (k: K), W * A -> list (W * (K * B))),
      mbinddt_list f nil = nil.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))),
mbinddt_list f [] = []
  Proof.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))),
mbinddt_list f [] = []
    intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
mbinddt_list f [] = [] easy.
  Qed.

  Lemma mbinddt_list_ret: forall
      `(f: forall (k: K), W * A -> list (W * (K * B))) (k: K) (a: A),
      mbinddt_list f (mret (fun k A => list (W * (K * A))) k a) =
        f k (Ƶ, a).
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))) (k : K)
  (a : A),
mbinddt_list f
  (mret
     (fun (_ : K) (A0 : Type) => list (W * (K * A0)))
     k a) = f k (Ƶ, a)
  Proof.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))) (k : K)
  (a : A),
mbinddt_list f
  (mret
     (fun (_ : K) (A0 : Type) => list (W * (K * A0)))
     k a) = f k (Ƶ, a)
    intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
mbinddt_list f
  (mret (fun (_ : K) (A : Type) => list (W * (K * A)))
     k a) = f k (Ƶ, a) cbn.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
map (incr Ƶ) (f k (Ƶ, a)) ++ [] = f k (Ƶ, a) List.simpl_list.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
map (incr Ƶ) (f k (Ƶ, a)) = f k (Ƶ, a)
    replace (incr (Ƶ: W)) with (@id (W * (K * B))).
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
map id (f k (Ƶ, a)) = f k (Ƶ, a)
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
id = incr Ƶ
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
map id (f k (Ƶ, a)) = f k (Ƶ, a) now rewrite (fun_map_id).
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
id = incr Ƶ ext [w [k2 b]].
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
w: W
k2: K
b: B
id (w, (k2, b)) = incr Ƶ (w, (k2, b)) cbn.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
k: K
a: A
w: W
k2: K
b: B
id (w, (k2, b)) = (Ƶ ● w, (k2, b)) now simpl_monoid.
  Qed.

  Lemma mbinddt_list_cons: forall
      `(f: forall (k: K), W * A -> list (W * (K * B)))
      (w: W) (k: K) (a: A)
      (l: list (W * (K * A))),
      mbinddt_list f ((w, (k, a)) :: l) =
        map (F := list) (incr w) (f k (w, a)) ++ mbinddt_list f l.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))) (w : W)
  (k : K) (a : A) (l : list (W * (K * A))),
mbinddt_list f ((w, (k, a)) :: l) =
map (incr w) (f k (w, a)) ++ mbinddt_list f l
  Proof.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))) (w : W)
  (k : K) (a : A) (l : list (W * (K * A))),
mbinddt_list f ((w, (k, a)) :: l) =
map (incr w) (f k (w, a)) ++ mbinddt_list f l
    intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
w: W
k: K
a: A
l: list (W * (K * A))
mbinddt_list f ((w, (k, a)) :: l) =
map (incr w) (f k (w, a)) ++ mbinddt_list f l easy.
  Qed.

  Lemma mbinddt_list_app: forall
      `(f: forall (k: K), W * A -> list (W * (K * B)))
      (l1 l2: list (W * (K * A))),
      mbinddt_list f (l1 ++ l2) =
        mbinddt_list f l1 ++ mbinddt_list f l2.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B)))
  (l1 l2 : list (W * (K * A))),
mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
  Proof.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B)))
  (l1 l2 : list (W * (K * A))),
mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
    intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
l1, l2: list (W * (K * A))
mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2 induction l1.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
l2: list (W * (K * A))
mbinddt_list f ([] ++ l2) =
mbinddt_list f [] ++ mbinddt_list f l2
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
a: W * (K * A)
l1, l2: list (W * (K * A))
IHl1: mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
mbinddt_list f ((a :: l1) ++ l2) =
mbinddt_list f (a :: l1) ++ mbinddt_list f l2
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
l2: list (W * (K * A))
mbinddt_list f ([] ++ l2) =
mbinddt_list f [] ++ mbinddt_list f l2 easy.
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
a: W * (K * A)
l1, l2: list (W * (K * A))
IHl1: mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
mbinddt_list f ((a :: l1) ++ l2) =
mbinddt_list f (a :: l1) ++ mbinddt_list f l2 destruct a as [w [k a]].
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
w: W
k: K
a: A
l1, l2: list (W * (K * A))
IHl1: mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
mbinddt_list f (((w, (k, a)) :: l1) ++ l2) =
mbinddt_list f ((w, (k, a)) :: l1) ++
mbinddt_list f l2
      cbn.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
w: W
k: K
a: A
l1, l2: list (W * (K * A))
IHl1: mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
map (incr w) (f k (w, a)) ++ mbinddt_list f (l1 ++ l2) =
(map (incr w) (f k (w, a)) ++ mbinddt_list f l1) ++
mbinddt_list f l2 rewrite IHl1.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
w: W
k: K
a: A
l1, l2: list (W * (K * A))
IHl1: mbinddt_list f (l1 ++ l2) =
mbinddt_list f l1 ++ mbinddt_list f l2
map (incr w) (f k (w, a)) ++
mbinddt_list f l1 ++ mbinddt_list f l2 =
(map (incr w) (f k (w, a)) ++ mbinddt_list f l1) ++
mbinddt_list f l2
      now rewrite List.app_assoc.
  Qed.

  #[global] Instance: forall `(f: forall (k: K), W * A -> list (W * (K * B))),
      ApplicativeMorphism (const (list (W * (K * A))))
                          (const (list (W * (K * B))))
                          (const (mbinddt_list f)).
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))),
ApplicativeMorphism (const (list (W * (K * A))))
  (const (list (W * (K * B))))
  (const (mbinddt_list f))
  Proof.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
forall (A B : Type)
  (f : K -> W * A -> list (W * (K * B))),
ApplicativeMorphism (const (list (W * (K * A))))
  (const (list (W * (K * B))))
  (const (mbinddt_list f))
    intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
ApplicativeMorphism (const (list (W * (K * A))))
  (const (list (W * (K * B))))
  (const (mbinddt_list f)) eapply ApplicativeMorphism_monoid_morphism.
    Unshelve.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
Monoid_Morphism (list (W * (K * A)))
  (list (W * (K * B))) (mbinddt_list f) constructor; try typeclasses eauto.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
mbinddt_list f Ƶ = Ƶ
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
forall a1 a2 : list (W * (K * A)),
mbinddt_list f (a1 ● a2) =
mbinddt_list f a1 ● mbinddt_list f a2
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
mbinddt_list f Ƶ = Ƶ easy.
    -
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
forall a1 a2 : list (W * (K * A)),
mbinddt_list f (a1 ● a2) =
mbinddt_list f a1 ● mbinddt_list f a2 intros.
ix: Index
W: Type
op: Monoid_op W
unit0: Monoid_unit W
H: Monoid W
A, B: Type
f: K -> W * A -> list (W * (K * B))
a1, a2: list (W * (K * A))
mbinddt_list f (a1 ● a2) =
mbinddt_list f a1 ● mbinddt_list f a2 apply mbinddt_list_app.
  Qed.

End list.


(** * Shape and Contents *)
(**********************************************************************)

(** ** Operations *)
(**********************************************************************)
Section shape_and_contents.

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

  Definition shape {A}: U A -> U unit :=
    mmap U (allK (const tt)).

  Definition tolistmd_gen_loc {A}:
    K -> W * A -> list (W * (K * A)) :=
    fun k '(w, a) => [(w, (k, a))].

  Definition tolistmd_gen {A} (fake: Type):
    U A -> list (W * (K * A)) :=
    mmapdt (B := fake) U (const (list (W * (K * A))))
           tolistmd_gen_loc.

  Definition tolistmd {A}:
    U A -> list (W * (K * A)) :=
    tolistmd_gen False.

  Definition tosetmd {A}:
    U A -> W * (K * A) -> Prop :=
    tosubset (F := list) ∘ tolistmd.

  Definition tolistm {A}:
    U A -> list (K * A) :=
    map (F := list) extract ∘ tolistmd.

  Definition tosetm {A}: U A -> K * A -> Prop :=
    tosubset (F := list) ∘ tolistm.

  Fixpoint filterk {A} (k: K) (l: list (W * (K * A))): list (W * A) :=
    match l with
    | nil => nil
    | cons (w, (j, a)) ts =>
      if k == j then (w, a) :: filterk k ts else filterk k ts
    end.

  Definition toklistd {A} (k: K): U A -> list (W * A) :=
    filterk k ∘ tolistmd.

  Definition toksetd {A} (k: K): U A -> W * A -> Prop :=
    tosubset (F := list) ∘ toklistd k.

  Definition toklist {A} (k: K): U A -> list A :=
    map (F := list) extract ∘ @toklistd A k.

End shape_and_contents.

(** ** Notations *)
(**********************************************************************)
Module Notations.

  Notation "x ∈md t" :=
    (tosetmd _ t x) (at level 50): tealeaves_multi_scope.

  Notation "x ∈m t" :=
    (tosetm _ t x) (at level 50): tealeaves_multi_scope.

End Notations.

Import Notations.

(** ** Rewriting Rules for <<filterk>> *)
(**********************************************************************)
Section rw_filterk.

  Context
    `{ix: Index} {W A: Type} (k: K).

  Implicit Types (l: list (W * (K * A))) (w: W) (a: A).

  Lemma filterk_nil: filterk k (nil: list (W * (K * A))) = nil.
ix: Index
W, A: Type
k: K
filterk k ([] : list (W * (K * A))) = []
  Proof.
ix: Index
W, A: Type
k: K
filterk k ([] : list (W * (K * A))) = []
    reflexivity.
  Qed.

  Lemma filterk_cons_eq:
    forall l w a, filterk k (cons (w, (k, a)) l) = (w, a) :: filterk k l.
ix: Index
W, A: Type
k: K
forall l w a,
filterk k ((w, (k, a)) :: l) = (w, a) :: filterk k l
  Proof.
ix: Index
W, A: Type
k: K
forall l w a,
filterk k ((w, (k, a)) :: l) = (w, a) :: filterk k l
    intros.
ix: Index
W, A: Type
k: K
l: list (W * (K * A))
w: W
a: A
filterk k ((w, (k, a)) :: l) = (w, a) :: filterk k l cbn.
ix: Index
W, A: Type
k: K
l: list (W * (K * A))
w: W
a: A
(if k == k then (w, a) :: filterk k l else filterk k l) =
(w, a) :: filterk k l compare values k and k.
  Qed.

  Lemma filterk_cons_neq:
    forall l w a j, j <> k -> filterk k (cons (w, (j, a)) l) = filterk k l.
ix: Index
W, A: Type
k: K
forall l w a (j : K),
j <> k -> filterk k ((w, (j, a)) :: l) = filterk k l
  Proof.
ix: Index
W, A: Type
k: K
forall l w a (j : K),
j <> k -> filterk k ((w, (j, a)) :: l) = filterk k l
    intros.
ix: Index
W, A: Type
k: K
l: list (W * (K * A))
w: W
a: A
j: K
H: j <> k
filterk k ((w, (j, a)) :: l) = filterk k l cbn.
ix: Index
W, A: Type
k: K
l: list (W * (K * A))
w: W
a: A
j: K
H: j <> k
(if k == j then (w, a) :: filterk k l else filterk k l) =
filterk k l compare values k and j.
  Qed.

  Lemma filterk_app:
    forall l1 l2, filterk k (l1 ++ l2) = filterk k l1 ++ filterk k l2.
ix: Index
W, A: Type
k: K
forall l1 l2,
filterk k (l1 ++ l2) = filterk k l1 ++ filterk k l2
  Proof.
ix: Index
W, A: Type
k: K
forall l1 l2,
filterk k (l1 ++ l2) = filterk k l1 ++ filterk k l2
    intros.
ix: Index
W, A: Type
k: K
l1, l2: list (W * (K * A))
filterk k (l1 ++ l2) = filterk k l1 ++ filterk k l2 induction l1.
ix: Index
W, A: Type
k: K
l2: list (W * (K * A))
filterk k ([] ++ l2) = filterk k [] ++ filterk k l2
ix: Index
W, A: Type
k: K
a: W * (K * A)
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
filterk k ((a :: l1) ++ l2) =
filterk k (a :: l1) ++ filterk k l2
    -
ix: Index
W, A: Type
k: K
l2: list (W * (K * A))
filterk k ([] ++ l2) = filterk k [] ++ filterk k l2 reflexivity.
    -
ix: Index
W, A: Type
k: K
a: W * (K * A)
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
filterk k ((a :: l1) ++ l2) =
filterk k (a :: l1) ++ filterk k l2 destruct a as [w [i a]].
ix: Index
W, A: Type
k: K
w: W
i: K
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
filterk k (((w, (i, a)) :: l1) ++ l2) =
filterk k ((w, (i, a)) :: l1) ++ filterk k l2
      compare values i and k.
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
filterk k (((w, (k, a)) :: l1) ++ l2) =
filterk k ((w, (k, a)) :: l1) ++ filterk k l2
ix: Index
W, A: Type
k: K
w: W
i: K
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_NEQ: i <> k
DESTR_NEQs: k <> i
filterk k (((w, (i, a)) :: l1) ++ l2) =
filterk k ((w, (i, a)) :: l1) ++ filterk k l2
      +
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
filterk k (((w, (k, a)) :: l1) ++ l2) =
filterk k ((w, (k, a)) :: l1) ++ filterk k l2 rewrite <- (List.app_comm_cons l1).
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
filterk k ((w, (k, a)) :: l1 ++ l2) =
filterk k ((w, (k, a)) :: l1) ++ filterk k l2
        rewrite filterk_cons_eq.
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
(w, a) :: filterk k (l1 ++ l2) =
filterk k ((w, (k, a)) :: l1) ++ filterk k l2
        rewrite filterk_cons_eq.
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
(w, a) :: filterk k (l1 ++ l2) =
((w, a) :: filterk k l1) ++ filterk k l2
        rewrite <- (List.app_comm_cons (filterk k l1)).
ix: Index
W, A: Type
k: K
w: W
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_EQs: k = k
(w, a) :: filterk k (l1 ++ l2) =
(w, a) :: filterk k l1 ++ filterk k l2
        now rewrite <- IHl1.
      +
ix: Index
W, A: Type
k: K
w: W
i: K
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_NEQ: i <> k
DESTR_NEQs: k <> i
filterk k (((w, (i, a)) :: l1) ++ l2) =
filterk k ((w, (i, a)) :: l1) ++ filterk k l2 rewrite <- (List.app_comm_cons l1).
ix: Index
W, A: Type
k: K
w: W
i: K
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_NEQ: i <> k
DESTR_NEQs: k <> i
filterk k ((w, (i, a)) :: l1 ++ l2) =
filterk k ((w, (i, a)) :: l1) ++ filterk k l2
        rewrite filterk_cons_neq; auto.
ix: Index
W, A: Type
k: K
w: W
i: K
a: A
l1, l2: list (W * (K * A))
IHl1: filterk k (l1 ++ l2) =
filterk k l1 ++ filterk k l2
DESTR_NEQ: i <> k
DESTR_NEQs: k <> i
filterk k (l1 ++ l2) =
filterk k ((w, (i, a)) :: l1) ++ filterk k l2
        rewrite filterk_cons_neq; auto.
  Qed.

End rw_filterk.

#[export] Hint Rewrite @filterk_nil @filterk_cons_eq
  @filterk_cons_neq @filterk_app: tea_list.

From Tealeaves Require Import
  Functors.List
  Functors.Constant.

Import Product.Notations.
Import Monoid.Notations.
Import List.ListNotations.

Open Scope list_scope.

(** ** Auxiliary Lemmas for constant Applicative Functors*)
(**********************************************************************)
Section lemmas.

  #[local] Generalizable Variable M.

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

  Lemma mbinddt_constant_applicative1
        `{Monoid M} {B: Type}
        `(f: forall (k: K), W * A -> const M (T k B)):
    mbinddt (B := B) U (const M) f =
    mbinddt (B := False) U (const M) (f: forall (k: K), W * A -> const M (T k False)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
B, A: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f =
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k False))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
B, A: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f =
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k False))
    change_right
      (map (F := const M) (B := U B) (mmap U (const exfalso))
         ∘ (mbinddt (B := False) U (const M)
              (f: forall (k: K), W * A -> const M (T k False)))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
B, A: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f =
map (mmap U (const exfalso))
∘ mbinddt U (const M)
    (f : forall k : K, W * A -> const M (T k False))
    rewrite (mmap_mbinddt U (F := const M)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
B, A: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f =
mbinddt U (const M)
  (fun k : K => map (mmap (T k) (const exfalso)) ∘ f k)
    reflexivity.
  Qed.

  Lemma mbinddt_constant_applicative2 (fake1 fake2: Type) `{Monoid M}
        `(f: forall (k: K), W * A -> const M (T k B)):
    mbinddt (B := fake1) U (const M)
            (f: forall (k: K), W * A -> const M (T k fake1))
    = mbinddt (B := fake2) U (const M)
              (f: forall (k: K), W * A -> const M (T k fake2)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
A, B: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake1)) =
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake2))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
A, B: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake1)) =
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake2))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
A, B: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake1)) =
mbinddt U (const M)
  (f : forall k : K, W * A -> const M (T k fake2))
    rewrite (mbinddt_constant_applicative1 (B := fake1)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
A, B: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f = mbinddt U (const M) f
    rewrite (mbinddt_constant_applicative1 (B := fake2)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H1: Monoid M
A, B: Type
f: forall k : K, W * A -> const M (T k B)
mbinddt U (const M) f = mbinddt U (const M) f easy.
  Qed.

  Lemma tolistmd_equiv1: forall (fake: Type) (A: Type),
      tolistmd_gen U (A := A) False = tolistmd_gen U fake.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall fake A : Type,
tolistmd_gen U False = tolistmd_gen U fake
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall fake A : Type,
tolistmd_gen U False = tolistmd_gen U fake
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A: Type
tolistmd_gen U False = tolistmd_gen U fake unfold tolistmd_gen at 2, mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A: Type
tolistmd_gen U False =
mbinddt U (const (list (W * (K * A))))
  (mapMret ◻ tolistmd_gen_loc)
    now rewrite (mbinddt_constant_applicative2 fake False).
  Qed.

  Lemma tolistmd_equiv: forall (fake1 fake2: Type) (A: Type),
      tolistmd_gen U (A := A) fake1 = tolistmd_gen U fake2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall fake1 fake2 A : Type,
tolistmd_gen U fake1 = tolistmd_gen U fake2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall fake1 fake2 A : Type,
tolistmd_gen U fake1 = tolistmd_gen U fake2
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, A: Type
tolistmd_gen U fake1 = tolistmd_gen U fake2 rewrite <- tolistmd_equiv1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, A: Type
tolistmd_gen U False = tolistmd_gen U fake2
    rewrite <- (tolistmd_equiv1 fake2).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake1, fake2, A: Type
tolistmd_gen U False = tolistmd_gen U False
    easy.
  Qed.

End lemmas.

(** ** Relating <<∈m>> and <<∈md>> *)
(**********************************************************************)
Section DTM_membership_lemmas.

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

  Lemma inmd_iff_in: forall (k: K) (A: Type) (a: A) (t: U A),
      (k, a) ∈m t <-> exists w, (w, (k, a)) ∈md t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (k : K) (A : Type) (a : A) (t : U A),
(k, a) ∈m t <-> (exists w : W, (w, (k, a)) ∈md t)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (k : K) (A : Type) (a : A) (t : U A),
(k, a) ∈m t <-> (exists w : W, (w, (k, a)) ∈md t)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
(k, a) ∈m t <-> (exists w : W, (w, (k, a)) ∈md t) unfold tosetm, tosetmd, tolistm, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
tosubset (map extract (tolistmd U t)) (k, a) <->
(exists w : W, tosubset (tolistmd U t) (w, (k, a)))
    induction (tolistmd U t).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
tosubset (map extract []) (k, a) <->
(exists w : W, tosubset [] (w, (k, a)))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
a0: W * (K * A)
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
tosubset (map extract (a0 :: l)) (k, a) <->
(exists w : W, tosubset (a0 :: l) (w, (k, a)))
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
tosubset (map extract []) (k, a) <->
(exists w : W, tosubset [] (w, (k, a))) cbv; split; intros []; easy.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
a0: W * (K * A)
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
tosubset (map extract (a0 :: l)) (k, a) <->
(exists w : W, tosubset (a0 :: l) (w, (k, a))) destruct a0 as [w' [k' a']].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
tosubset (map extract ((w', (k', a')) :: l)) (k, a) <->
(exists w : W,
   tosubset ((w', (k', a')) :: l) (w, (k, a)))
      rewrite map_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
tosubset (extract (w', (k', a')) :: map extract l)
  (k, a) <->
(exists w : W,
   tosubset ((w', (k', a')) :: l) (w, (k, a)))
      rewrite tosubset_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
({{extract (w', (k', a'))}} ∪ tosubset (map extract l))
  (k, a) <->
(exists w : W,
   tosubset ((w', (k', a')) :: l) (w, (k, a)))
      rewrite tosubset_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
({{extract (w', (k', a'))}} ∪ tosubset (map extract l))
  (k, a) <->
(exists w : W,
   ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)))
      rewrite subset_in_add.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
{{extract (w', (k', a'))}} (k, a) \/
tosubset (map extract l) (k, a) <->
(exists w : W,
   ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)))
      rewrite IHl.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
{{extract (w', (k', a'))}} (k, a) \/
(exists w : W, tosubset l (w, (k, a))) <->
(exists w : W,
   ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)))
      split; [ intros [Hfst|[w Hrest]] | intros [w [rest1|rest2]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
Hfst: {{extract (w', (k', a'))}} (k, a)
exists w : W,
  ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
Hrest: tosubset l (w, (k, a))
exists w : W,
  ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest1: {{(w', (k', a'))}} (w, (k, a))
{{extract (w', (k', a'))}} (k, a) \/
(exists w : W, tosubset l (w, (k, a)))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
{{extract (w', (k', a'))}} (k, a) \/
(exists w : W, tosubset l (w, (k, a)))
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
Hfst: {{extract (w', (k', a'))}} (k, a)
exists w : W,
  ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)) exists w'.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
Hfst: {{extract (w', (k', a'))}} (k, a)
({{(w', (k', a'))}} ∪ tosubset l) (w', (k, a)) left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
Hfst: {{extract (w', (k', a'))}} (k, a)
{{(w', (k', a'))}} (w', (k, a))
        rewrite Hfst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
Hfst: {{extract (w', (k', a'))}} (k, a)
{{(w', (k, a))}} (w', (k, a)) easy.
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
Hrest: tosubset l (w, (k, a))
exists w : W,
  ({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)) exists w.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
Hrest: tosubset l (w, (k, a))
({{(w', (k', a'))}} ∪ tosubset l) (w, (k, a)) now right.
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest1: {{(w', (k', a'))}} (w, (k, a))
{{extract (w', (k', a'))}} (k, a) \/
(exists w : W, tosubset l (w, (k, a))) left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest1: {{(w', (k', a'))}} (w, (k, a))
{{extract (w', (k', a'))}} (k, a)
        cbv in rest1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest1: (w', (k', a')) = (w, (k, a))
{{extract (w', (k', a'))}} (k, a)
        now inversion rest1.
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
{{extract (w', (k', a'))}} (k, a) \/
(exists w : W, tosubset l (w, (k, a))) right.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
exists w : W, tosubset l (w, (k, a)) rewrite <- IHl.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
tosubset (map extract l) (k, a)
        compose near l.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
(tosubset ∘ map extract) l (k, a)
        rewrite tosubset_list_map.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
(map extract ∘ tosubset) l (k, a)
        unfold compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
map extract (tosubset l) (k, a)
        exists (w, (k, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
t: U A
w': W
k': K
a': A
l: list (W * (K * A))
IHl: tosubset (map extract l) (k, a) <->
(exists w : W, tosubset l (w, (k, a)))
w: W
rest2: tosubset l (w, (k, a))
tosubset l (w, (k, a)) /\ extract (w, (k, a)) = (k, a)
        easy.
  Qed.

  Corollary inmd_implies_in:
    forall (k: K) (A: Type) (a: A) (w: W) (t: U A),
      (w, (k, a)) ∈md t -> (k, a) ∈m t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (k : K) (A : Type) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t -> (k, a) ∈m t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (k : K) (A : Type) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t -> (k, a) ∈m t
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
w: W
t: U A
H1: (w, (k, a)) ∈md t
(k, a) ∈m t rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
k: K
A: Type
a: A
w: W
t: U A
H1: (w, (k, a)) ∈md t
exists w : W, (w, (k, a)) ∈md t eauto.
  Qed.

End DTM_membership_lemmas.

(** ** Characterizing Membership in List Operations *)
(**********************************************************************)
Section DTM_tolist.

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

  Lemma in_filterk_iff:
    forall (A: Type) (l: list (W * (K * A))) (k: K) (a: A) (w: W),
      (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (l : list (W * (K * A))) (k : K)
  (a : A) (w : W),
(w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (l : list (W * (K * A))) (k : K)
  (a : A) (w : W),
(w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
l: list (W * (K * A))
k: K
a: A
w: W
(w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l induction l.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
(w, a) ∈ filterk k [] <-> (w, (k, a)) ∈ []
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a0: W * (K * A)
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
(w, a) ∈ filterk k (a0 :: l) <->
(w, (k, a)) ∈ (a0 :: l)
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
(w, a) ∈ filterk k [] <-> (w, (k, a)) ∈ [] cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
False <-> False easy.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a0: W * (K * A)
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
(w, a) ∈ filterk k (a0 :: l) <->
(w, (k, a)) ∈ (a0 :: l) destruct a0 as [w' [j a']].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
(w, a) ∈ filterk k ((w', (j, a')) :: l) <->
(w, (k, a)) ∈ ((w', (j, a')) :: l) cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
List.In (w, a)
  (if k == j
   then (w', a') :: filterk k l
   else filterk k l) <->
(w', (j, a')) = (w, (k, a)) \/ List.In (w, (k, a)) l compare values k and j.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
IHl: (w, a) ∈ filterk j l <-> (w, (j, a)) ∈ l
DESTR_EQs: j = j
List.In (w, a) ((w', a') :: filterk j l) <->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
List.In (w, a) (filterk k l) <->
(w', (j, a')) = (w, (k, a)) \/ List.In (w, (k, a)) l
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
IHl: (w, a) ∈ filterk j l <-> (w, (j, a)) ∈ l
DESTR_EQs: j = j
List.In (w, a) ((w', a') :: filterk j l) <->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
IHl: (w, a) ∈ filterk j l <-> (w, (j, a)) ∈ l
DESTR_EQs: j = j
(w', a') = (w, a) \/ List.In (w, a) (filterk j l) <->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l rewrite IHl.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
IHl: (w, a) ∈ filterk j l <-> (w, (j, a)) ∈ l
DESTR_EQs: j = j
(w', a') = (w, a) \/ (w, (j, a)) ∈ l <->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l clear.
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', a') = (w, a) \/ (w, (j, a)) ∈ l <->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l split.
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', a') = (w, a) \/ (w, (j, a)) ∈ l ->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l ->
(w', a') = (w, a) \/ (w, (j, a)) ∈ l
        {
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', a') = (w, a) \/ (w, (j, a)) ∈ l ->
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l intros [hyp1 | hyp2].
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp1: (w', a') = (w, a)
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp2: (w, (j, a)) ∈ l
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l
          -
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp1: (w', a') = (w, a)
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l inverts hyp1.
ix: Index
W, A: Type
j: K
l: list (W * (K * A))
a: A
w: W
(w, (j, a)) = (w, (j, a)) \/ List.In (w, (j, a)) l now left.
          -
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp2: (w, (j, a)) ∈ l
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l now right.
        }
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l ->
(w', a') = (w, a) \/ (w, (j, a)) ∈ l
        {
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
(w', (j, a')) = (w, (j, a)) \/ List.In (w, (j, a)) l ->
(w', a') = (w, a) \/ (w, (j, a)) ∈ l intros [hyp1 | hyp2].
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp1: (w', (j, a')) = (w, (j, a))
(w', a') = (w, a) \/ (w, (j, a)) ∈ l
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp2: List.In (w, (j, a)) l
(w', a') = (w, a) \/ (w, (j, a)) ∈ l
          -
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp1: (w', (j, a')) = (w, (j, a))
(w', a') = (w, a) \/ (w, (j, a)) ∈ l inverts hyp1.
ix: Index
W, A: Type
j: K
l: list (W * (K * A))
a: A
w: W
(w, a) = (w, a) \/ (w, (j, a)) ∈ l now left.
          -
ix: Index
W, A: Type
w': W
j: K
a': A
l: list (W * (K * A))
a: A
w: W
hyp2: List.In (w, (j, a)) l
(w', a') = (w, a) \/ (w, (j, a)) ∈ l now right. }
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
List.In (w, a) (filterk k l) <->
(w', (j, a')) = (w, (k, a)) \/ List.In (w, (k, a)) l rewrite <- IHl.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
List.In (w, a) (filterk k l) <->
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
List.In (w, a) (filterk k l) ->
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l ->
List.In (w, a) (filterk k l)
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
List.In (w, a) (filterk k l) ->
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l intro hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
hyp: List.In (w, a) (filterk k l)
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l now right. }
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l ->
List.In (w, a) (filterk k l)
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w', (j, a')) = (w, (k, a)) \/ (w, a) ∈ filterk k l ->
List.In (w, a) (filterk k l) intros [hyp1 | hyp2].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
hyp1: (w', (j, a')) = (w, (k, a))
List.In (w, a) (filterk k l)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
hyp2: (w, a) ∈ filterk k l
List.In (w, a) (filterk k l)
          -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
hyp1: (w', (j, a')) = (w, (k, a))
List.In (w, a) (filterk k l) inverts hyp1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQs, DESTR_NEQ: k <> k
List.In (w, a) (filterk k l) contradiction.
          -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
w': W
j: K
a': A
l: list (W * (K * A))
k: K
a: A
w: W
IHl: (w, a) ∈ filterk k l <-> (w, (k, a)) ∈ l
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
hyp2: (w, a) ∈ filterk k l
List.In (w, a) (filterk k l) auto. }
  Qed.

  Lemma inmd_iff_in_tolistmd:
    forall (A: Type) (k: K) (a: A) (w: W) (t: U A),
      (w, (k, a)) ∈md t <-> (w, (k, a)) ∈ tolistmd U t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t <-> (w, (k, a)) ∈ tolistmd U t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t <-> (w, (k, a)) ∈ tolistmd U t
    reflexivity.
  Qed.

  Lemma in_iff_in_tolistmd:
    forall (A: Type) (k: K) (a: A) (t: U A),
      (k, a) ∈m t <-> (k, a) ∈ tolistm U t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (t : U A),
(k, a) ∈m t <-> (k, a) ∈ tolistm U t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (t : U A),
(k, a) ∈m t <-> (k, a) ∈ tolistm U t
    reflexivity.
  Qed.

  Lemma inmd_iff_in_toklistd:
    forall (A: Type) (k: K) (a: A) (w: W) (t: U A),
      (w, (k, a)) ∈md t <-> (w, a) ∈ toklistd U k t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t <-> (w, a) ∈ toklistd U k t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (w : W) (t : U A),
(w, (k, a)) ∈md t <-> (w, a) ∈ toklistd U k t
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
t: U A
(w, (k, a)) ∈md t <-> (w, a) ∈ toklistd U k t unfold toklistd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
t: U A
(w, (k, a)) ∈md t <->
(w, a) ∈ (filterk k ∘ tolistmd U) t unfold compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
t: U A
(w, (k, a)) ∈md t <->
(w, a) ∈ filterk k (tolistmd U t)
    rewrite in_filterk_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
w: W
t: U A
(w, (k, a)) ∈md t <-> (w, (k, a)) ∈ tolistmd U t reflexivity.
  Qed.

  Lemma in_iff_in_toklist:
    forall (A: Type) (k: K) (a: A) (t: U A),
      (k, a) ∈m t <-> a ∈ toklist U k t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (t : U A),
(k, a) ∈m t <-> a ∈ toklist U k t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (k : K) (a : A) (t : U A),
(k, a) ∈m t <-> a ∈ toklist U k t
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t <-> a ∈ toklist U k t unfold toklist.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t <-> a ∈ (map extract ∘ toklistd U k) t unfold compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t <-> a ∈ map extract (toklistd U k t)
    rewrite (in_map_iff list).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t <->
(exists a0 : W * A,
   a0 ∈ toklistd U k t /\ extract a0 = a) split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t ->
exists a0 : W * A,
  a0 ∈ toklistd U k t /\ extract a0 = a
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(exists a0 : W * A,
   a0 ∈ toklistd U k t /\ extract a0 = a) ->
(k, a) ∈m t
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(k, a) ∈m t ->
exists a0 : W * A,
  a0 ∈ toklistd U k t /\ extract a0 = a intro hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
hyp: (k, a) ∈m t
exists a0 : W * A,
  a0 ∈ toklistd U k t /\ extract a0 = a rewrite inmd_iff_in in hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
hyp: exists w : W, (w, (k, a)) ∈md t
exists a0 : W * A,
  a0 ∈ toklistd U k t /\ extract a0 = a
      destruct hyp as [w' hyp].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
hyp: (w', (k, a)) ∈md t
exists a0 : W * A,
  a0 ∈ toklistd U k t /\ extract a0 = a
      exists (w', a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
hyp: (w', (k, a)) ∈md t
(w', a) ∈ toklistd U k t /\ extract (w', a) = a rewrite inmd_iff_in_toklistd in hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
hyp: (w', a) ∈ toklistd U k t
(w', a) ∈ toklistd U k t /\ extract (w', a) = a
      auto.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
(exists a0 : W * A,
   a0 ∈ toklistd U k t /\ extract a0 = a) ->
(k, a) ∈m t intros [[w' a'] [hyp1 hyp2]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
a': A
hyp1: (w', a') ∈ toklistd U k t
hyp2: extract (w', a') = a
(k, a) ∈m t rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
a': A
hyp1: (w', a') ∈ toklistd U k t
hyp2: extract (w', a') = a
exists w : W, (w, (k, a)) ∈md t
      exists w'.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
a': A
hyp1: (w', a') ∈ toklistd U k t
hyp2: extract (w', a') = a
(w', (k, a)) ∈md t rewrite <- inmd_iff_in_toklistd in hyp1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
a': A
hyp1: (w', (k, a')) ∈md t
hyp2: extract (w', a') = a
(w', (k, a)) ∈md t cbn in hyp2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
a: A
t: U A
w': W
a': A
hyp1: (w', (k, a')) ∈md t
hyp2: a' = a
(w', (k, a)) ∈md t
      now subst.
  Qed.

End DTM_tolist.

(** ** Interaction between <<tolistmd>> and <<mret>>/<<mbindd>> *)
(**********************************************************************)
Section DTM_tolist.

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

  Lemma tolistmd_gen_mret: forall (A B: Type) (a: A) (k: K),
      tolistmd_gen (T k) B (mret T k a) = [ (Ƶ, (k, a)) ].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (a : A) (k : K),
tolistmd_gen (T k) B (mret T k a) = [(Ƶ, (k, a))]
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (a : A) (k : K),
tolistmd_gen (T k) B (mret T k a) = [(Ƶ, (k, a))]
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
a: A
k: K
tolistmd_gen (T k) B (mret T k a) = [(Ƶ, (k, a))] unfold tolistmd_gen.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
a: A
k: K
mmapdt (T k) (const (list (W * (K * A))))
  tolistmd_gen_loc (mret T k a) = [(Ƶ, (k, a))]
    compose near a on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
a: A
k: K
(mmapdt (T k) (const (list (W * (K * A))))
   tolistmd_gen_loc ∘ mret T k) a = [(Ƶ, (k, a))]
    now rewrite mmapdt_comp_mret.
  Qed.

  Corollary tolistmd_mret: forall (A: Type) (a: A) (k: K),
      tolistmd (T k) (mret T k a) = [ (Ƶ, (k, a)) ].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tolistmd (T k) (mret T k a) = [(Ƶ, (k, a))]
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tolistmd (T k) (mret T k a) = [(Ƶ, (k, a))]
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tolistmd (T k) (mret T k a) = [(Ƶ, (k, a))] unfold tolistmd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tolistmd_gen (T k) False (mret T k a) = [(Ƶ, (k, a))] apply tolistmd_gen_mret.
  Qed.

  Corollary tosetmd_mret: forall (A: Type) (a: A) (k: K),
      tosetmd (T k) (mret T k a) = {{ (Ƶ, (k, a)) }}.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tosetmd (T k) (mret T k a) = {{(Ƶ, (k, a))}}
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tosetmd (T k) (mret T k a) = {{(Ƶ, (k, a))}}
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosetmd (T k) (mret T k a) = {{(Ƶ, (k, a))}} unfold tosetmd, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosubset (tolistmd (T k) (mret T k a)) =
{{(Ƶ, (k, a))}} rewrite tolistmd_mret.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosubset [(Ƶ, (k, a))] = {{(Ƶ, (k, a))}}
    rewrite tosubset_list_one.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
{{(Ƶ, (k, a))}} = {{(Ƶ, (k, a))}}
    reflexivity.
  Qed.

  Corollary tolistm_mret: forall (A: Type) (a: A) (k: K),
      tolistm (T k) (mret T k a) = [ (k, a) ].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tolistm (T k) (mret T k a) = [(k, a)]
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tolistm (T k) (mret T k a) = [(k, a)]
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tolistm (T k) (mret T k a) = [(k, a)] unfold tolistm, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
map extract (tolistmd (T k) (mret T k a)) = [(k, a)]
    rewrite tolistmd_mret.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
map extract [(Ƶ, (k, a))] = [(k, a)] easy.
  Qed.

  Corollary tosetm_mret: forall (A: Type) (a: A) (k: K),
      tosetm (T k) (mret T k a) = {{ (k, a) }}.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tosetm (T k) (mret T k a) = {{(k, a)}}
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a : A) (k : K),
tosetm (T k) (mret T k a) = {{(k, a)}}
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosetm (T k) (mret T k a) = {{(k, a)}} unfold tosetm, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosubset (tolistm (T k) (mret T k a)) = {{(k, a)}}
    rewrite tolistm_mret.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a: A
k: K
tosubset [(k, a)] = {{(k, a)}}
    apply tosubset_list_ret.
  Qed.

  Lemma tolistmd_gen_mbindd:
    forall (fake: Type)
      `(f: forall k, W * A -> T k B) (t: U A),
      tolistmd_gen U fake (mbindd U f t) =
        mbinddt_list
          (fun k '(w, a) =>
             tolistmd_gen (T k) fake (f k (w, a)))
          (tolistmd_gen U fake t).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (fake A B : Type) (f : W * A ~k~> T B)
  (t : U A),
tolistmd_gen U fake (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) =>
   tolistmd_gen (T k) fake (f k (w, a)))
  (tolistmd_gen U fake t)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (fake A B : Type) (f : W * A ~k~> T B)
  (t : U A),
tolistmd_gen U fake (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) =>
   tolistmd_gen (T k) fake (f k (w, a)))
  (tolistmd_gen U fake t)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
tolistmd_gen U fake (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) =>
   tolistmd_gen (T k) fake (f k (w, a)))
  (tolistmd_gen U fake t) unfold tolistmd_gen, mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (mapMret ◻ tolistmd_gen_loc) (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a)))
  (mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc) t)
    compose near t on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
(mbinddt U (const (list (W * (K * B))))
   (mapMret ◻ tolistmd_gen_loc) ∘ mbindd U f) t =
mbinddt_list
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a)))
  (mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc) t)
    rewrite (mbinddt_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
mbinddt_list
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a)))
  (mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc) t)
    compose near t on right.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
(mbinddt_list
   (fun (k : K) '(w, a) =>
    mbinddt (T k) (const (list (W * (K * B))))
      (mapMret ◻ tolistmd_gen_loc) (f k (w, a)))
 ∘ mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc)) t
    change (mbinddt_list ?f) with (const (mbinddt_list f) (U fake)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
(const
   (mbinddt_list
      (fun (k : K) '(w, a) =>
       mbinddt (T k) (const (list (W * (K * B))))
         (mapMret ◻ tolistmd_gen_loc) (f k (w, a))))
   (U fake)
 ∘ mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc)) t
    #[local] Set Keyed Unification.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
(const
   (mbinddt_list
      (fun (k : K) '(w, a) =>
       mbinddt (T k) (const (list (W * (K * B))))
         (mapMret ◻ tolistmd_gen_loc) (f k (w, a))))
   (U fake)
 ∘ mbinddt U (const (list (W * (K * A))))
     (mapMret ◻ tolistmd_gen_loc)) t
    rewrite (dtp_mbinddt_morphism W T U
               (const (list (W * (K * A))))
               (const (list (W * (K * B))))
               (A := A) (B := fake)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
mbinddt U (const (list (W * (K * B))))
  ((fun k : K =>
    const
      (mbinddt_list
         (fun (k0 : K) '(w, a) =>
          mbinddt (T k0) (const (list (W * (K * B))))
            (mapMret ◻ tolistmd_gen_loc) (f k0 (w, a))))
      (T k fake)) ◻ (mapMret ◻ tolistmd_gen_loc)) t
    #[local] Unset Keyed Unification.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
mbinddt U (const (list (W * (K * B))))
  (fun (k : K) '(w, a) =>
   mbinddt (T k) (const (list (W * (K * B))))
     ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
     (f k (w, a))) t =
mbinddt U (const (list (W * (K * B))))
  ((fun k : K =>
    const
      (mbinddt_list
         (fun (k0 : K) '(w, a) =>
          mbinddt (T k0) (const (list (W * (K * B))))
            (mapMret ◻ tolistmd_gen_loc) (f k0 (w, a))))
      (T k fake)) ◻ (mapMret ◻ tolistmd_gen_loc)) t
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
(fun (k : K) '(w, a) =>
 mbinddt (T k) (const (list (W * (K * B))))
   ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
   (f k (w, a))) =
(fun k : K =>
 const
   (mbinddt_list
      (fun (k0 : K) '(w, a) =>
       mbinddt (T k0) (const (list (W * (K * B))))
         (mapMret ◻ tolistmd_gen_loc) (f k0 (w, a))))
   (T k fake)) ◻ (mapMret ◻ tolistmd_gen_loc) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
((fun k : K =>
  const
    (mbinddt_list
       (fun (k0 : K) '(w, a) =>
        mbinddt (T k0) (const (list (W * (K * B))))
          (mapMret ◻ tolistmd_gen_loc) (f k0 (w, a))))
    (T k fake)) ◻ (mapMret ◻ tolistmd_gen_loc)) k
  (w, a)
    cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
map (incr w)
  (mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a))) ++ []
    change (map (F := list) ?f) with (const (map (F := list) f) (U B)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
const (map (incr w)) (U B)
  (mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a))) ++ []
    List.simpl_list.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
const (map (incr w)) (U B)
  (mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc) (f k (w, a)))
    compose near (f k (w, a)) on right.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
(const (map (incr w)) (U B)
 ∘ mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc)) (f k (w, a))
    (* for some reason I can't rewrite without posing first. *)
    pose (rw := dtp_mbinddt_morphism
                  W T (T k)
                  (const (list (W * (K * B))))
                  (const (list (W * (K * B))))
                  (ϕ := (const (map (F := list) (incr w))))
                  (A := B) (B := fake)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
rw:= dtp_mbinddt_morphism W T 
  (T k) (const (list (W * (K * B))))
  (const (list (W * (K * B)))): forall
  f : forall k : K,
      W * B ->
      const (list (W * (K * B))) (T k fake),
const (map (incr w)) (T k fake)
∘ mbinddt (T k) (const (list (W * (K * B)))) f =
mbinddt (T k) 
  (const (list (W * (K * B))))
  ((fun k : K => const (map (incr w)) (T k fake))
   ◻ f)
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
(const (map (incr w)) (U B)
 ∘ mbinddt (T k) (const (list (W * (K * B))))
     (mapMret ◻ tolistmd_gen_loc)) (f k (w, a))
    rewrite rw.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
rw:= dtp_mbinddt_morphism W T 
  (T k) (const (list (W * (K * B))))
  (const (list (W * (K * B)))): forall
  f : forall k : K,
      W * B ->
      const (list (W * (K * B))) (T k fake),
const (map (incr w)) (T k fake)
∘ mbinddt (T k) (const (list (W * (K * B)))) f =
mbinddt (T k) 
  (const (list (W * (K * B))))
  ((fun k : K => const (map (incr w)) (T k fake))
   ◻ f)
mbinddt (T k) (const (list (W * (K * B))))
  ((mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w))
  (f k (w, a)) =
mbinddt (T k) (const (list (W * (K * B))))
  ((fun k : K => const (map (incr w)) (T k fake))
   ◻ (mapMret ◻ tolistmd_gen_loc)) (f k (w, a)) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
fake, A, B: Type
f: W * A ~k~> T B
t: U A
k: K
w: W
a: A
rw:= dtp_mbinddt_morphism W T 
  (T k) (const (list (W * (K * B))))
  (const (list (W * (K * B)))): forall
  f : forall k : K,
      W * B ->
      const (list (W * (K * B))) (T k fake),
const (map (incr w)) (T k fake)
∘ mbinddt (T k) (const (list (W * (K * B)))) f =
mbinddt (T k) 
  (const (list (W * (K * B))))
  ((fun k : K => const (map (incr w)) (T k fake))
   ◻ f)
(mapMret ◻ tolistmd_gen_loc) ◻ allK (incr w) =
(fun k : K => const (map (incr w)) (T k fake))
◻ (mapMret ◻ tolistmd_gen_loc) now ext k2 [w2 b].
  Qed.

  Corollary tolistmd_mbindd: forall
      `(f: forall k, W * A -> T k B) (t: U A),
      tolistmd U (mbindd U f t) =
        mbinddt_list (fun k '(w, a) =>
                        tolistmd (T k) (f k (w, a))) (tolistmd U t).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A),
tolistmd U (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) => tolistmd (T k) (f k (w, a)))
  (tolistmd U t)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A),
tolistmd U (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) => tolistmd (T k) (f k (w, a)))
  (tolistmd U t)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
tolistmd U (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) => tolistmd (T k) (f k (w, a)))
  (tolistmd U t) unfold tolistmd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
tolistmd_gen U False (mbindd U f t) =
mbinddt_list
  (fun (k : K) '(w, a) =>
   tolistmd_gen (T k) False (f k (w, a)))
  (tolistmd_gen U False t) apply tolistmd_gen_mbindd.
  Qed.

End DTM_tolist.

(** ** Characterizing Occurrences Post-Operation *)
(**********************************************************************)
Section DTM_membership.

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

  (** *** Occurrences in <<mret>> *)
  (********************************************************************)
  Lemma inmd_mret_iff: forall (A: Type) (a1 a2: A) (k1 k2: K) (w: W),
      (w, (k2, a2)) ∈md mret T k1 a1 <-> w = Ƶ /\ k1 = k2 /\ a1 = a2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k1 k2 : K) (w : W),
(w, (k2, a2)) ∈md mret T k1 a1 <->
w = Ƶ /\ k1 = k2 /\ a1 = a2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k1 k2 : K) (w : W),
(w, (k2, a2)) ∈md mret T k1 a1 <->
w = Ƶ /\ k1 = k2 /\ a1 = a2
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
(w, (k2, a2)) ∈md mret T k1 a1 <->
w = Ƶ /\ k1 = k2 /\ a1 = a2 rewrite (tosetmd_mret).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
{{(Ƶ, (k1, a1))}} (w, (k2, a2)) <->
w = Ƶ /\ k1 = k2 /\ a1 = a2
    autorewrite with tea_set.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
(Ƶ, (k1, a1)) = (w, (k2, a2)) <->
w = Ƶ /\ k1 = k2 /\ a1 = a2
    split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
(Ƶ, (k1, a1)) = (w, (k2, a2)) ->
w = Ƶ /\ k1 = k2 /\ a1 = a2
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
w = Ƶ /\ k1 = k2 /\ a1 = a2 ->
(Ƶ, (k1, a1)) = (w, (k2, a2))
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
(Ƶ, (k1, a1)) = (w, (k2, a2)) ->
w = Ƶ /\ k1 = k2 /\ a1 = a2 inversion 1; now subst.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
w = Ƶ /\ k1 = k2 /\ a1 = a2 ->
(Ƶ, (k1, a1)) = (w, (k2, a2)) introv [? [? ?]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
w: W
H1: w = Ƶ
H2: k1 = k2
H3: a1 = a2
(Ƶ, (k1, a1)) = (w, (k2, a2)) now subst.
  Qed.

  Corollary in_mret_iff: forall (A: Type) (a1 a2: A) (k1 k2: K),
      (k2, a2) ∈m mret T k1 a1 <-> k1 = k2 /\ a1 = a2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k1 k2 : K),
(k2, a2) ∈m mret T k1 a1 <-> k1 = k2 /\ a1 = a2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k1 k2 : K),
(k2, a2) ∈m mret T k1 a1 <-> k1 = k2 /\ a1 = a2
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
(k2, a2) ∈m mret T k1 a1 <-> k1 = k2 /\ a1 = a2 rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
(exists w : W, (w, (k2, a2)) ∈md mret T k1 a1) <->
k1 = k2 /\ a1 = a2 setoid_rewrite inmd_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k1, k2: K
(exists w : W, w = Ƶ /\ k1 = k2 /\ a1 = a2) <->
k1 = k2 /\ a1 = a2
    firstorder.
  Qed.

  Lemma inmd_mret_eq_iff: forall (A: Type) (a1 a2: A) (k: K) (w: W),
      (w, (k, a2)) ∈md mret T k a1 <-> w = Ƶ /\ a1 = a2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k : K) (w : W),
(w, (k, a2)) ∈md mret T k a1 <-> w = Ƶ /\ a1 = a2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k : K) (w : W),
(w, (k, a2)) ∈md mret T k a1 <-> w = Ƶ /\ a1 = a2
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k: K
w: W
(w, (k, a2)) ∈md mret T k a1 <-> w = Ƶ /\ a1 = a2 rewrite inmd_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k: K
w: W
w = Ƶ /\ k = k /\ a1 = a2 <-> w = Ƶ /\ a1 = a2 clear.
ix: Index
W: Type
mn_unit: Monoid_unit W
A: Type
a1, a2: A
k: K
w: W
w = Ƶ /\ k = k /\ a1 = a2 <-> w = Ƶ /\ a1 = a2 firstorder.
  Qed.

  Lemma inmd_mret_neq_iff: forall (A: Type) (a1 a2: A) (k j: K) (w: W),
      k <> j ->
      (w, (j, a2)) ∈md mret T k a1 <-> False.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k j : K) (w : W),
k <> j -> (w, (j, a2)) ∈md mret T k a1 <-> False
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k j : K) (w : W),
k <> j -> (w, (j, a2)) ∈md mret T k a1 <-> False
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k, j: K
w: W
H1: k <> j
(w, (j, a2)) ∈md mret T k a1 <-> False rewrite inmd_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k, j: K
w: W
H1: k <> j
w = Ƶ /\ k = j /\ a1 = a2 <-> False firstorder.
  Qed.

  Corollary in_mret_eq_iff: forall (A: Type) (a1 a2: A) (k: K),
      (k, a2) ∈m mret T k a1 <-> a1 = a2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k : K),
(k, a2) ∈m mret T k a1 <-> a1 = a2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k : K),
(k, a2) ∈m mret T k a1 <-> a1 = a2
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k: K
(k, a2) ∈m mret T k a1 <-> a1 = a2 rewrite in_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k: K
k = k /\ a1 = a2 <-> a1 = a2 firstorder.
  Qed.

  Corollary in_mret_neq_iff: forall (A: Type) (a1 a2: A) (k j: K),
      k <> j ->
      (j, a2) ∈m mret T k a1 <-> False.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k j : K),
k <> j -> (j, a2) ∈m mret T k a1 <-> False
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (a1 a2 : A) (k j : K),
k <> j -> (j, a2) ∈m mret T k a1 <-> False
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k, j: K
H1: k <> j
(j, a2) ∈m mret T k a1 <-> False rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k, j: K
H1: k <> j
(exists w : W, (w, (j, a2)) ∈md mret T k a1) <-> False setoid_rewrite inmd_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
a1, a2: A
k, j: K
H1: k <> j
(exists w : W, w = Ƶ /\ k = j /\ a1 = a2) <-> False
    firstorder.
  Qed.

  Lemma tosubset_map_iff: forall {A B:Type} (l: list A) (f: A -> B),
      tosubset (map f l) = map f (tosubset l).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (l : list A) (f : A -> B),
tosubset (map f l) = map f (tosubset l)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (l : list A) (f : A -> B),
tosubset (map f l) = map f (tosubset l)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
l: list A
f: A -> B
tosubset (map f l) = map f (tosubset l)
    compose near l.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
l: list A
f: A -> B
(tosubset ∘ map f) l = (map f ∘ tosubset) l
    rewrite tosubset_list_map.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
l: list A
f: A -> B
(map f ∘ tosubset) l = (map f ∘ tosubset) l
    reflexivity.
  Qed.

  (** *** Occurrences in <<mbindd>> with context *)
  (********************************************************************)
  Lemma inmd_mbindd_iff1:
    forall `(f: forall k, W * A -> T k B) (t: U A) (k2: K) (wtotal: W) (b: B),
      (wtotal, (k2, b)) ∈md mbindd U f t ->
      exists (k1: K) (w1 w2: W) (a: A),
        (w1, (k1, a)) ∈md t /\ (w2, (k2, b)) ∈md f k1 (w1, a)
        /\ wtotal = w1 ● w2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbindd U f t ->
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbindd U f t ->
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2
    introv hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
hyp: (wtotal, (k2, b)) ∈md mbindd U f t
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2 unfold tosetmd, compose in *.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
hyp: tosubset (tolistmd U (mbindd U f t))
  (wtotal, (k2, b))
exists (k1 : K) (w1 w2 : W) (a : A),
  tosubset (tolistmd U t) (w1, (k1, a)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
    rewrite (tolistmd_mbindd U) in hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a)))
     (tolistmd U t)) (
  wtotal, (k2, b))
exists (k1 : K) (w1 w2 : W) (a : A),
  tosubset (tolistmd U t) (w1, (k1, a)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 induction (tolistmd U t).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) [])
  (wtotal, (k2, b))
exists (k1 : K) (w1 w2 : W) (a : A),
  tosubset [] (w1, (k1, a)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
a: W * (K * A)
l: list (W * (K * A))
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) 
     (a :: l)) (
  wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) 
(w1 w2 : W) 
(a0 : A),
  tosubset (a :: l) (w1, (k1, a0)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a0))) (
    w2, (k2, b)) /\ 
  wtotal = w1 ● w2
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) [])
  (wtotal, (k2, b))
exists (k1 : K) (w1 w2 : W) (a : A),
  tosubset [] (w1, (k1, a)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 inversion hyp.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
a: W * (K * A)
l: list (W * (K * A))
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) 
     (a :: l)) (
  wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset (a :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 destruct a as [w [k a]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a)))
     ((w, (k, a)) :: l)) (
  wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 rewrite mbinddt_list_cons in hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp: tosubset
  (map (incr w) (tolistmd (T k) (f k (w, a))) ++
   mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
      rewrite tosubset_list_app in hyp.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp: (tosubset
   (map (incr w) (tolistmd (T k) (f k (w, a))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 destruct hyp as [hyp1 | hyp2].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp1: tosubset
  (map (incr w) (tolistmd (T k) (f k (w, a))))
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp2: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) 
(w1 w2 : W) 
(a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a0))) (
    w2, (k2, b)) /\ 
  wtotal = w1 ● w2
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp1: tosubset
  (map (incr w) (tolistmd (T k) (f k (w, a))))
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 rewrite tosubset_map_iff in hyp1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp1: map (incr w)
  (tosubset (tolistmd (T k) (f k (w, a))))
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
        destruct hyp1 as [[w2 [k2' b']] [hyp1 hyp2]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
k2': K
b': B
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2', b'))
hyp2: incr w (w2, (k2', b')) = (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
        inversion hyp2; subst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
exists (k1 : K) (w1 w3 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w3, (k2, b)) /\ w ● w2 = w1 ● w3 exists k w w2 a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset ((w, (k, a)) :: l) (w, (k, a)) /\
tosubset (tolistmd (T k) (f k (w, a))) (w2, (k2, b)) /\
w ● w2 = w ● w2 splits.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset ((w, (k, a)) :: l) (w, (k, a))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset (tolistmd (T k) (f k (w, a))) (w2, (k2, b))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
w ● w2 = w ● w2
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset ((w, (k, a)) :: l) (w, (k, a)) rewrite tosubset_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
({{(w, (k, a))}} ∪ tosubset l) (w, (k, a)) now left. }
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset (tolistmd (T k) (f k (w, a))) (w2, (k2, b))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
w ● w2 = w ● w2
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
tosubset (tolistmd (T k) (f k (w, a))) (w2, (k2, b)) auto. }
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
w ● w2 = w ● w2
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
w2: W
hyp1: tosubset (tolistmd (T k) (f k (w, a)))
  (w2, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w ● w2, (k2, b)) ->
exists (k1 : K) 
(w1 w3 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w3, (k2, b)) /\ 
  w ● w2 = w1 ● w3
hyp2: incr w (w2, (k2, b)) = (w ● w2, (k2, b))
w ● w2 = w ● w2 easy. }
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp2: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b))
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 apply IHl in hyp2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp2: exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
IHl: tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (wtotal, (k2, b)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2 clear IHl.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
hyp2: exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  tosubset l (w1, (k1, a)) /\
  tosubset 
    (tolistmd (T k1) (f k1 (w1, a)))
    (w2, (k2, b)) /\ 
  wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
        destruct hyp2 as [k1 [w1 [w2 [a' [hyp1 [hyp2 hyp3]] ]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
hyp3: wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w1, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w1, a0)))
    (w2, (k2, b)) /\ wtotal = w1 ● w2
        subst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
exists (k1 : K) (w3 w4 : W) (a0 : A),
  tosubset ((w, (k, a)) :: l) (w3, (k1, a0)) /\
  tosubset (tolistmd (T k1) (f k1 (w3, a0)))
    (w4, (k2, b)) /\ w1 ● w2 = w3 ● w4 repeat eexists.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset ((w, (k, a)) :: l) (w1, (?k1, ?a))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset (tolistmd (T ?k1) (f ?k1 (w1, ?a)))
  (w2, (k2, b))
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset ((w, (k, a)) :: l) (w1, (?k1, ?a)) rewrite tosubset_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
({{(w, (k, a))}} ∪ tosubset l) (w1, (?k1, ?a)) right.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset l (w1, (?k1, ?a)) eauto. }
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
        {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
w: W
k: K
a: A
l: list (W * (K * A))
k1: K
w1, w2: W
a': A
hyp1: tosubset l (w1, (k1, a'))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b))
tosubset (tolistmd (T k1) (f k1 (w1, a')))
  (w2, (k2, b)) auto. }
  Qed.

  Lemma inmd_mbindd_iff2:
    forall `(f: forall k, W * A -> T k B) (t: U A) (k2: K) (wtotal: W) (b: B),
    (exists (k1: K) (w1 w2: W) (a: A),
      (w1, (k1, a)) ∈md t /\ (w2, (k2, b)) ∈md f k1 (w1, a)
        /\ wtotal = w1 ● w2) ->
      (wtotal, (k2, b)) ∈md mbindd U f t.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2) ->
(wtotal, (k2, b)) ∈md mbindd U f t
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2) ->
(wtotal, (k2, b)) ∈md mbindd U f t
    introv [k1 [w1 [w2 [a [hyp1 [hyp2 hyp3]]]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k2, b)) ∈md f k1 (w1, a)
hyp3: wtotal = w1 ● w2
(wtotal, (k2, b)) ∈md mbindd U f t subst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k2, b)) ∈md f k1 (w1, a)
(w1 ● w2, (k2, b)) ∈md mbindd U f t
    unfold tosetmd, compose in *.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
hyp1: tosubset (tolistmd U t) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
tosubset (tolistmd U (mbindd U f t))
  (w1 ● w2, (k2, b)) rewrite (tolistmd_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
hyp1: tosubset (tolistmd U t) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) (tolistmd U t))
  (w1 ● w2, (k2, b))
    induction (tolistmd U t).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
hyp1: tosubset [] (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) [])
  (w1 ● w2, (k2, b))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
a0: W * (K * A)
l: list (W * (K * A))
hyp1: tosubset (a0 :: l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) 
     (a0 :: l)) (
  w1 ● w2, (k2, b))
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
hyp1: tosubset [] (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) [])
  (w1 ● w2, (k2, b)) inversion hyp1.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
a0: W * (K * A)
l: list (W * (K * A))
hyp1: tosubset (a0 :: l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) (a0 :: l))
  (w1 ● w2, (k2, b)) destruct a0 as [w [k' a']].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset ((w, (k', a')) :: l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a)))
     ((w, (k', a')) :: l)) (w1 ● w2, (k2, b)) rewrite mbinddt_list_cons.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset ((w, (k', a')) :: l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (map (incr w) (tolistmd (T k') (f k' (w, a'))) ++
   mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
      simpl_list.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset ((w, (k', a')) :: l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b)) rewrite tosubset_list_cons in hyp1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: ({{(w, (k', a'))}} ∪ tosubset l) (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b))
      destruct hyp1 as [hyp1 | hyp1].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: {{(w, (k', a'))}} (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset l (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b))
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: {{(w, (k', a'))}} (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b)) inverts hyp1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
l: list (W * (K * A))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w1) (tolistmd (T k1) (f k1 (w1, a))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b)) left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
l: list (W * (K * A))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (map (incr w1) (tolistmd (T k1) (f k1 (w1, a))))
  (w1 ● w2, (k2, b)) rewrite (tosubset_map_iff).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
l: list (W * (K * A))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
map (incr w1)
  (tosubset (tolistmd (T k1) (f k1 (w1, a))))
  (w1 ● w2, (k2, b))
        exists (w2, (k2, b)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
l: list (W * (K * A))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b)) /\
incr w1 (w2, (k2, b)) = (w1 ● w2, (k2, b)) now splits.
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset l (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
(tosubset
   (map (incr w) (tolistmd (T k') (f k' (w, a'))))
 ∪ tosubset
     (mbinddt_list
        (fun (k : K) '(w, a) =>
         tolistmd (T k) (f k (w, a))) l))
  (w1 ● w2, (k2, b)) right.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1, w2: W
a: A
w: W
k': K
a': A
l: list (W * (K * A))
hyp1: tosubset l (w1, (k1, a))
hyp2: tosubset (tolistmd (T k1) (f k1 (w1, a)))
  (w2, (k2, b))
IHl: tosubset l (w1, (k1, a)) ->
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b))
tosubset
  (mbinddt_list
     (fun (k : K) '(w, a) =>
      tolistmd (T k) (f k (w, a))) l)
  (w1 ● w2, (k2, b)) now apply IHl in hyp1.
  Qed.

  Theorem inmd_mbindd_iff:
    forall `(f: forall k, W * A -> T k B) (t: U A) (k2: K) (wtotal: W) (b: B),
      (wtotal, (k2, b)) ∈md mbindd U f t <->
      exists (k1: K) (w1 w2: W) (a: A),
        (w1, (k1, a)) ∈md t /\ (w2, (k2, b)) ∈md f k1 (w1, a)
        /\ wtotal = w1 ● w2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbindd U f t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbindd U f t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ wtotal = w1 ● w2)
    split; auto using inmd_mbindd_iff1, inmd_mbindd_iff2.
  Qed.

  (** *** Corollaries for other operations *)
  (********************************************************************)
  Corollary inmd_mbind_iff:
    forall `(f: forall k, A -> T k B) (t: U A) (k2: K) (wtotal: W) (b: B),
      (wtotal, (k2, b)) ∈md mbind U f t <->
      exists (k1: K) (w1 w2: W) (a: A),
        (w1, (k1, a)) ∈md t /\ (w2, (k2, b)) ∈md f k1 a
        /\ wtotal = w1 ● w2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbind U f t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 a /\ wtotal = w1 ● w2)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : A ~k~> T B) (t : U A)
  (k2 : K) (wtotal : W) (b : B),
(wtotal, (k2, b)) ∈md mbind U f t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 a /\ wtotal = w1 ● w2)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
(wtotal, (k2, b)) ∈md mbind U f t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 a /\ wtotal = w1 ● w2)
    rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
(wtotal, (k2, b)) ∈md mbindd U (f ◻ allK extract) t <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 a /\ wtotal = w1 ● w2)
    rewrite inmd_mbindd_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
wtotal: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
   wtotal = w1 ● w2) <->
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 a /\ wtotal = w1 ● w2)
    reflexivity.
  Qed.

  Corollary inmd_mmapd_iff:
    forall `(f: forall k, W * A -> B) (t: U A) (k: K) (w: W) (b: B),
      (w, (k, b)) ∈md mmapd U f t <->
      exists (a: A), (w, (k, a)) ∈md t /\ b = f k (w, a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> W * A -> B) (t : U A)
  (k : K) (w : W) (b : B),
(w, (k, b)) ∈md mmapd U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> W * A -> B) (t : U A)
  (k : K) (w : W) (b : B),
(w, (k, b)) ∈md mmapd U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(w, (k, b)) ∈md mmapd U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)) unfold mmapd, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(w, (k, b)) ∈md mbindd U (mret T ◻ f) t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)) setoid_rewrite inmd_mbindd_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k, b)) ∈md (mret T ◻ f) k1 (w1, a) /\
   w = w1 ● w2) <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a))
    unfold_ops @Map_I.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k, b)) ∈md (mret T ◻ f) k1 (w1, a) /\
   w = w1 ● w2) <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)) setoid_rewrite inmd_mret_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
   w = w1 ● w2) <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a))
    split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
   w = w1 ● w2) ->
exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)) ->
exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  (w1, (k1, a)) ∈md t /\
  (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
  w = w1 ● w2
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
   w = w1 ● w2) ->
exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a) intros [k1 [w1 [w2 [a [hyp1 [[hyp2 [hyp2' hyp2'']] hyp3]]]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: w2 = Ƶ
hyp2': k1 = k
hyp2'': f k1 (w1, a) = b
hyp3: w = w1 ● w2
exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)
      subst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w1: W
a: A
hyp1: (w1, (k, a)) ∈md t
exists a0 : A,
  (w1 ● Ƶ, (k, a0)) ∈md t /\
  f k (w1, a) = f k (w1 ● Ƶ, a0) exists a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w1: W
a: A
hyp1: (w1, (k, a)) ∈md t
(w1 ● Ƶ, (k, a)) ∈md t /\
f k (w1, a) = f k (w1 ● Ƶ, a) simpl_monoid.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w1: W
a: A
hyp1: (w1, (k, a)) ∈md t
(w1, (k, a)) ∈md t /\ f k (w1, a) = f k (w1, a) auto.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
(exists a : A, (w, (k, a)) ∈md t /\ b = f k (w, a)) ->
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
  w = w1 ● w2 intros [a [hyp1 hyp2]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
b: B
a: A
hyp1: (w, (k, a)) ∈md t
hyp2: b = f k (w, a)
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a) = b) /\
  w = w1 ● w2 subst.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
a: A
hyp1: (w, (k, a)) ∈md t
exists (k1 : K) (w1 w2 : W) (a0 : A),
  (w1, (k1, a0)) ∈md t /\
  (w2 = Ƶ /\ k1 = k /\ f k1 (w1, a0) = f k (w, a)) /\
  w = w1 ● w2 repeat eexists.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
a: A
hyp1: (w, (k, a)) ∈md t
(w, (k, a)) ∈md t
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
a: A
hyp1: (w, (k, a)) ∈md t
w = w ● Ƶ
      easy.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
w: W
a: A
hyp1: (w, (k, a)) ∈md t
w = w ● Ƶ now simpl_monoid.
  Qed.

  Corollary inmd_mmap_iff:
    forall `(f: K -> A -> B) (t: U A) (k: K) (w: W) (b: B),
      (w, (k, b)) ∈md mmap U f t <->
      exists (a: A), (w, (k, a)) ∈md t /\ b = f k a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> A -> B) (t : U A)
  (k : K) (w : W) (b : B),
(w, (k, b)) ∈md mmap U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> A -> B) (t : U A)
  (k : K) (w : W) (b : B),
(w, (k, b)) ∈md mmap U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
w: W
b: B
(w, (k, b)) ∈md mmap U f t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k a) rewrite (mmap_to_mmapd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
w: W
b: B
(w, (k, b)) ∈md mmapd U (f ◻ allK extract) t <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k a)
    rewrite inmd_mmapd_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
w: W
b: B
(exists a : A,
   (w, (k, a)) ∈md t /\
   b = (f ◻ allK extract) k (w, a)) <->
(exists a : A, (w, (k, a)) ∈md t /\ b = f k a) easy.
  Qed.

  (** *** Occurrences without context *)
  (********************************************************************)
  Theorem in_mbindd_iff:
    forall `(f: forall k, W * A -> T k B) (t: U A) (k2: K) (b: B),
      (k2, b) ∈m mbindd U f t <->
      exists (k1: K) (w1: W) (a: A),
        (w1, (k1, a)) ∈md t
        /\ (k2, b) ∈m (f k1 (w1, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (b : B),
(k2, b) ∈m mbindd U f t <->
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : W * A ~k~> T B) (t : U A)
  (k2 : K) (b : B),
(k2, b) ∈m mbindd U f t <->
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(k2, b) ∈m mbindd U f t <->
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a))
    rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists w : W, (w, (k2, b)) ∈md mbindd U f t) <->
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)) setoid_rewrite inmd_mbindd_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2) <->
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)) split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2) ->
exists (k1 : K) (w1 : W) (a : A),
  (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists (k1 : K) 
 (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)) ->
exists (w : W) 
(k1 : K) (w1 w2 : W) 
(a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2) ->
exists (k1 : K) (w1 : W) (a : A),
  (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a) intros [wtotal [k1 [w1 [w2 [a [hyp1 [hyp2 hyp3]]]]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
wtotal: W
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k2, b)) ∈md f k1 (w1, a)
hyp3: wtotal = w1 ● w2
exists (k1 : K) (w1 : W) (a : A),
  (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)
      exists k1 w1 a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
wtotal: W
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k2, b)) ∈md f k1 (w1, a)
hyp3: wtotal = w1 ● w2
(w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a) split; [auto|].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
wtotal: W
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k2, b)) ∈md f k1 (w1, a)
hyp3: wtotal = w1 ● w2
(k2, b) ∈m f k1 (w1, a)
      apply (inmd_implies_in) in hyp2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
wtotal: W
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (k2, b) ∈m f k1 (w1, a)
hyp3: wtotal = w1 ● w2
(k2, b) ∈m f k1 (w1, a) auto.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
(exists (k1 : K) (w1 : W) (a : A),
   (w1, (k1, a)) ∈md t /\ (k2, b) ∈m f k1 (w1, a)) ->
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2 intros [k1 [w1 [a [hyp1 hyp2]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (k2, b) ∈m f k1 (w1, a)
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2
      rewrite inmd_iff_in in hyp2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: exists w : W, (w, (k2, b)) ∈md f k1 (w1, a)
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2 destruct hyp2 as [w2 rest].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1: W
a: A
hyp1: (w1, (k1, a)) ∈md t
w2: W
rest: (w2, (k2, b)) ∈md f k1 (w1, a)
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md f k1 (w1, a) /\ w = w1 ● w2
      exists (w1 ● w2) k1 w1 w2 a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: W * A ~k~> T B
t: U A
k2: K
b: B
k1: K
w1: W
a: A
hyp1: (w1, (k1, a)) ∈md t
w2: W
rest: (w2, (k2, b)) ∈md f k1 (w1, a)
(w1, (k1, a)) ∈md t /\
(w2, (k2, b)) ∈md f k1 (w1, a) /\ w1 ● w2 = w1 ● w2 intuition.
  Qed.

  (** *** Corollaries for other operations *)
  (********************************************************************)
  Corollary in_mbind_iff:
    forall `(f: forall k, A -> T k B) (t: U A) (k2: K) (b: B),
      (k2, b) ∈m mbind U f t <->
      exists (k1: K) (a: A), (k1, a) ∈m t /\ (k2, b) ∈m f k1 a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : A ~k~> T B) (t : U A)
  (k2 : K) (b : B),
(k2, b) ∈m mbind U f t <->
(exists (k1 : K) (a : A),
   (k1, a) ∈m t /\ (k2, b) ∈m f k1 a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : A ~k~> T B) (t : U A)
  (k2 : K) (b : B),
(k2, b) ∈m mbind U f t <->
(exists (k1 : K) (a : A),
   (k1, a) ∈m t /\ (k2, b) ∈m f k1 a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(k2, b) ∈m mbind U f t <->
(exists (k1 : K) (a : A),
   (k1, a) ∈m t /\ (k2, b) ∈m f k1 a) unfold mbind, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(k2, b) ∈m mbindd U (f ◻ allK extract) t <->
(exists (k1 : K) (a : A),
   (k1, a) ∈m t /\ (k2, b) ∈m f k1 a) setoid_rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists w : W,
   (w, (k2, b)) ∈md mbindd U (f ◻ allK extract) t) <->
(exists (k1 : K) (a : A),
   (exists w : W, (w, (k1, a)) ∈md t) /\
   (exists w : W, (w, (k2, b)) ∈md f k1 a))
    setoid_rewrite inmd_mbindd_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
   w = w1 ● w2) <->
(exists (k1 : K) (a : A),
   (exists w : W, (w, (k1, a)) ∈md t) /\
   (exists w : W, (w, (k2, b)) ∈md f k1 a)) cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
   w = w1 ● w2) <->
(exists (k1 : K) (a : A),
   (exists w : W, (w, (k1, a)) ∈md t) /\
   (exists w : W, (w, (k2, b)) ∈md f k1 a)) split.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
   w = w1 ● w2) ->
exists (k1 : K) (a : A),
  (exists w : W, (w, (k1, a)) ∈md t) /\
  (exists w : W, (w, (k2, b)) ∈md f k1 a)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (k1 : K) 
 (a : A),
   (exists w : W, (w, (k1, a)) ∈md t) /\
   (exists w : W, (w, (k2, b)) ∈md f k1 a)) ->
exists (w : W) 
(k1 : K) (w1 w2 : W) 
(a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
  w = w1 ● w2
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
   (w1, (k1, a)) ∈md t /\
   (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
   w = w1 ● w2) ->
exists (k1 : K) (a : A),
  (exists w : W, (w, (k1, a)) ∈md t) /\
  (exists w : W, (w, (k2, b)) ∈md f k1 a) firstorder.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
(exists (k1 : K) (a : A),
   (exists w : W, (w, (k1, a)) ∈md t) /\
   (exists w : W, (w, (k2, b)) ∈md f k1 a)) ->
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
  w = w1 ● w2 intros [k1 [a [[w1 hyp1] [w hyp2]]]].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: A ~k~> T B
t: U A
k2: K
b: B
k1: K
a: A
w1: W
hyp1: (w1, (k1, a)) ∈md t
w: W
hyp2: (w, (k2, b)) ∈md f k1 a
exists (w : W) (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (k2, b)) ∈md (f ◻ allK extract) k1 (w1, a) /\
  w = w1 ● w2
      repeat eexists; eauto.
  Qed.

  Corollary in_mmapd_iff:
    forall `(f: forall k, W * A -> B) (t: U A) (k: K) (b: B),
      (k, b) ∈m mmapd U f t <->
      exists (w: W) (a: A), (w, (k, a)) ∈md t /\ b = f k (w, a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> W * A -> B) (t : U A)
  (k : K) (b : B),
(k, b) ∈m mmapd U f t <->
(exists (w : W) (a : A),
   (w, (k, a)) ∈md t /\ b = f k (w, a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> W * A -> B) (t : U A)
  (k : K) (b : B),
(k, b) ∈m mmapd U f t <->
(exists (w : W) (a : A),
   (w, (k, a)) ∈md t /\ b = f k (w, a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
b: B
(k, b) ∈m mmapd U f t <->
(exists (w : W) (a : A),
   (w, (k, a)) ∈md t /\ b = f k (w, a)) setoid_rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> W * A -> B
t: U A
k: K
b: B
(exists w : W, (w, (k, b)) ∈md mmapd U f t) <->
(exists (w : W) (a : A),
   (w, (k, a)) ∈md t /\ b = f k (w, a))
    now setoid_rewrite inmd_mmapd_iff.
  Qed.

  Corollary in_mmap_iff:
    forall `(f: forall k, A -> B) (t: U A) (k: K) (b: B),
      (k, b) ∈m mmap U f t <->
      exists (a: A), (k, a) ∈m t /\ b = f k a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> A -> B) (t : U A)
  (k : K) (b : B),
(k, b) ∈m mmap U f t <->
(exists a : A, (k, a) ∈m t /\ b = f k a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A B : Type) (f : K -> A -> B) (t : U A)
  (k : K) (b : B),
(k, b) ∈m mmap U f t <->
(exists a : A, (k, a) ∈m t /\ b = f k a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
b: B
(k, b) ∈m mmap U f t <->
(exists a : A, (k, a) ∈m t /\ b = f k a) setoid_rewrite inmd_iff_in.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
b: B
(exists w : W, (w, (k, b)) ∈md mmap U f t) <->
(exists a : A,
   (exists w : W, (w, (k, a)) ∈md t) /\ b = f k a)
    setoid_rewrite inmd_mmap_iff.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
t: U A
k: K
b: B
(exists (w : W) (a : A),
   (w, (k, a)) ∈md t /\ b = f k a) <->
(exists a : A,
   (exists w : W, (w, (k, a)) ∈md t) /\ b = f k a)
    firstorder.
  Qed.

End DTM_membership.