From Tealeaves Require Import
  Classes.Multisorted.DecoratedTraversableMonad
  Classes.Multisorted.Theory.Container
  Categories.TypeFamily
  Functors.List
  Functors.Constant.

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

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

(** * Targeted substitution-building combinators: [btg] and [btgd] *)
(**********************************************************************)
(* TODO: Define a version that works for applicative effects. *)
(*
  #[program] Definition btga `{ix: Index} `{Map F} `{Pure F} `{Mult F}
  {A W: Type} (T: K -> Type -> Type) `{! MReturn T}
  (j: K) (f: W * A -> F (T j A)): forall (k: K), W * A -> F (T k A) :=
  fun k '(w, a) => if k == j then f (w, a) else pure ∘ mret T k a.
 *)

Require Import Coq.Program.Equality.

#[program] Definition btgd `{ix: Index} {A W: Type}
  {T: K -> Type -> Type} `{! MReturn T}
  (j: K) (f: W * A -> T j A): forall (k: K), W * A -> T k A :=
  fun k '(w, a) => if k == j then f (w, a) else mret T k a.

#[program] Definition btg `{ix: Index} {A: Type}
  {T: K -> Type -> Type} `{! MReturn T}
  (j: K) (f: A -> T j A): forall (k: K), A -> T k A :=
  fun k => if k == j then f else mret T k.

Require Import Coq.Program.Equality.

Section btg_lemmas.

  Context
    `{ix: Index}.

  Context
    `{MReturn T}
    {W A: Type}.

  Lemma btgd_eq: forall k (f: W * A -> T k A),
      btgd k f k = f.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k : K) (f : W * A -> T k A), btgd k f k = f
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k : K) (f : W * A -> T k A), btgd k f k = f
    introv.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
btgd k f k = f unfold btgd.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
(fun pat : W * A =>
 (let
  '(w, a) as anonymous' := pat
   return (anonymous' = pat -> T k A) in
   fun Heq_anonymous : (w, a) = pat =>
   match k == k with
   | left x =>
       rew [fun H : K => T H A]
           btgd_obligation_1 ix0 A W k k pat w a
             Heq_anonymous x in f (w, a)
   | right _ => mret T k a
   end) eq_refl) = f ext [w a].
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
w: W
a: A
match k == k with
| left x =>
    rew [fun H : K => T H A]
        btgd_obligation_1 ix0 A W k k (w, a) w a
          eq_refl x in f (w, a)
| right _ => mret T k a
end = f (w, a)
    compare values k and k.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQ, DESTR_EQs: k = k
rew [fun H : K => T H A]
    btgd_obligation_1 ix0 A W k k (w, a) w a eq_refl
      DESTR_EQ in f (w, a) = f (w, a)
    dependent destruction DESTR_EQ.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
rew [fun H : K => T H A]
    btgd_obligation_1 ix0 A W k k (w, a) w a eq_refl
      eq_refl in f (w, a) = f (w, a)
    cbn.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
f (w, a) = f (w, a) reflexivity.
  Qed.

  Lemma btgd_neq: forall {k j} (f: W * A -> T j A),
      k <> j -> btgd j f k = mret T k ∘ extract (W := (W ×)).
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k j : K) (f : W * A -> T j A),
k <> j -> btgd j f k = mret T k ∘ extract
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k j : K) (f : W * A -> T j A),
k <> j -> btgd j f k = mret T k ∘ extract
    introv.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: W * A -> T j A
k <> j -> btgd j f k = mret T k ∘ extract unfold btgd.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: W * A -> T j A
k <> j ->
(fun pat : W * A =>
 (let
  '(w, a) as anonymous' := pat
   return (anonymous' = pat -> T k A) in
   fun Heq_anonymous : (w, a) = pat =>
   match k == j with
   | left x =>
       rew [fun H : K => T H A]
           btgd_obligation_1 ix0 A W j k pat w a
             Heq_anonymous x in f (w, a)
   | right _ => mret T k a
   end) eq_refl) = mret T k ∘ extract intro hyp.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: W * A -> T j A
hyp: k <> j
(fun pat : W * A =>
 (let
  '(w, a) as anonymous' := pat
   return (anonymous' = pat -> T k A) in
   fun Heq_anonymous : (w, a) = pat =>
   match k == j with
   | left x =>
       rew [fun H : K => T H A]
           btgd_obligation_1 ix0 A W j k pat w a
             Heq_anonymous x in f (w, a)
   | right _ => mret T k a
   end) eq_refl) = mret T k ∘ extract ext [w a].
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: W * A -> T j A
hyp: k <> j
w: W
a: A
match k == j with
| left x =>
    rew [fun H : K => T H A]
        btgd_obligation_1 ix0 A W j k (w, a) w a
          eq_refl x in f (w, a)
| right _ => mret T k a
end = (mret T k ∘ extract) (w, a)
    compare values k and j.
  Qed.

  Lemma btgd_id (j: K):
    btgd (A := A) j
      (mret T j ∘ extract (W := (W ×))) = mret T ◻ allK extract.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
btgd j (mret T j ∘ extract) = mret T ◻ allK extract
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
btgd j (mret T j ∘ extract) = mret T ◻ allK extract
    unfold btgd.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
(fun (k : K) (pat : W * A) =>
 (let
  '(w, a) as anonymous' := pat
   return (anonymous' = pat -> T k A) in
   fun Heq_anonymous : (w, a) = pat =>
   match k == j with
   | left x =>
       rew [fun H : K => T H A]
           btgd_obligation_1 ix0 A W j k pat w a
             Heq_anonymous x in
       (mret T j ∘ extract) (w, a)
   | right _ => mret T k a
   end) eq_refl) = mret T ◻ allK extract ext k [w a].
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j, k: K
w: W
a: A
match k == j with
| left x =>
    rew [fun H : K => T H A]
        btgd_obligation_1 ix0 A W j k (w, a) w a
          eq_refl x in (mret T j ∘ extract) (w, a)
| right _ => mret T k a
end = (mret T ◻ allK extract) k (w, a) compare values k and j.
  Qed.

  Lemma btg_eq: forall k (f: A -> T k A),
      btg k f k = f.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k : K) (f : A -> T k A), btg k f k = f
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k : K) (f : A -> T k A), btg k f k = f
    introv.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: A -> T k A
btg k f k = f unfold btg.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: A -> T k A
match k == k with
| left x =>
    fun x0 : A =>
    rew [fun H : K => T H A]
        btg_obligation_1 ix0 k k x in f x0
| right _ => mret T k
end = f
    compare values k and k.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: A -> T k A
DESTR_EQ, DESTR_EQs: k = k
(fun x : A =>
 rew [fun H : K => T H A]
     btg_obligation_1 ix0 k k DESTR_EQ in f x) = f
    dependent destruction DESTR_EQ.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: A -> T k A
DESTR_EQs: k = k
(fun x : A =>
 rew [fun H : K => T H A]
     btg_obligation_1 ix0 k k eq_refl in f x) = f
    cbn.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k: K
f: A -> T k A
DESTR_EQs: k = k
(fun x : A => f x) = f reflexivity.
  Qed.

  Lemma btg_neq: forall {k j} (f: A -> T j A),
      k <> j -> btg j f k = mret T k.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k j : K) (f : A -> T j A),
k <> j -> btg j f k = mret T k
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
forall (k j : K) (f : A -> T j A),
k <> j -> btg j f k = mret T k
    introv.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: A -> T j A
k <> j -> btg j f k = mret T k unfold btg.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: A -> T j A
k <> j ->
match k == j with
| left x =>
    fun x0 : A =>
    rew [fun H : K => T H A]
        btg_obligation_1 ix0 j k x in f x0
| right _ => mret T k
end = mret T k intro hyp.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
k, j: K
f: A -> T j A
hyp: k <> j
match k == j with
| left x =>
    fun x0 : A =>
    rew [fun H : K => T H A]
        btg_obligation_1 ix0 j k x in f x0
| right _ => mret T k
end = mret T k
    compare values k and j.
  Qed.

  Lemma btg_id (j: K):
    btg (A := A) j (mret T j) = mret T.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
btg j (mret T j) = mret T
  Proof.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
btg j (mret T j) = mret T
    unfold btg.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j: K
(fun k : K =>
 match k == j with
 | left x =>
     fun x0 : A =>
     rew [fun H : K => T H A]
         btg_obligation_1 ix0 j k x in mret T j x0
 | right _ => mret T k
 end) = mret T ext k.
ix, ix0: Index
T: K -> Type -> Type
H: MReturn T
W, A: Type
j, k: K
match k == j with
| left x =>
    fun x0 : A =>
    rew [fun H : K => T H A]
        btg_obligation_1 ix0 j k x in mret T j x0
| right _ => mret T k
end = mret T k compare values k and j.
  Qed.

End btg_lemmas.

(** ** Rewrite Hint registration *)
(**********************************************************************)
#[export] Hint Rewrite @btg_eq @btg_id @btgd_eq @btgd_id: tea_tgt.
#[export] Hint Rewrite @tgtd_eq @tgtd_eq @tgtd_id: tea_tgt.
#[export] Hint Rewrite @btgd_neq @btg_neq using auto: tea_tgt.

#[export] Hint Rewrite @btgd_eq @btg_eq @btg_id @btgd_id: tea_tgt_eq.
#[export] Hint Rewrite @tgtd_eq @tgt_eq @tgt_id: tea_tgt_eq.
#[export] Hint Rewrite @btgd_neq @btg_neq using auto: tea_tgt_neq.
#[export] Hint Rewrite @tgtd_neq: tea_tgt_neq.

(** ** Derived targeted DTM operations *)
(**********************************************************************)
Section DTM_targeted.

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

  (** *** Definitions *)
  (* For now we ignore traversals because we don't need them for
     System F. *)
  (********************************************************************)
  Definition kbindd {A} `(f: W * A -> T j A): U A -> U A
    := mbindd U (btgd j f).

  Definition kbind `(f: A -> T j A): U A -> U A
    := mbind U (btg j f).

  Definition kmapd `(f: W * A -> A): U A -> U A :=
    mmapd U (tgtd j f).

  Definition kmap `(f: A -> A): U A -> U A :=
    mmap U (tgt j f).

  Section traversals.

    Context `{Applicative G}.

    Definition tgtdt
      {A} (k: K) (f: W * A -> G A): W * A -k-> G A :=
      fun j '(w, a) => if k == j then f (w, a) else pure a.

    Definition tgtdt_def
      {A B} (k: K) (f def: W * A -> G B): W * A -k-> G B :=
      fun j => if k == j then f else def.

    Definition tgtt {A} (k: K) (f: A -> G A): A -k-> G A :=
      fun j => if k == j then f else pure.

    Definition tgtt_def {A B} (k: K) (f def: A -> G B): A -k-> G B :=
      fun j => if k == j then f else def.

    Definition kmapdt `(f: W * A -> G A): U A -> G (U A) :=
      mmapdt U G (tgtdt j f).

    Definition ktraverse `(f: A -> G A): U A -> G (U A) :=
      mmapt U G (tgtt j f).

    Lemma kmapdt_to_mmapdt `(f: W * A -> G A):
      kmapdt f = mmapdt U G (tgtdt j f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
A: Type
f: W * A -> G A
kmapdt f = mmapdt U G (tgtdt j f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
A: Type
f: W * A -> G A
kmapdt f = mmapdt U G (tgtdt j f)
      reflexivity.
    Qed.

    Lemma kmapt_to_mtraverse `(f: A -> G A):
      ktraverse f = mmapt U G (tgtt j f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
A: Type
f: A -> G A
ktraverse f = mmapt U G (tgtt j f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
A: Type
f: A -> G A
ktraverse f = mmapt U G (tgtt j f)
      reflexivity.
    Qed.

  End traversals.

  Section special_cases.

    Context
      {A: Type}.

    (** *** Rewriting rules for special cases of <<kbindd>> *)
    (******************************************************************)
    Lemma kbind_to_kbindd (f: A -> T j A):
      kbind f = kbindd (f ∘ extract (W := (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
j: K
A: Type
f: A -> T j A
kbind f = kbindd (f ∘ extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
kbind f = kbindd (f ∘ extract)
      unfold kbind, kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
mbind U (btg j f) = mbindd U (btgd j (f ∘ extract)) 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
j: K
A: Type
f: A -> T j A
mbindd U (btg j f ◻ allK extract) =
mbindd U (btgd j (f ∘ extract))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
btg j f ◻ allK extract = btgd j (f ∘ extract) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
k: K
w: W
a: A
(btg j f ◻ allK extract) k (w, a) =
btgd j (f ∘ extract) k (w, a)
      unfold vec_compose, compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
k: K
w: W
a: A
btg j f k a =
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in 
    f a
| right _ => mret T k a
end
      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
j: K
A: Type
f: A -> T j A
w: W
a: A
DESTR_EQs: j = j
btg j f j a =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
f 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
j: K
A: Type
f: A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
btg j f k a = mret T 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
j: K
A: Type
f: A -> T j A
w: W
a: A
DESTR_EQs: j = j
btg j f j a =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
f a autorewrite  with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
w: W
a: A
DESTR_EQs: j = j
f a =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
f 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
j: K
A: Type
f: A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
btg j f k a = mret T k a autorewrite  with tea_tgt_neq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mret T k a = mret T k a easy.
    Qed.

    Lemma kmapd_to_kbindd (f: W * A -> A):
      kmapd f = kbindd (mret T j ∘ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: W * A -> A
kmapd f = kbindd (mret T j ∘ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: W * A -> A
kmapd f = kbindd (mret T j ∘ f)
      unfold kmapd, kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: W * A -> A
mmapd U (tgtd j f) = mbindd U (btgd j (mret T j ∘ f)) rewrite mmapd_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: W * A -> A
mbindd U (mret T ◻ tgtd j f) =
mbindd U (btgd j (mret T j ∘ f))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: W * A -> A
mret T ◻ tgtd j f = btgd j (mret T j ∘ f) 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
j: K
A: Type
f: W * A -> A
k: K
w: W
a: A
(mret T ◻ tgtd j f) k (w, a) =
btgd j (mret T j ∘ f) k (w, a)
      unfold vec_compose, 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
j: K
A: Type
f: W * A -> A
k: K
w: W
a: A
mret T k (tgtd j f k (w, a)) =
btgd j (mret T j ○ f) 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
j: K
A: Type
f: W * A -> A
k: K
w: W
a: A
mret T k (if k == j then f (w, a) else a) =
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in
    mret T j (f (w, a))
| right _ => mret T k a
end compare values k and j.
    Qed.

    Lemma kmap_to_kbindd (f: A -> A):
      kmap f = kbindd (mret T j ∘ f ∘ extract (W := (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
j: K
A: Type
f: A -> A
kmap f = kbindd (mret T j ∘ f ∘ extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
kmap f = kbindd (mret T j ∘ f ∘ extract)
      unfold kmap, kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mmap U (tgt j f) =
mbindd U (btgd j (mret T j ∘ f ∘ extract)) rewrite mmap_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
j: K
A: Type
f: A -> A
mbindd U ((mret T ◻ tgt j f) ◻ allK extract) =
mbindd U (btgd j (mret T j ∘ f ∘ extract))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
(mret T ◻ tgt j f) ◻ allK extract =
btgd j (mret T j ∘ f ∘ extract) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
w: W
a: A
((mret T ◻ tgt j f) ◻ allK extract) k (w, a) =
btgd j (mret T j ∘ f ∘ extract) k (w, a)
      unfold vec_compose, 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
j: K
A: Type
f: A -> A
k: K
w: W
a: A
mret T k (tgt j f k (allK extract k (w, a))) =
btgd j (fun a : W * A => mret T j (f (extract a))) 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
j: K
A: Type
f: A -> A
k: K
w: W
a: A
mret T k (tgt j f k a) =
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in
    mret T j (f a)
| right _ => mret T k a
end
      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
j: K
A: Type
f: A -> A
w: W
a: A
DESTR_EQs: j = j
mret T j (tgt j f j a) =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
mret T j (f 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
j: K
A: Type
f: A -> A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mret T k (tgt j f k a) = mret T 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
j: K
A: Type
f: A -> A
w: W
a: A
DESTR_EQs: j = j
mret T j (tgt j f j a) = mret T j (f 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
j: K
A: Type
f: A -> A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mret T k (tgt j f k a) = mret T k a
      now autorewrite with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mret T k (tgt j f k a) = mret T k a
      now autorewrite with tea_tgt_neq.
    Qed.

    (** *** Rewriting rules for special cases of <<kmapd>> *)
    (******************************************************************)
    Lemma kmap_to_kmapd (f: A -> A):
      kmap f = kmapd (f ∘ extract (W := (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
j: K
A: Type
f: A -> A
kmap f = kmapd (f ∘ extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
kmap f = kmapd (f ∘ extract)
      unfold kmap, kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mmap U (tgt j f) = kmapd (f ∘ extract)
      unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mmap U (tgt j f) = mmapd U (tgtd j (f ∘ extract)) rewrite mmap_to_mmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mmapd U (tgt j f ◻ allK extract) =
mmapd U (tgtd j (f ∘ extract))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
tgt j f ◻ allK extract = tgtd j (f ∘ extract) ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
(tgt j f ◻ allK extract) k = tgtd j (f ∘ extract) k
      unfold vec_compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
tgt j f k ∘ allK extract k = tgtd j (f ∘ extract) k
      compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A: Type
f: A -> A
k: K
DESTR_EQs: k = k
tgt k f k ∘ allK extract k = tgtd k (f ∘ extract) k
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgt j f k ∘ allK extract k = tgtd j (f ∘ extract) k
      now autorewrite with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgt j f k ∘ allK extract k = tgtd j (f ∘ extract) k
      now autorewrite with tea_tgt_neq.
    Qed.

    (** *** Rewriting rules for special cases of <<kbind>> *)
    (******************************************************************)
    Lemma kmap_to_kbind (f: A -> A):
      kmap f = kbind (mret T j ∘ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
kmap f = kbind (mret T j ∘ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
kmap f = kbind (mret T j ∘ f)
      unfold kmap, kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mmap U (tgt j f) = mbind U (btg j (mret T j ∘ f))
      rewrite mmap_to_mbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mbind U (mret T ◻ tgt j f) =
mbind U (btg j (mret T j ∘ f))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
mret T ◻ tgt j f = btg j (mret T j ∘ f) ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
(mret T ◻ tgt j f) k = btg j (mret T j ∘ f) k
      unfold vec_compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
mret T k ∘ tgt j f k = btg j (mret T j ∘ f) k
      compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A: Type
f: A -> A
k: K
DESTR_EQs: k = k
mret T k ∘ tgt k f k = btg k (mret T k ∘ f) k
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mret T k ∘ tgt j f k = btg j (mret T j ∘ f) k
      now autorewrite with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
j: K
A: Type
f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mret T k ∘ tgt j f k = btg j (mret T j ∘ f) k
      now autorewrite with tea_tgt_neq.
    Qed.

  End special_cases.

End DTM_targeted.

(** ** Decorated monad (<<kbindd>>) *)
(**********************************************************************)

Definition compose_kdm
  `{ix: Index}
  {W: Type}
  {T: K -> Type -> Type}
  `{mn_op: Monoid_op W}
  `{mn_unit: Monoid_unit W}
  `{forall k, MBind W T (T k)}
  `{! MReturn T}
  {j: K}
  {A: Type}
  (g: W * A -> T j A)
  (f: W * A -> T j A): W * A -> T j A :=
  fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f (w, a)).

Infix "⋆kdm" := compose_kdm (at level 40).

Section DecoratedMonad.

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

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem kbindd_id:
    kbindd U j (mret T j ∘ extract) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kbindd U j (mret T j ∘ extract) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kbindd U j (mret T j ∘ extract) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kbindd U j (mret T j ∘ extract) = id unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mbindd U (btgd j (mret T j ∘ extract)) = id rewrite <- (mbindd_id U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mbindd U (btgd j (mret T j ∘ extract)) =
mbindd U (mret T ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
btgd j (mret T j ∘ extract) = mret T ◻ allK extract ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
w: W
a: A
btgd j (mret T j ∘ extract) k (w, a) =
(mret T ◻ allK extract) k (w, a) cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
w: W
a: A
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in
    (mret T j ∘ extract) (w, a)
| right _ => mret T k a
end = (mret T ◻ allK extract) k (w, a) compare values k and j.
  Qed.

  Theorem kbindd_kbindd_eq: forall (g: W * A -> T j A) (f: W * A -> T j A),
      kbindd U j g ∘ kbindd U j f =
        kbindd U j (g ⋆kdm f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall g f : W * A -> T j A,
kbindd U j g ∘ kbindd U j f = kbindd U j (g ⋆kdm f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall g f : W * A -> T j A,
kbindd U j g ∘ kbindd U j f = kbindd U j (g ⋆kdm f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> T j A
kbindd U j g ∘ kbindd U j f = kbindd U j (g ⋆kdm f) unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> T j A
mbindd U (btgd j g) ∘ mbindd U (btgd j f) =
mbindd U (btgd j (g ⋆kdm f)) rewrite (mbindd_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> T j A
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (btgd j g ◻ allK (incr w))
     (btgd j f k (w, a))) =
mbindd U (btgd j (g ⋆kdm f))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> T j A
(fun (k : K) '(w, a) =>
 mbindd (T k) (btgd j g ◻ allK (incr w))
   (btgd j f k (w, a))) = btgd j (g ⋆kdm f) 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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
mbindd (T k) (btgd j g ◻ allK (incr w))
  (btgd j f k (w, a)) = btgd j (g ⋆kdm f) 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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
mbindd (T k) (btgd j g ◻ allK (incr w))
  match k == j with
  | left x =>
      rew [fun H : K => T H A]
          eq_ind_r (fun k : K => j = k) eq_refl x in
      f (w, a)
  | right _ => mret T k a
  end =
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in
    kbindd (T j) j (g ∘ incr w) (f (w, a))
| right _ => mret T k a
end 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
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
mbindd (T j) (btgd j g ◻ allK (incr w))
  (rew [fun H : K => T H A]
       eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
   f (w, a)) =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
kbindd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mbindd (T k) (btgd j g ◻ allK (incr w)) (mret T k a) =
mret T 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
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
mbindd (T j) (btgd j g ◻ allK (incr w))
  (rew [fun H : K => T H A]
       eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
   f (w, a)) =
rew [fun H : K => T H A]
    eq_ind_r (fun k : K => j = k) eq_refl eq_refl in
kbindd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
mbindd (T j) (btgd j g ◻ allK (incr w)) (f (w, a)) =
kbindd (T j) j (g ∘ incr w) (f (w, a)) unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
mbindd (T j) (btgd j g ◻ allK (incr w)) (f (w, a)) =
mbindd (T j) (btgd j (g ∘ incr w)) (f (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
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
btgd j g ◻ allK (incr w) = btgd j (g ∘ incr w) 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
j: K
A: Type
g, f: W * A -> T j A
w: W
a: A
DESTR_EQs: j = j
k: K
w': W
a': A
(btgd j g ◻ allK (incr w)) k (w', a') =
btgd j (g ∘ incr w) k (w', a')
      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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
mbindd (T k) (btgd j g ◻ allK (incr w)) (mret T k a) =
mret T 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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(mbindd (T k) (btgd j g ◻ allK (incr w)) ∘ mret T k) a =
mret T k a rewrite mbindd_comp_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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
((btgd j g ◻ allK (incr w)) k ∘ Monad.ret) a =
mret T 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
j: K
A: Type
g, f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
match k == j with
| left x =>
    rew [fun H : K => T H A]
        eq_ind_r (fun k : K => j = k) eq_refl x in
    g (w ● Ƶ, a)
| right _ => mret T k a
end = mret T k a compare values k and j.
  Qed.

  Theorem kbindd_kbindd_neq:
    forall {i: K} (Hneq: j <> i)
           (g: W * A -> T i A) (f: W * A -> T j A),
      kbindd U i g ∘ kbindd U j f =
        mbindd U (btgd i g ⋆dm btgd j f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall i : K,
j <> i ->
forall (g : W * A -> T i A) (f : W * A -> T j A),
kbindd U i g ∘ kbindd U j f =
mbindd U (btgd i g ⋆dm btgd j f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall i : K,
j <> i ->
forall (g : W * A -> T i A) (f : W * A -> T j A),
kbindd U i g ∘ kbindd U j f =
mbindd U (btgd i g ⋆dm btgd j f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
g: W * A -> T i A
f: W * A -> T j A
kbindd U i g ∘ kbindd U j f =
mbindd U (btgd i g ⋆dm btgd j f) unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
g: W * A -> T i A
f: W * A -> T j A
mbindd U (btgd i g) ∘ mbindd U (btgd j f) =
mbindd U (btgd i g ⋆dm btgd j f) now rewrite (mbindd_mbindd U).
  Qed.

  (** *** Right unit law for monads *)
  (********************************************************************)
  Theorem kbindd_comp_mret_eq: forall (f: W * A -> T j A) (a: A),
      kbindd (T j) j f (mret T j a) = f (Ƶ, a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : W * A -> T j A) (a : A),
kbindd (T j) j f (mret T j a) = f (Ƶ, a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : W * A -> T j A) (a : A),
kbindd (T j) j f (mret T j a) = f (Ƶ, a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
a: A
kbindd (T j) j f (mret T j a) = f (Ƶ, a) unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
a: A
mbindd (T j) (btgd j f) (mret T j a) = f (Ƶ, a) compose near a on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
a: A
(mbindd (T j) (btgd j f) ∘ mret T j) a = f (Ƶ, a)
    rewrite (mbindd_comp_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
j: K
A: Type
f: W * A -> T j A
a: A
(btgd j f j ∘ Monad.ret) a = f (Ƶ, a)
    now autorewrite with tea_tgt_eq.
  Qed.

  Theorem kbindd_comp_mret_neq:
    forall (i: K) (Hneq: j <> i)
           (f: W * A -> T j A) (a: A),
      kbindd (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (a : A),
kbindd (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (a : A),
kbindd (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
a: A
kbindd (T i) j f (mret T i a) = mret T i a unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
a: A
mbindd (T i) (btgd j f) (mret T i a) = mret T i 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
a: A
(mbindd (T i) (btgd j f) ∘ mret T i) a = mret T i a
    rewrite (mbindd_comp_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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
a: A
(btgd j f i ∘ Monad.ret) a = mret T i a
    now autorewrite with tea_tgt_neq.
  Qed.

  (** *** Composition with special cases *)
  (********************************************************************)
  Lemma kmapd_kbindd: forall
      (g: W * A -> A) (f: W * A -> T j A),
      kmapd U j g ∘ kbindd U j f =
        kbindd U j (fun '(w, a) => kmapd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
forall (g : W * A -> A) (f : W * A -> T j A),
kmapd U j g ∘ kbindd U j f =
kbindd U j
  (fun '(w, a) =>
   kmapd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
forall (g : W * A -> A) (f : W * A -> T j A),
kmapd U j g ∘ kbindd U j f =
kbindd U j
  (fun '(w, a) =>
   kmapd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
kmapd U j g ∘ kbindd U j f =
kbindd U j
  (fun '(w, a) =>
   kmapd (T j) j (g ∘ incr w) (f (w, a))) rewrite kmapd_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
kbindd U j (mret T j ∘ g) ∘ kbindd U j f =
kbindd U j
  (fun '(w, a) =>
   kmapd (T j) j (g ∘ incr w) (f (w, a)))
    rewrite kbindd_kbindd_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
kbindd U j ((mret T j ∘ g) ⋆kdm f) =
kbindd U j
  (fun '(w, a) =>
   kmapd (T j) j (g ∘ incr w) (f (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
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
(mret T j ∘ g) ⋆kdm f =
(fun '(w, a) => kmapd (T j) j (g ∘ incr w) (f (w, a)))
    unfold compose_kdm.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
(fun '(w, a) =>
 kbindd (T j) j (mret T j ∘ g ∘ incr w) (f (w, a))) =
(fun '(w, a) => kmapd (T j) j (g ∘ incr w) (f (w, a))) ext [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
j: K
A: Type
g: W * A -> A
f: W * A -> T j A
w: W
a: A
kbindd (T j) j (mret T j ∘ g ∘ incr w) (f (w, a)) =
kmapd (T j) j (g ∘ incr w) (f (w, a))
    now rewrite kmapd_to_kbindd.
  Qed.

  Lemma kbind_kbindd: forall
      (g: A -> T j A) (f: W * A -> T j A),
      kbind U j g ∘ kbindd U j f = kbindd U j (kbind (T j) j g ∘ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (g : A -> T j A) (f : W * A -> T j A),
kbind U j g ∘ kbindd U j f =
kbindd U j (kbind (T j) j g ∘ f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (g : A -> T j A) (f : W * A -> T j A),
kbind U j g ∘ kbindd U j f =
kbindd U j (kbind (T j) j g ∘ f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
kbind U j g ∘ kbindd U j f =
kbindd U j (kbind (T j) j g ∘ f) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
kbindd U j (g ∘ extract) ∘ kbindd U j f =
kbindd U j (kbind (T j) j g ∘ f) rewrite kbindd_kbindd_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
kbindd U j ((g ∘ extract) ⋆kdm f) =
kbindd U j (kbind (T j) j g ∘ f)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
(g ∘ extract) ⋆kdm f = kbind (T j) j g ∘ f unfold compose_kdm.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
(fun '(w, a) =>
 kbindd (T j) j (g ∘ extract ∘ incr w) (f (w, a))) =
kbind (T j) j g ∘ f ext [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
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
w: W
a: A
kbindd (T j) j (g ∘ extract ∘ incr w) (f (w, a)) =
(kbind (T j) j g ∘ f) (w, a)
    reassociate ->.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
w: W
a: A
kbindd (T j) j (g ∘ (extract ∘ incr w)) (f (w, a)) =
(kbind (T j) j g ∘ f) (w, a) rewrite extract_incr.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> T j A
f: W * A -> T j A
w: W
a: A
kbindd (T j) j (g ∘ extract) (f (w, a)) =
(kbind (T j) j g ∘ f) (w, a) now rewrite kbind_to_kbindd.
  Qed.

  Lemma kmap_kbindd: forall
      (g: A -> A) (f: W * A -> T j A),
      kmap U j g ∘ kbindd U j f =
        kbindd U j (fun '(w, a) => kmap (T j) j g (f (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
j: K
A: Type
forall (g : A -> A) (f : W * A -> T j A),
kmap U j g ∘ kbindd U j f =
kbindd U j (fun '(w, a) => kmap (T j) j g (f (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
j: K
A: Type
forall (g : A -> A) (f : W * A -> T j A),
kmap U j g ∘ kbindd U j f =
kbindd U j (fun '(w, a) => kmap (T j) j g (f (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
j: K
A: Type
g: A -> A
f: W * A -> T j A
kmap U j g ∘ kbindd U j f =
kbindd U j (fun '(w, a) => kmap (T j) j g (f (w, a))) unfold kmap, kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
mmap U (tgt j g) ∘ mbindd U (btgd j f) =
mbindd U
  (btgd j
     (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a)))) rewrite mmap_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
j: K
A: Type
g: A -> A
f: W * A -> T j A
mbindd U ((mret T ◻ tgt j g) ◻ allK extract)
∘ mbindd U (btgd j f) =
mbindd U
  (btgd j
     (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a))))
    rewrite (mbindd_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k)
     (((mret T ◻ tgt j g) ◻ allK extract)
      ◻ allK (incr w)) (btgd j f k (w, a))) =
mbindd U
  (btgd j
     (fun '(w, a) => mmap (T j) (tgt j g) (f (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
j: K
A: Type
g: A -> A
f: W * A -> T j A
(fun (k : K) '(w, a) =>
 mbindd (T k)
   (((mret T ◻ tgt j g) ◻ allK extract)
    ◻ allK (incr w)) (btgd j f k (w, a))) =
btgd j
  (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a))) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
mbindd (T k)
  (((mret T ◻ tgt j g) ◻ allK extract) ◻ allK (incr w))
  (btgd j f k (w, a)) =
btgd j
  (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a))) k
  (w, a)
    compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
mbindd (T k)
  (((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  (btgd k f k (w, a)) =
btgd k
  (fun '(w, a) => mmap (T k) (tgt k g) (f (w, a))) 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
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbindd (T k)
  (((mret T ◻ tgt j g) ◻ allK extract) ◻ allK (incr w))
  (btgd j f k (w, a)) =
btgd j
  (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a))) 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: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
mbindd (T k)
  (((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  (btgd k f k (w, a)) =
btgd k
  (fun '(w, a) => mmap (T k) (tgt k g) (f (w, a))) k
  (w, a) autorewrite with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind 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
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
mbindd (T k)
  (((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  (f (w, a)) = mmap (T k) (tgt k g) (f (w, a))
      rewrite mmap_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
mbindd (T k)
  (((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  (f (w, a)) =
mbindd (T k) ((mret T ◻ tgt k g) ◻ allK extract)
  (f (w, a)) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w) =
(mret T ◻ tgt k g) ◻ allK extract
      ext k' [w' a'].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
k': K
w': W
a': A
(((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  k' (w', a') =
((mret T ◻ tgt k g) ◻ allK extract) k' (w', a') unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g: A -> A
k: K
f: W * A -> T k A
w: W
a: A
DESTR_EQs: k = k
k': K
w': W
a': A
(((mret T ◻ tgt k g) ◻ allK extract) ◻ allK (incr w))
  k' (w', a') =
((mret T ◻ tgt k g) ◻ allK extract) k' (w', a') reflexivity.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbindd (T k)
  (((mret T ◻ tgt j g) ◻ allK extract) ◻ allK (incr w))
  (btgd j f k (w, a)) =
btgd j
  (fun '(w, a) => mmap (T j) (tgt j g) (f (w, a))) k
  (w, a) autorewrite with tea_tgt_neq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbindd (T k)
  (((mret T ◻ tgt j g) ◻ allK extract) ◻ allK (incr w))
  ((mret T k ∘ extract) (w, a)) =
(mret T k ∘ extract) (w, a)
      unfold vec_compose, compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbindd (T k)
  (fun (k : K) (a : W * A) =>
   mret T k
     (tgt j g k (allK extract k (allK (incr w) k a))))
  (mret T k a) = mret T 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
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
(mbindd (T k)
   (fun (k : K) (a : W * A) =>
    mret T k
      (tgt j g k (allK extract k (allK (incr w) k a))))
 ∘ mret T k) a = mret T k a
      rewrite (mbindd_comp_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
j: K
A: Type
g: A -> A
f: W * A -> T j A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
((fun a : W * A =>
  mret T k
    (tgt j g k (allK extract k (allK (incr w) k a))))
 ∘ Monad.ret) a = mret T k a
      rewrite tgt_neq; auto.
  Qed.

  Lemma kbindd_kmapd: forall
      (g: W * A -> T j A) (f: W * A -> A),
      kbindd U j g ∘ kmapd U j f =
        kbindd U j (fun '(w, a) => g (w, f (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
j: K
A: Type
forall (g : W * A -> T j A) (f : W * A -> A),
kbindd U j g ∘ kmapd U j f =
kbindd U j (fun '(w, a) => g (w, f (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
j: K
A: Type
forall (g : W * A -> T j A) (f : W * A -> A),
kbindd U j g ∘ kmapd U j f =
kbindd U j (fun '(w, a) => g (w, f (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
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
kbindd U j g ∘ kmapd U j f =
kbindd U j (fun '(w, a) => g (w, f (w, a))) rewrite kmapd_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
kbindd U j g ∘ kbindd U j (mret T j ∘ f) =
kbindd U j (fun '(w, a) => g (w, f (w, a)))
    rewrite kbindd_kbindd_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
kbindd U j (g ⋆kdm (mret T j ∘ f)) =
kbindd U j (fun '(w, a) => g (w, f (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
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
g ⋆kdm (mret T j ∘ f) =
(fun '(w, a) => g (w, f (w, a)))
    ext (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
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
w: W
a: A
(g ⋆kdm (mret T j ∘ f)) (w, a) = g (w, f (w, a)) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
w: W
a: A
kbindd (T j) j (g ∘ incr w) (mret T j (f (w, a))) =
g (w, f (w, a))
    rewrite kbindd_comp_mret_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
w: W
a: A
(g ∘ incr w) (Ƶ, f (w, a)) = g (w, f (w, a)) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: W * A -> A
w: W
a: A
g (w ● Ƶ, f (w, a)) = g (w, f (w, a))
    now simpl_monoid.
  Qed.

  Lemma kbindd_kbind: forall
      (g: W * A -> T j A) (f: A -> T j A),
      kbindd U j g ∘ kbind U j f =
        kbindd U j (fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f 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
j: K
A: Type
forall (g : W * A -> T j A) (f : A -> T j A),
kbindd U j g ∘ kbind U j f =
kbindd U j
  (fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f 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
j: K
A: Type
forall (g : W * A -> T j A) (f : A -> T j A),
kbindd U j g ∘ kbind U j f =
kbindd U j
  (fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f 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
j: K
A: Type
g: W * A -> T j A
f: A -> T j A
kbindd U j g ∘ kbind U j f =
kbindd U j
  (fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f a)) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g: W * A -> T j A
f: A -> T j A
kbindd U j g ∘ kbindd U j (f ∘ extract) =
kbindd U j
  (fun '(w, a) => kbindd (T j) j (g ∘ incr w) (f a)) now rewrite kbindd_kbindd_eq.
  Qed.

  (* TODO <<kbindd_kmap>> *)

End DecoratedMonad.

(** ** Mixed structure composition laws *)
(** Composition laws involving <<kbind>> and <<kmapd>> *)
(**********************************************************************)

(* TODO <<kbind_kmapd>> *)

(* TODO <<kmapd_kbind>> *)

(** ** Decorated functors (<<kmapd>>) *)
(**********************************************************************)
Section DecoratedFunctor.

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

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem kmapd_id: forall A,
      kmapd U j extract = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kmapd U j extract = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kmapd U j extract = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kmapd U j extract = id unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mmapd U (tgtd j extract) = id
    rewrite <- (mmapd_id U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mmapd U (tgtd j extract) = mmapd U (allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
tgtd j extract = allK extract ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
tgtd j extract k = allK extract k compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
DESTR_EQs: k = k
tgtd k extract k = allK extract k
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgtd j extract k = allK extract k
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
DESTR_EQs: k = k
tgtd k extract k = allK extract k now autorewrite with tea_tgt.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgtd j extract k = allK extract k now autorewrite with tea_tgt.
  Qed.

  Theorem kmapd_kmapd: forall A,
    forall (g: W * A -> A) (f: W * A -> A),
      kmapd U j g ∘ kmapd U j f =
        kmapd U j (fun '(w, a) => g (w, f (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
j: K
forall (A : Type) (g f : W * A -> A),
kmapd U j g ∘ kmapd U j f =
kmapd U j (fun '(w, a) => g (w, f (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
j: K
forall (A : Type) (g f : W * A -> A),
kmapd U j g ∘ kmapd U j f =
kmapd U j (fun '(w, a) => g (w, f (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
j: K
A: Type
g, f: W * A -> A
kmapd U j g ∘ kmapd U j f =
kmapd U j (fun '(w, a) => g (w, f (w, a))) unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> A
mmapd U (tgtd j g) ∘ mmapd U (tgtd j f) =
mmapd U (tgtd j (fun '(w, a) => g (w, f (w, a))))
    rewrite (mmapd_mmapd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> A
mmapd U
  (fun (k : K) '(w, a) =>
   tgtd j g k (w, tgtd j f k (w, a))) =
mmapd U (tgtd j (fun '(w, a) => g (w, f (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
j: K
A: Type
g, f: W * A -> A
(fun (k : K) '(w, a) =>
 tgtd j g k (w, tgtd j f k (w, a))) =
tgtd j (fun '(w, a) => g (w, f (w, a)))
    ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> A
k: K
w: W
a: A
tgtd j g k (w, tgtd j f k (w, a)) =
tgtd j (fun '(w, a) => g (w, f (w, a))) k (w, a) compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g, f: W * A -> A
k: K
w: W
a: A
DESTR_EQs: k = k
tgtd k g k (w, tgtd k f k (w, a)) =
tgtd k (fun '(w, a) => g (w, f (w, a))) 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
j: K
A: Type
g, f: W * A -> A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgtd j g k (w, tgtd j f k (w, a)) =
tgtd j (fun '(w, a) => g (w, f (w, a))) 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: Type
g, f: W * A -> A
k: K
w: W
a: A
DESTR_EQs: k = k
tgtd k g k (w, tgtd k f k (w, a)) =
tgtd k (fun '(w, a) => g (w, f (w, a))) k (w, a) now autorewrite with tea_tgt.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: W * A -> A
k: K
w: W
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgtd j g k (w, tgtd j f k (w, a)) =
tgtd j (fun '(w, a) => g (w, f (w, a))) k (w, a) now autorewrite with tea_tgt_neq.
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma kmapd_comp_mret_eq: forall A,
    forall (f: W * A -> A) (a: A),
      kmapd (T j) j f (mret T j a) = mret T j (f (Ƶ, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (f : W * A -> A) (a : A),
kmapd (T j) j f (mret T j a) = mret T j (f (Ƶ, a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (f : W * A -> A) (a : A),
kmapd (T j) j f (mret T j a) = mret T j (f (Ƶ, a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> A
a: A
kmapd (T j) j f (mret T j a) = mret T j (f (Ƶ, a)) unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> A
a: A
mmapd (T j) (tgtd j f) (mret T j a) =
mret T j (f (Ƶ, a)) rewrite mmapd_comp_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
j: K
A: Type
f: W * A -> A
a: A
mret T j (tgtd j f j (Ƶ, a)) = mret T j (f (Ƶ, a))
    now autorewrite with tea_tgt.
  Qed.

  Lemma kmapd_comp_mret_neq: forall A,
    forall (k: K) (neq: k <> j) (f: W * A -> A) (a: A),
      kmapd (T k) j f (mret T k a) = mret T 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
j: K
forall (A : Type) (k : K),
k <> j ->
forall (f : W * A -> A) (a : A),
kmapd (T k) j f (mret T k a) = mret T 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
j: K
forall (A : Type) (k : K),
k <> j ->
forall (f : W * A -> A) (a : A),
kmapd (T k) j f (mret T k a) = mret T 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
j: K
A: Type
k: K
neq: k <> j
f: W * A -> A
a: A
kmapd (T k) j f (mret T k a) = mret T k a unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
neq: k <> j
f: W * A -> A
a: A
mmapd (T k) (tgtd j f) (mret T k a) = mret T k a rewrite mmapd_comp_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
j: K
A: Type
k: K
neq: k <> j
f: W * A -> A
a: A
mret T k (tgtd j f k (Ƶ, a)) = mret T k a
    now autorewrite with tea_tgt_neq.
  Qed.

  (* TODO <<kmap_kmapd>> *)

  (* TODO <<kmapd_kmap>> *)

End DecoratedFunctor.

(** ** Monads (<<kbind>>) *)
(**********************************************************************)
Definition compose_km
  `{ix: Index}
  {W: Type}
  {T: K -> Type -> Type}
  `{forall k, MBind W T (T k)}
  `{! MReturn T}
  {j: K}
  {A: Type}
  (g: A -> T j A)
  (f: A -> T j A): A -> T j A :=
  (kbind (T j) j g ∘ f).

Infix "⋆km" := compose_km (at level 40).

Section Monad.

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

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem kbind_id: forall A,
      kbind U j (mret T j) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kbind U j (mret T j) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kbind U j (mret T j) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kbind U j (mret T j) = id unfold kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mbind U (btg j (mret T j)) = id
    rewrite <- (mbind_id U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mbind U (btg j (mret T j)) =
mbind U (fun k : K => mret T k) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
btg j (mret T j) = (fun k : K => mret T k)
    ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
btg j (mret T j) k = mret T k compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
DESTR_EQs: k = k
btg k (mret T k) k = mret T k
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
btg j (mret T j) k = mret T k
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
DESTR_EQs: k = k
btg k (mret T k) k = mret T k now autorewrite with tea_tgt_eq.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
btg j (mret T j) k = mret T k now autorewrite with tea_tgt_neq.
  Qed.

  Theorem kbind_kbind: forall A,
    forall (g f: A -> T j A),
      kbind U j g ∘ kbind U j f =
        kbind U j (g ⋆km f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (g f : A -> T j A),
kbind U j g ∘ kbind U j f = kbind U j (g ⋆km f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (g f : A -> T j A),
kbind U j g ∘ kbind U j f = kbind U j (g ⋆km f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
kbind U j g ∘ kbind U j f = kbind U j (g ⋆km f) unfold kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
mbind U (btg j g) ∘ mbind U (btg j f) =
mbind U (btg j (g ⋆km f))
    rewrite (mbind_mbind 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
j: K
A: Type
g, f: A -> T j A
mbind U
  (fun k : K => mbind (T k) (btg j g) ○ btg j f k) =
mbind U (btg j (g ⋆km f)) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
(fun k : K => mbind (T k) (btg j g) ○ btg j f k) =
btg j (g ⋆km f)
    ext 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
j: K
A: Type
g, f: A -> T j A
k: K
a: A
mbind (T k) (btg j g) (btg j f k a) =
btg j (g ⋆km f) k a compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
f, g: A -> T k A
a: A
DESTR_EQs: k = k
mbind (T k) (btg k g) (btg k f k a) =
btg k (g ⋆km 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
j: K
A: Type
g, f: A -> T j A
k: K
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbind (T k) (btg j g) (btg j f k a) =
btg j (g ⋆km f) k a
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
k: K
f, g: A -> T k A
a: A
DESTR_EQs: k = k
mbind (T k) (btg k g) (btg k f k a) =
btg k (g ⋆km f) k a now autorewrite with tea_tgt_eq.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
k: K
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbind (T k) (btg j g) (btg j f k a) =
btg j (g ⋆km f) k a autorewrite with tea_tgt_neq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
k: K
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
mbind (T k) (btg j g) (mret T k a) = mret T k a
      rewrite (mbind_comp_mret k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> T j A
k: K
a: A
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
btg j g k a = mret T k a
      now autorewrite with tea_tgt_neq.
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma kbind_comp_mret_eq: forall A,
    forall (f: A -> T j A) (a: A),
      kbind (T j) j f (mret T j a) = f 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
j: K
forall (A : Type) (f : A -> T j A) (a : A),
kbind (T j) j f (mret T j a) = f 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
j: K
forall (A : Type) (f : A -> T j A) (a : A),
kbind (T j) j f (mret T j a) = f 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
j: K
A: Type
f: A -> T j A
a: A
kbind (T j) j f (mret T j a) = f a unfold kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> T j A
a: A
mbind (T j) (btg j f) (mret T j a) = f a rewrite mbind_comp_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
j: K
A: Type
f: A -> T j A
a: A
btg j f j a = f a
    now autorewrite with tea_tgt_eq.
  Qed.

  Lemma kbind_comp_mret_neq: forall A,
    forall (i: K) (Hneq: j <> i) (f: A -> T j A) (a: A),
      kbind (T i) j f (mret T i a) = mret T i 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
j: K
forall (A : Type) (i : K),
j <> i ->
forall (f : A -> T j A) (a : A),
kbind (T i) j f (mret T i a) = mret T i 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
j: K
forall (A : Type) (i : K),
j <> i ->
forall (f : A -> T j A) (a : A),
kbind (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
a: A
kbind (T i) j f (mret T i a) = mret T i a unfold kbind.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
a: A
mbind (T i) (btg j f) (mret T i a) = mret T i a rewrite mbind_comp_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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
a: A
btg j f i a = mret T i a
    now autorewrite with tea_tgt_neq.
  Qed.

  (* TODO <<kmap_kbind>> *)

  (* TODO <<kbind_kmap>> *)

End Monad.

(** ** Functors (<<kmap>>) *)
(**********************************************************************)
Section Functor.

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

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem kmap_id: forall A,
      kmap U j (@id A) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kmap U j id = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall A : Type, kmap U j id = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
kmap U j id = id unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mmap U (tgt j id) = id
    rewrite <- (mmap_id U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
mmap U (tgt j id) = mmap U (allK id)
    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
j: K
A: Type
tgt j id = allK id ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
tgt j id k = allK id k 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
j: K
A: Type
DESTR_EQs: j = j
tgt j id j = allK id 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
j: K
A: Type
k: K
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
tgt j id k = allK id k
    now autorewrite with tea_tgt_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
k: K
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
tgt j id k = allK id k
    now autorewrite with tea_tgt_neq.
  Qed.

  Theorem kmap_kmap: forall (A: Type) (g f: A -> A),
      kmap U j g ∘ kmap U j f = kmap U j (g ∘ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (g f : A -> A),
kmap U j g ∘ kmap U j f = kmap U j (g ∘ f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
forall (A : Type) (g f : A -> A),
kmap U j g ∘ kmap U j f = kmap U j (g ∘ f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
kmap U j g ∘ kmap U j f = kmap U j (g ∘ f) unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
mmap U (tgt j g) ∘ mmap U (tgt j f) =
mmap U (tgt j (g ∘ f))
    rewrite (mmap_mmap 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
j: K
A: Type
g, f: A -> A
mmap U (tgt j g ◻ tgt j f) = mmap U (tgt j (g ∘ f)) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
tgt j g ◻ tgt j f = tgt j (g ∘ f)
    ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
k: K
(tgt j g ◻ tgt j f) k = tgt j (g ∘ f) k
    rewrite vec_compose_k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
k: K
tgt j g k ∘ tgt j f k = tgt j (g ∘ f) k
    compare values j and k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g, f: A -> A
k: K
DESTR_EQs: k = k
tgt k g k ∘ tgt k f k = tgt k (g ∘ f) k
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgt j g k ∘ tgt j f k = tgt j (g ∘ f) k
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
g, f: A -> A
k: K
DESTR_EQs: k = k
tgt k g k ∘ tgt k f k = tgt k (g ∘ f) k now autorewrite with tea_tgt_eq.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
g, f: A -> A
k: K
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
tgt j g k ∘ tgt j f k = tgt j (g ∘ f) k now autorewrite with tea_tgt_neq.
  Qed.

  (** *** Naturality w.r.t. <<mret>> *)
  (********************************************************************)
  Lemma kmap_comp_kret_eq {A}:
    forall (f: A -> A) (a: A),
      kmap (T j) j f (mret T j a) = mret T j (f 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
j: K
A: Type
forall (f : A -> A) (a : A),
kmap (T j) j f (mret T j a) = mret T j (f 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
j: K
A: Type
forall (f : A -> A) (a : A),
kmap (T j) j f (mret T j a) = mret T j (f 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
j: K
A: Type
f: A -> A
a: A
kmap (T j) j f (mret T j a) = mret T j (f a) unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> A
a: A
mmap (T j) (tgt j f) (mret T j a) = mret T j (f a) rewrite mmap_comp_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
j: K
A: Type
f: A -> A
a: A
mret T j (tgt j f j a) = mret T j (f a)
    now rewrite tgt_eq.
  Qed.

  Lemma kmap_comp_kret_neq {A}:
    forall (i: K) (Hneq: j <> i) (f: A -> A) (a: A),
      kmap (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (a : A),
kmap (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (a : A),
kmap (T i) j f (mret T i a) = mret T i 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
a: A
kmap (T i) j f (mret T i a) = mret T i a unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
a: A
mmap (T i) (tgt j f) (mret T i a) = mret T i a rewrite mmap_comp_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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
a: A
mret T i (tgt j f i a) = mret T i a
    rewrite tgt_neq; auto.
  Qed.

End Functor.

(** ** Notations **)
(**********************************************************************)
Module Notations.
  Infix "⋆dtm" := compose_dtm (at level 40): tealeaves_scope.
  Infix "⋆kdm" := compose_kdm (at level 40): tealeaves_scope.
  Infix "⋆km" := compose_km (at level 40): tealeaves_scope.
End Notations.

Import Container.Notations.

(** * Characterizing occurrences post-operation (targetted operations) *)
(**********************************************************************)
Section DTM_membership_targetted.

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

  Context
    (j: K)
    {A: Type}.

  (** *** Occurrences in <<kbindd>> with context *)
  (********************************************************************)
  Lemma inmd_kbindd_eq_iff1:
    forall `(f: W * A -> T j A) (t: U A) (wtotal: W) (a2: A),
      (wtotal, (j, a2)) ∈md kbindd U j f t ->
      exists (w1 w2: W) (a1: A),
        (w1, (j, a1)) ∈md t /\ (w2, (j, a2)) ∈md f (w1, a1)
        /\ 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbindd U j f t ->
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbindd U j f t ->
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (j, a2)) ∈md kbindd U j f t
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2 unfold kbindd 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (j, a2)) ∈md mbindd U (btgd j f) t
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2
    apply (inmd_mbindd_iff1 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (j, a2)) ∈md btgd j f k1 (w1, a) /\
  wtotal = w1 ● w2
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2
    destruct hyp 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md btgd j f k1 (w1, a)
hyp3: wtotal = w1 ● w2
exists (w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md btgd j f k1 (w1, a)
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
    compare values j and k1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind 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
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
exists (w3 w4 : W) (a1 : A),
  (w3, (k1, a1)) ∈md t /\
  (w4, (k1, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md btgd j f k1 (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) 
(a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
    +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind 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
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
exists (w3 w4 : W) (a1 : A),
  (w3, (k1, a1)) ∈md t /\
  (w4, (k1, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4 exists 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1, (k1, a)) ∈md t /\
(w2, (k1, a2)) ∈md f (w1, a) /\ w1 ● w2 = w1 ● 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1, (k1, 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w2, (k1, a2)) ∈md f (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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
w1 ● w2 = 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1, (k1, a)) ∈md t 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
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w2, (k1, a2)) ∈md f (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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
w1 ● w2 = 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w2, (k1, a2)) ∈md f (w1, a) rewrite btgd_eq 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
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md f (w1, a)
DESTR_EQs: k1 = k1
(w2, (k1, a2)) ∈md f (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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
w1 ● w2 = 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: Type
k1: K
f: W * A -> T k1 A
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (k1, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
w1 ● w2 = w1 ● w2 reflexivity. }
    +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md btgd j f k1 (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4 rewrite btgd_neq in hyp2; 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md (mret T k1 ∘ extract) (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
      unfold compose in hyp2; 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (j, a2)) ∈md mret T k1 a
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
      rewrite inmd_mret_iff 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: w2 = Ƶ /\ k1 = j /\ a = a2
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4 destructs 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
H1: w2 = Ƶ
H2: k1 = j
H3: a = a2
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
exists (w3 w4 : W) (a1 : A),
  (w3, (j, a1)) ∈md t /\
  (w4, (j, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4
      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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
w1: W
hyp1: (w1, (j, a2)) ∈md t
DESTR_NEQs, DESTR_NEQ: j <> j
exists (w2 w3 : W) (a1 : A),
  (w2, (j, a1)) ∈md t /\
  (w3, (j, a2)) ∈md f (w2, a1) /\ w1 ● Ƶ = w2 ● w3 contradiction.
  Qed.

  Lemma inmd_kbindd_eq_iff2:
    forall `(f: W * A -> T j A) (t: U A) (wtotal: W) (a2: A),
      (exists (w1 w2: W) (a1: A),
          (w1, (j, a1)) ∈md t /\ (w2, (j, a2)) ∈md f (w1, a1)
          /\ wtotal = w1 ● w2) ->
      (wtotal, (j, a2)) ∈md kbindd U j 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2) ->
(wtotal, (j, a2)) ∈md kbindd U j 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2) ->
(wtotal, (j, a2)) ∈md kbindd U j f t
    introv [w1 [w2 [a1 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
hyp: (w1, (j, a1)) ∈md t /\
(w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2
(wtotal, (j, a2)) ∈md kbindd U j f t destructs 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
H1: (w1, (j, a1)) ∈md t
H2: (w2, (j, a2)) ∈md f (w1, a1)
H3: wtotal = w1 ● w2
(wtotal, (j, a2)) ∈md kbindd U j f t unfold kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
H1: (w1, (j, a1)) ∈md t
H2: (w2, (j, a2)) ∈md f (w1, a1)
H3: wtotal = w1 ● w2
(wtotal, (j, a2)) ∈md mbindd U (btgd j f) t
    apply (inmd_mbindd_iff2 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
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
H1: (w1, (j, a1)) ∈md t
H2: (w2, (j, a2)) ∈md f (w1, a1)
H3: wtotal = w1 ● w2
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (j, a2)) ∈md btgd j f k1 (w1, a) /\
  wtotal = w1 ● w2
    exists j w1 w2 a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
H1: (w1, (j, a1)) ∈md t
H2: (w2, (j, a2)) ∈md f (w1, a1)
H3: wtotal = w1 ● w2
(w1, (j, a1)) ∈md t /\
(w2, (j, a2)) ∈md btgd j f j (w1, a1) /\
wtotal = w1 ● w2 rewrite btgd_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
H1: (w1, (j, a1)) ∈md t
H2: (w2, (j, a2)) ∈md f (w1, a1)
H3: wtotal = w1 ● w2
(w1, (j, a1)) ∈md t /\
(w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2 auto.
  Qed.

  Theorem inmd_kbindd_eq_iff:
    forall `(f: W * A -> T j A) (t: U A) (wtotal: W) (a2: A),
      (wtotal, (j, a2)) ∈md kbindd U j f t <->
        exists (w1 w2: W) (a1: A),
          (w1, (j, a1)) ∈md t /\ (w2, (j, a2)) ∈md f (w1, a1)
          /\ 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbindd U j f t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbindd U j f t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2)
    split; auto using inmd_kbindd_eq_iff1, inmd_kbindd_eq_iff2.
  Qed.

  Lemma inmd_kbindd_neq_iff1:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> T j A)
      (t: U A) (wtotal: W) (a2: A),
      (wtotal, (i, a2)) ∈md kbindd U j f t ->
      (wtotal, (i, a2)) ∈md t \/
        (exists (w1 w2: W) (a1: A),
            (w1, (j, a1)) ∈md t /\
              (w2, (i, a2)) ∈md f (w1, a1) /\
              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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbindd U j f t ->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbindd U j f t ->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md kbindd U j f t
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2) unfold kbindd 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md mbindd U (btgd j f) t
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2)
    apply (inmd_mbindd_iff1 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: exists (k1 : K) 
(w1 w2 : W) 
(a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (i, a2)) ∈md btgd j f k1 (w1, a) /\
  wtotal = w1 ● w2
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2)
    destruct hyp 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd j f k1 (w1, a)
hyp3: wtotal = w1 ● w2
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd j f k1 (w1, a)
(w1 ● w2, (i, a2)) ∈md t \/
(exists (w3 w4 : W) (a1 : A),
   (w3, (j, a1)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4)
    compare values j and k1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind 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
i, k1: K
f: W * A -> T k1 A
Hneq: k1 <> i
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1 ● w2, (i, a2)) ∈md t \/
(exists (w3 w4 : W) (a1 : A),
   (w3, (k1, a1)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd j f k1 (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t \/
(exists (w3 w4 : W) 
 (a1 : A),
   (w3, (j, a1)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4)
    +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind 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
i, k1: K
f: W * A -> T k1 A
Hneq: k1 <> i
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1 ● w2, (i, a2)) ∈md t \/
(exists (w3 w4 : W) (a1 : A),
   (w3, (k1, a1)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4) 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
i, k1: K
f: W * A -> T k1 A
Hneq: k1 <> i
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
exists (w3 w4 : W) (a1 : A),
  (w3, (k1, a1)) ∈md t /\
  (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4 exists 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: Type
i, k1: K
f: W * A -> T k1 A
Hneq: k1 <> i
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd k1 f k1 (w1, a)
DESTR_EQs: k1 = k1
(w1, (k1, a)) ∈md t /\
(w2, (i, a2)) ∈md f (w1, a) /\ w1 ● w2 = w1 ● w2 rewrite btgd_eq 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
i, k1: K
f: W * A -> T k1 A
Hneq: k1 <> i
t: U A
a2: A
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a)
DESTR_EQs: k1 = k1
(w1, (k1, a)) ∈md t /\
(w2, (i, a2)) ∈md f (w1, a) /\ w1 ● w2 = w1 ● w2 splits; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd j f k1 (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t \/
(exists (w3 w4 : W) (a1 : A),
   (w3, (j, a1)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a1) /\ w1 ● w2 = w3 ● w4) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md btgd j f k1 (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t rewrite btgd_neq in hyp2; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md (mret T k1 ∘ extract) (w1, a)
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t
      unfold compose 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md mret T k1 (extract (w1, a))
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: (w2, (i, a2)) ∈md mret T k1 a
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t
      rewrite inmd_mret_iff 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
k1: K
w1, w2: W
a: A
hyp1: (w1, (k1, a)) ∈md t
hyp2: w2 = Ƶ /\ k1 = i /\ a = a2
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(w1 ● w2, (i, a2)) ∈md t destructs 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1: W
hyp1: (w1, (i, a2)) ∈md t
DESTR_NEQs: i <> j
DESTR_NEQ: j <> i
(w1 ● Ƶ, (i, a2)) ∈md t
      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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1: W
hyp1: (w1, (i, a2)) ∈md t
DESTR_NEQs: i <> j
DESTR_NEQ: j <> i
(w1, (i, a2)) ∈md t auto.
  Qed.

  Lemma inmd_kbindd_neq_iff2:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> T j A)
      (t: U A) (wtotal: W) (a2: A),
      (wtotal, (i, a2)) ∈md t \/
        (exists (w1 w2: W) (a1: A),
            (w1, (j, a1)) ∈md t /\
              (w2, (i, a2)) ∈md f (w1, a1) /\
              wtotal = w1 ● w2) ->
      (wtotal, (i, a2)) ∈md kbindd U j 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2) ->
(wtotal, (i, a2)) ∈md kbindd U j 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2) ->
(wtotal, (i, a2)) ∈md kbindd U j f t
    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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\
   wtotal = w1 ● w2)
(wtotal, (i, a2)) ∈md kbindd U j f t destruct hyp as [hyp | 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(wtotal, (i, a2)) ∈md kbindd U j 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: exists (w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (i, a2)) ∈md f (w1, a1) /\
  wtotal = w1 ● w2
(wtotal, (i, a2)) ∈md kbindd U j 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(wtotal, (i, a2)) ∈md kbindd U j f t apply (inmd_mbindd_iff2 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
exists (k1 : K) (w1 w2 : W) (a : A),
  (w1, (k1, a)) ∈md t /\
  (w2, (i, a2)) ∈md btgd j f k1 (w1, a) /\
  wtotal = w1 ● w2 exists i wtotal Ƶ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(wtotal, (i, a2)) ∈md t /\
(Ƶ, (i, a2)) ∈md btgd j f i (wtotal, a2) /\
wtotal = wtotal ● Ƶ
      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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(wtotal, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(Ƶ, (i, a2)) ∈md btgd j f i (wtotal, 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
wtotal = wtotal ● Ƶ
      {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(wtotal, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(Ƶ, (i, a2)) ∈md btgd j f i (wtotal, 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
wtotal = wtotal ● Ƶ
      {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(Ƶ, (i, a2)) ∈md btgd j f i (wtotal, a2) rewrite btgd_neq; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(Ƶ, (i, a2)) ∈md (mret T i ∘ extract) (wtotal, a2) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
(Ƶ, (i, a2)) ∈md mret T i a2
        rewrite inmd_mret_iff; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
wtotal = wtotal ● Ƶ
      {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: (wtotal, (i, a2)) ∈md t
wtotal = wtotal ● Ƶ now 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
hyp: exists (w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (i, a2)) ∈md f (w1, a1) /\
  wtotal = w1 ● w2
(wtotal, (i, a2)) ∈md kbindd U j f t destruct hyp as [w1 [w2 [a1 [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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
wtotal: W
a2: A
w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
hyp3: wtotal = w1 ● w2
(wtotal, (i, a2)) ∈md kbindd U j 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
(w1 ● w2, (i, a2)) ∈md kbindd U j f t
      apply (inmd_mbindd_iff2 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
exists (k1 : K) (w3 w4 : W) (a : A),
  (w3, (k1, a)) ∈md t /\
  (w4, (i, a2)) ∈md btgd j f k1 (w3, a) /\
  w1 ● w2 = w3 ● w4
      exists j w1 w2 a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
(w1, (j, a1)) ∈md t /\
(w2, (i, a2)) ∈md btgd j f j (w1, a1) /\
w1 ● w2 = w1 ● w2 rewrite btgd_eq.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
(w1, (j, a1)) ∈md t /\
(w2, (i, a2)) ∈md f (w1, a1) /\ w1 ● w2 = w1 ● w2 auto.
  Qed.

  Theorem inmd_kbindd_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> T j A)
      (t: U A) (wtotal: W) (a2: A),
      (wtotal, (i, a2)) ∈md kbindd U j f t <->
        (wtotal, (i, a2)) ∈md t \/
          (exists (w1 w2: W) (a1: A),
              (w1, (j, a1)) ∈md t /\
                (w2, (i, a2)) ∈md f (w1, a1) /\
                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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbindd U j f t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbindd U j f t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ wtotal = w1 ● w2)
    split; auto using inmd_kbindd_neq_iff1, inmd_kbindd_neq_iff2.
  Qed.

  (** *** Corollaries for <<kbind>>, <<kmapd>>, and <<kmap>>*)
  (********************************************************************)
  Corollary inmd_kbind_eq_iff:
    forall `(f: A -> T j A) (t: U A) (wtotal: W) (a2: A),
      (wtotal, (j, a2)) ∈md kbind U j f t <->
        exists (w1 w2: W) (a1: A),
          (w1, (j, a1)) ∈md t /\ (w2, (j, a2)) ∈md f a1
          /\ 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
j: K
A: Type
forall (f : A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbind U j f t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f a1 /\ 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
j: K
A: Type
forall (f : A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (j, a2)) ∈md kbind U j f t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f a1 /\ 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
j: K
A: Type
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (j, a2)) ∈md kbind U j f t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f a1 /\ wtotal = w1 ● w2) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (j, a2)) ∈md kbindd U j (f ∘ extract) t <->
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f a1 /\ wtotal = w1 ● w2) now rewrite (inmd_kbindd_eq_iff).
  Qed.

  Corollary inmd_kbind_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: A -> T j A) (t: U A) (wtotal: W) (a2: A),
      (wtotal, (i, a2)) ∈md kbind U j f t <->
        (wtotal, (i, a2)) ∈md t \/
          (exists (w1 w2: W) (a1: A),
              (w1, (j, a1)) ∈md t /\ (w2, (i, a2)) ∈md f a1
              /\ 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbind U j f t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ 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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> T j A) (t : U A) (wtotal : W)
  (a2 : A),
(wtotal, (i, a2)) ∈md kbind U j f t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (i, a2)) ∈md kbind U j f t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (i, a2)) ∈md kbindd U j (f ∘ extract) t <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2) rewrite inmd_kbindd_neq_iff; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md (f ∘ extract) (w1, a1) /\
   wtotal = w1 ● w2) <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2)
    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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (extract (w1, a1)) /\
   wtotal = w1 ● w2) <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2) 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
wtotal: W
a2: A
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2) <->
(wtotal, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f a1 /\ wtotal = w1 ● w2) easy.
  Qed.

  Corollary inmd_kmapd_eq_iff:
    forall `(f: W * A -> A) (t: U A) (w: W) (a2: A),
      (w, (j, a2)) ∈md kmapd U j f t <->
        exists (a1: A), (w, (j, a1)) ∈md t /\ a2 = f (w, a1).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : W * A -> A) (t : U A) (w : W) (a2 : A),
(w, (j, a2)) ∈md kmapd U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
  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
j: K
A: Type
forall (f : W * A -> A) (t : U A) (w : W) (a2 : A),
(w, (j, a2)) ∈md kmapd U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
    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
j: K
A: Type
f: W * A -> A
t: U A
w: W
a2: A
(w, (j, a2)) ∈md kmapd U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f (w, a1)) unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> A
t: U A
w: W
a2: A
(w, (j, a2)) ∈md mmapd U (tgtd j f) t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f (w, a1)) rewrite (inmd_mmapd_iff 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
j: K
A: Type
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A,
   (w, (j, a)) ∈md t /\ a2 = tgtd j f j (w, a)) <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
    now rewrite tgtd_eq.
  Qed.

  Corollary inmd_kmapd_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> A) (t: U A) (w: W) (a2: A),
      (w, (i, a2)) ∈md kmapd U j f t <->
        (w, (i, a2)) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> A) (t : U A) (w : W) (a2 : A),
(w, (i, a2)) ∈md kmapd U j f t <-> (w, (i, a2)) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> A) (t : U A) (w : W) (a2 : A),
(w, (i, a2)) ∈md kmapd U j f t <-> (w, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md kmapd U j f t <-> (w, (i, a2)) ∈md t unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md mmapd U (tgtd j f) t <->
(w, (i, a2)) ∈md t rewrite (inmd_mmapd_iff 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A,
   (w, (i, a)) ∈md t /\ a2 = tgtd j f i (w, a)) <->
(w, (i, a2)) ∈md t
    rewrite tgtd_neq; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A,
   (w, (i, a)) ∈md t /\ a2 = extract (w, a)) <->
(w, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = a) <->
(w, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = a) ->
(w, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md t ->
exists a : A, (w, (i, a)) ∈md t /\ a2 = 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = a) ->
(w, (i, a2)) ∈md t intros [a [hyp eq]]; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a: A
hyp: (w, (i, a)) ∈md t
(w, (i, a)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md t ->
exists a : A, (w, (i, a)) ∈md t /\ a2 = a intros 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a2: A
hyp: (w, (i, a2)) ∈md t
exists a : A, (w, (i, a)) ∈md t /\ a2 = a now (exists a2).
  Qed.

  Corollary inmd_kmap_eq_iff:
    forall `(f: A -> A) (t: U A) (w: W) (a2: A),
      (w, (j, a2)) ∈md kmap U j f t <->
        exists (a1: A), (w, (j, a1)) ∈md t /\ a2 = f a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : A -> A) (t : U A) (w : W) (a2 : A),
(w, (j, a2)) ∈md kmap U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f a1)
  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
j: K
A: Type
forall (f : A -> A) (t : U A) (w : W) (a2 : A),
(w, (j, a2)) ∈md kmap U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f a1)
    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
j: K
A: Type
f: A -> A
t: U A
w: W
a2: A
(w, (j, a2)) ∈md kmap U j f t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f a1) unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> A
t: U A
w: W
a2: A
(w, (j, a2)) ∈md mmap U (tgt j f) t <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f a1) rewrite (inmd_mmap_iff 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
j: K
A: Type
f: A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (j, a)) ∈md t /\ a2 = tgt j f j a) <->
(exists a1 : A, (w, (j, a1)) ∈md t /\ a2 = f a1)
    now rewrite tgt_eq.
  Qed.

  Corollary inmd_kmap_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: A -> A) (t: U A) (w: W) (a2: A),
      (w, (i, a2)) ∈md kmap U j f t <->
        (w, (i, a2)) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (t : U A) (w : W) (a2 : A),
(w, (i, a2)) ∈md kmap U j f t <-> (w, (i, a2)) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (t : U A) (w : W) (a2 : A),
(w, (i, a2)) ∈md kmap U j f t <-> (w, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md kmap U j f t <-> (w, (i, a2)) ∈md t unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md mmap U (tgt j f) t <->
(w, (i, a2)) ∈md t rewrite (inmd_mmap_iff 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = tgt j f i a) <->
(w, (i, a2)) ∈md t
    rewrite tgt_neq; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = id a) <->
(w, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = id a) ->
(w, (i, a2)) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md t ->
exists a : A, (w, (i, a)) ∈md t /\ a2 = id 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(exists a : A, (w, (i, a)) ∈md t /\ a2 = id a) ->
(w, (i, a2)) ∈md t intros [a [hyp eq]]; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a: A
hyp: (w, (i, a)) ∈md t
(w, (i, id a)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
(w, (i, a2)) ∈md t ->
exists a : A, (w, (i, a)) ∈md t /\ a2 = id a intros 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
w: W
a2: A
hyp: (w, (i, a2)) ∈md t
exists a : A, (w, (i, a)) ∈md t /\ a2 = id a now (exists a2).
  Qed.

  (** *** Occurrences without context *)
  (********************************************************************)
  Theorem in_kbindd_eq_iff:
    forall `(f: W * A -> T j A) (t: U A) (a2: A),
      (j, a2) ∈m kbindd U j f t <->
        exists (w1: W) (a1: A),
          (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f (w1, a1).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (a2 : A),
(j, a2) ∈m kbindd U j f t <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f (w1, a1))
  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
j: K
A: Type
forall (f : W * A -> T j A) (t : U A) (a2 : A),
(j, a2) ∈m kbindd U j f t <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f (w1, a1))
    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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(j, a2) ∈m kbindd U j f t <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f (w1, a1)) 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (j, a2)) ∈md kbindd U j f t) <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f (w1, a1))
    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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (j, a2)) ∈md kbindd U j f t) <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (j, a2)) ∈md f (w1, a1)))
    setoid_rewrite inmd_kbindd_eq_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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists (w w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) <->
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (j, a2)) ∈md f (w1, a1)))
    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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists (w w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) ->
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (j, a2)) ∈md f (w1, a1))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists (w1 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (j, a2)) ∈md f (w1, a1))) ->
exists (w w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists (w w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (j, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) ->
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (j, a2)) ∈md f (w1, a1)) intros [w [w1 [w2 [a1 [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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
w, w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (j, a2)) ∈md f (w1, a1)
hyp3: w = w1 ● w2
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (j, a2)) ∈md f (w1, a1))
      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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
w, w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (j, a2)) ∈md f (w1, a1)
hyp3: w = w1 ● w2
exists a3 : A,
  (?w1, (j, a3)) ∈md t /\
  (exists w0 : W, (w0, (j, a2)) ∈md f (?w1, a3)) 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
w, w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (j, a2)) ∈md f (w1, a1)
hyp3: w = w1 ● w2
(?w1, (j, ?a1)) ∈md t /\
(exists w0 : W, (w0, (j, a2)) ∈md f (?w1, ?a1)) split; 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (j, a2)) ∈md f (w1, a1))) ->
exists (w w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ w = w1 ● w2 intros [w [a1 [hyp1 [w2 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
j: K
A: Type
f: W * A -> T j A
t: U A
a2: A
w: W
a1: A
hyp1: (w, (j, a1)) ∈md t
w2: W
hyp2: (w2, (j, a2)) ∈md f (w, a1)
exists (w w1 w2 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (j, a2)) ∈md f (w1, a1) /\ w = w1 ● w2
      repeat eexists; eauto.
  Qed.

  Theorem in_kbindd_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> T j A) (t: U A) (a2: A),
      (i, a2) ∈m kbindd U j f t <->
        (i, a2) ∈m t \/
          (exists (w1: W) (a1: A),
              (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f (w1, a1)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (a2 : A),
(i, a2) ∈m kbindd U j f t <->
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f (w1, a1))
  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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> T j A) (t : U A) (a2 : A),
(i, a2) ∈m kbindd U j f t <->
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f (w1, a1))
    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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(i, a2) ∈m kbindd U j f t <->
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f (w1, a1)) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (i, a2)) ∈md kbindd U j f t) <->
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f (w1, a1))
    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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (i, a2)) ∈md kbindd U j f t) <->
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1)))
    setoid_rewrite inmd_kbindd_neq_iff; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W,
   (w, (i, a2)) ∈md t \/
   (exists (w1 w2 : W) (a1 : A),
      (w1, (j, a1)) ∈md t /\
      (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2)) <->
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1)))
    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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W,
   (w, (i, a2)) ∈md t \/
   (exists (w1 w2 : W) (a1 : A),
      (w1, (j, a1)) ∈md t /\
      (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2)) ->
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1)))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1))) ->
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) 
   (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W,
   (w, (i, a2)) ∈md t \/
   (exists (w1 w2 : W) (a1 : A),
      (w1, (j, a1)) ∈md t /\
      (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2)) ->
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1))) intros [w [hyp | 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1)))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: exists (w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1)))
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1))) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
exists w : W, (w, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: exists (w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1))) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: exists (w1 w2 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (i, a2)) ∈md f (w1, a1)) destruct hyp as [w1 [w2 [a1 [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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w, w1, w2: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
hyp3: w = w1 ● w2
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (i, a2)) ∈md f (w1, a1))
        repeat eexists; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
(exists w : W, (w, (i, a2)) ∈md t) \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (exists w : W, (w, (i, a2)) ∈md f (w1, a1))) ->
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) intros [hyp | 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
hyp: exists w : W, (w, (i, a2)) ∈md t
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (i, a2)) ∈md f (w1, a1))
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) 
   (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
hyp: exists w : W, (w, (i, a2)) ∈md t
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
(?w, (i, a2)) ∈md t \/
(exists (w1 w2 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (w2, (i, a2)) ∈md f (w1, a1) /\ ?w = w1 ● w2) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
(?w, (i, a2)) ∈md t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (exists w : W, (w, (i, a2)) ∈md f (w1, a1))
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2) destruct hyp as [w1 [a1 [hyp1 [w2 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
w2: W
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
exists w : W,
  (w, (i, a2)) ∈md t \/
  (exists (w1 w2 : W) (a1 : A),
     (w1, (j, a1)) ∈md t /\
     (w2, (i, a2)) ∈md f (w1, a1) /\ w = w1 ● w2)
        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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
w2: W
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
(?w, (i, a2)) ∈md t \/
(exists (w3 w4 : W) (a3 : A),
   (w3, (j, a3)) ∈md t /\
   (w4, (i, a2)) ∈md f (w3, a3) /\ ?w = w3 ● w4) 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> T j A
t: U A
a2: A
w1: W
a1: A
hyp1: (w1, (j, a1)) ∈md t
w2: W
hyp2: (w2, (i, a2)) ∈md f (w1, a1)
exists (w3 w4 : W) (a3 : A),
  (w3, (j, a3)) ∈md t /\
  (w4, (i, a2)) ∈md f (w3, a3) /\ ?w = w3 ● w4 repeat eexists; eauto.
  Qed.

  Corollary in_kbind_eq_iff:
    forall `(f: A -> T j A) (t: U A) (a2: A),
      (j, a2) ∈m kbind U j f t <->
        exists (a1: A),
          (j, a1) ∈m t /\ (j, a2) ∈m f a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : A -> T j A) (t : U A) (a2 : A),
(j, a2) ∈m kbind U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ (j, a2) ∈m f a1)
  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
j: K
A: Type
forall (f : A -> T j A) (t : U A) (a2 : A),
(j, a2) ∈m kbind U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ (j, a2) ∈m f a1)
    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
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(j, a2) ∈m kbind U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ (j, a2) ∈m f a1) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(j, a2) ∈m kbindd U j (f ∘ extract) t <->
(exists a1 : A, (j, a1) ∈m t /\ (j, a2) ∈m f a1) rewrite (in_kbindd_eq_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
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (j, a2) ∈m (f ∘ extract) (w1, a1)) <->
(exists a1 : A, (j, a1) ∈m t /\ (j, a2) ∈m f a1)
    setoid_rewrite inmd_iff_in at 2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (j, a2) ∈m (f ∘ extract) (w1, a1)) <->
(exists a1 : A,
   (exists w : W, (w, (j, a1)) ∈md t) /\
   (j, a2) ∈m f a1)
    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
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (j, a2) ∈m f (extract (w1, a1))) <->
(exists a1 : A,
   (exists w : W, (w, (j, a1)) ∈md t) /\
   (j, a2) ∈m f a1) 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
j: K
A: Type
f: A -> T j A
t: U A
a2: A
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\ (j, a2) ∈m f a1) <->
(exists a1 : A,
   (exists w : W, (w, (j, a1)) ∈md t) /\
   (j, a2) ∈m f a1) firstorder.
  Qed.

  Corollary in_kbind_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: A -> T j A) (t: U A) (a2: A),
      (i, a2) ∈m kbind U j f t <->
        (i, a2) ∈m t \/
          (exists (a1: A),
              (j, a1) ∈m t /\ (i, a2) ∈m f a1).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> T j A) (t : U A) (a2 : A),
(i, a2) ∈m kbind U j f t <->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
  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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> T j A) (t : U A) (a2 : A),
(i, a2) ∈m kbind U j f t <->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
    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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m kbind U j f t <->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) rewrite kbind_to_kbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m kbindd U j (f ∘ extract) t <->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) rewrite in_kbindd_neq_iff; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) <->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
    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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) ->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) ->
(i, a2) ∈m t \/
(exists (w1 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1))
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) ->
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) intros [hyp|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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: (i, a2) ∈m t
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1)
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: (i, a2) ∈m t
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1 unfold compose 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m f (extract (w1, a1))
exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1 cbn 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists (w1 : W) 
(a1 : A), (w1, (j, a1)) ∈md t /\ (i, a2) ∈m f a1
exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1
        destruct hyp as [? [a1 [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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
x: W
a1: A
hyp1: (x, (j, a1)) ∈md t
hyp2: (i, a2) ∈m f a1
exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1
        apply inmd_implies_in 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
x: W
a1: A
hyp1: (j, a1) ∈m t
hyp2: (i, a2) ∈m f a1
exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
(i, a2) ∈m t \/
(exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1) ->
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) intros [hyp|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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: (i, a2) ∈m t
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1))
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1
(i, a2) ∈m t \/
(exists (w1 : W) 
 (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1))
      +
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: (i, a2) ∈m t
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1
(i, a2) ∈m t \/
(exists (w1 : W) (a1 : A),
   (w1, (j, a1)) ∈md t /\
   (i, a2) ∈m (f ∘ extract) (w1, a1)) 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2: A
hyp: exists a1 : A, (j, a1) ∈m t /\ (i, a2) ∈m f a1
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
        destruct hyp as [a1 [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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2, a1: A
hyp1: (j, a1) ∈m t
hyp2: (i, a2) ∈m f a1
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
        rewrite inmd_iff_in 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2, a1: A
hyp1: exists w : W, (w, (j, a1)) ∈md t
hyp2: (i, a2) ∈m f a1
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1) destruct hyp1 as [w1 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2, a1: A
w1: W
hyp1: (w1, (j, a1)) ∈md t
hyp2: (i, a2) ∈m f a1
exists (w1 : W) (a1 : A),
  (w1, (j, a1)) ∈md t /\
  (i, a2) ∈m (f ∘ extract) (w1, a1)
        exists w1 a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> T j A
t: U A
a2, a1: A
w1: W
hyp1: (w1, (j, a1)) ∈md t
hyp2: (i, a2) ∈m f a1
(w1, (j, a1)) ∈md t /\
(i, a2) ∈m (f ∘ extract) (w1, a1) auto.
  Qed.

  Corollary in_kmapd_eq_iff:
    forall `(f: W * A -> A) (t: U A) (a2: A),
      (j, a2) ∈m kmapd U j f t <->
        exists (w: W) (a1: A), (w, (j, a1)) ∈md t /\ a2 = f (w, a1).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : W * A -> A) (t : U A) (a2 : A),
(j, a2) ∈m kmapd U j f t <->
(exists (w : W) (a1 : A),
   (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
  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
j: K
A: Type
forall (f : W * A -> A) (t : U A) (a2 : A),
(j, a2) ∈m kmapd U j f t <->
(exists (w : W) (a1 : A),
   (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
    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
j: K
A: Type
f: W * A -> A
t: U A
a2: A
(j, a2) ∈m kmapd U j f t <->
(exists (w : W) (a1 : A),
   (w, (j, a1)) ∈md t /\ a2 = f (w, a1)) unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: W * A -> A
t: U A
a2: A
(j, a2) ∈m mmapd U (tgtd j f) t <->
(exists (w : W) (a1 : A),
   (w, (j, a1)) ∈md t /\ a2 = f (w, a1)) rewrite (in_mmapd_iff 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
j: K
A: Type
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A),
   (w, (j, a)) ∈md t /\ a2 = tgtd j f j (w, a)) <->
(exists (w : W) (a1 : A),
   (w, (j, a1)) ∈md t /\ a2 = f (w, a1))
    now rewrite tgtd_eq.
  Qed.

  Corollary in_kmapd_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: W * A -> A) (t: U A) (a2: A),
      (i, a2) ∈m kmapd U j f t <->
        (i, a2) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> A) (t : U A) (a2 : A),
(i, a2) ∈m kmapd U j f t <-> (i, a2) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : W * A -> A) (t : U A) (a2 : A),
(i, a2) ∈m kmapd U j f t <-> (i, a2) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(i, a2) ∈m kmapd U j f t <-> (i, a2) ∈m t unfold kmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(i, a2) ∈m mmapd U (tgtd j f) t <-> (i, a2) ∈m t rewrite (in_mmapd_iff 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A),
   (w, (i, a)) ∈md t /\ a2 = tgtd j f i (w, a)) <->
(i, a2) ∈m t
    rewrite tgtd_neq; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A),
   (w, (i, a)) ∈md t /\ a2 = extract (w, a)) <->
(i, a2) ∈m t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = a) <->
(i, a2) ∈m t 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = a) ->
(i, a2) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(i, a2) ∈m t ->
exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = a) ->
(i, a2) ∈m t intros [w [a [hyp eq]]]; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
w: W
a: A
hyp: (w, (i, a)) ∈md t
(i, a) ∈m t
      eapply inmd_implies_in; 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
(i, a2) ∈m t ->
exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = a intros 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
hyp: (i, a2) ∈m t
exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
hyp: exists w : W, (w, (i, a2)) ∈md t
exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = 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
j: K
A: Type
i: K
Hneq: j <> i
f: W * A -> A
t: U A
a2: A
w: W
hyp: (w, (i, a2)) ∈md t
exists (w : W) (a : A), (w, (i, a)) ∈md t /\ a2 = a eauto.
  Qed.

  Corollary in_kmap_eq_iff:
    forall `(f: A -> A) (t: U A) (a2: A),
      (j, a2) ∈m kmap U j f t <->
        exists (a1: A), (j, a1) ∈m t /\ a2 = f a1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
forall (f : A -> A) (t : U A) (a2 : A),
(j, a2) ∈m kmap U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ a2 = f a1)
  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
j: K
A: Type
forall (f : A -> A) (t : U A) (a2 : A),
(j, a2) ∈m kmap U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ a2 = f a1)
    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
j: K
A: Type
f: A -> A
t: U A
a2: A
(j, a2) ∈m kmap U j f t <->
(exists a1 : A, (j, a1) ∈m t /\ a2 = f a1) unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
f: A -> A
t: U A
a2: A
(j, a2) ∈m mmap U (tgt j f) t <->
(exists a1 : A, (j, a1) ∈m t /\ a2 = f a1) rewrite (in_mmap_iff 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
j: K
A: Type
f: A -> A
t: U A
a2: A
(exists a : A, (j, a) ∈m t /\ a2 = tgt j f j a) <->
(exists a1 : A, (j, a1) ∈m t /\ a2 = f a1)
    now rewrite tgt_eq.
  Qed.

  Corollary in_kmap_neq_iff:
    forall (i: K) (Hneq: j <> i) `(f: A -> A) (t: U A) (a2: A),
      (i, a2) ∈m kmap U j f t <->
        (i, a2) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (t : U A) (a2 : A),
(i, a2) ∈m kmap U j f t <-> (i, a2) ∈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
j: K
A: Type
forall i : K,
j <> i ->
forall (f : A -> A) (t : U A) (a2 : A),
(i, a2) ∈m kmap U j f t <-> (i, a2) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(i, a2) ∈m kmap U j f t <-> (i, a2) ∈m t unfold kmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(i, a2) ∈m mmap U (tgt j f) t <-> (i, a2) ∈m t rewrite (in_mmap_iff 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(exists a : A, (i, a) ∈m t /\ a2 = tgt j f i a) <->
(i, a2) ∈m t
    rewrite tgt_neq; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(exists a : A, (i, a) ∈m t /\ a2 = id a) <->
(i, a2) ∈m t 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(exists a : A, (i, a) ∈m t /\ a2 = id a) ->
(i, a2) ∈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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(i, a2) ∈m t -> exists a : A, (i, a) ∈m t /\ a2 = id 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(exists a : A, (i, a) ∈m t /\ a2 = id a) ->
(i, a2) ∈m t intros [a [hyp ?]]; 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
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a: A
hyp: (i, a) ∈m t
(i, id a) ∈m t assumption.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
j: K
A: Type
i: K
Hneq: j <> i
f: A -> A
t: U A
a2: A
(i, a2) ∈m t -> exists a : A, (i, a) ∈m t /\ a2 = id a intros; now (exists a2).
  Qed.

End DTM_membership_targetted.