Multisorted locally nameless infrastructure

Basic properties of operations

From Tealeaves.Backends.Common Require Import
  Names AtomSet.
From Tealeaves.Misc Require Import
  NaturalNumbers.
From Tealeaves.Theory Require Import
  DecoratedTraversableMonad (* Mostly for the imports *)
  Multisorted.DecoratedTraversableMonad.

Import
  Product.Notations
  Monoid.Notations
  List.ListNotations
  Subset.Notations
  Kleisli.Monad.Notations
  Kleisli.Comonad.Notations
  ContainerFunctor.Notations
  DecoratedMonad.Notations
  DecoratedContainerFunctor.Notations
  DecoratedTraversableMonad.Notations
  AtomSet.Notations
  DecoratedTraversableMonad.Notations
  Functors.Multisorted.Batch.Notations
  Theory.Container.Notations
  Categories.TypeFamily.Notations.

#[local] Open Scope nat_scope. (* Let <<x + y>> be addition, not sum type. *)
#[local] Open Scope set_scope. (* Don't confuse with Functors/Subset.v *)
#[local] Generalizable Variable T.

(** * Binders contexts **)
(** We assume that binder contexts are lists of tagged values of type
    <<unit>>, which are just tags themselves. *)
(******************************************************************************)
Section operations_on_context.

  Context
    `{Index}.

  Fixpoint countk (j : K) (l : list K) : nat :=
    match l with
    | nil => 0
    | cons k rest =>
      (if j == k then 1 else 0) + countk j rest
    end.

  Lemma countk_nil : forall j, countk j nil = 0.H: Index
forall j : K, countk j [] = 0
  Proof.H: Index
forall j : K, countk j [] = 0
    easy.
  Qed.

  Lemma countk_one_eq : forall j, countk j [j] = 1.H: Index
forall j : K, countk j [j] = 1
  Proof.H: Index
forall j : K, countk j [j] = 1
    intros.H: Index
j: K
countk j [j] = 1 cbn.H: Index
j: K
(if j == j then 1 else 0) + 0 = 1 compare values j and j.
  Qed.

  Lemma countk_one_neq : forall j k, j <> k -> countk j [k] = 0.H: Index
forall j k : K, j <> k -> countk j [k] = 0
  Proof.H: Index
forall j k : K, j <> k -> countk j [k] = 0
    intros.H: Index
j, k: K
H0: j <> k
countk j [k] = 0 cbn.H: Index
j, k: K
H0: j <> k
(if j == k then 1 else 0) + 0 = 0 compare values j and k.
  Qed.

  Lemma countk_cons_eq : forall j l, countk j (j :: l) = S (countk j l).H: Index
forall (j : K) (l : list K),
countk j (j :: l) = S (countk j l)
  Proof.H: Index
forall (j : K) (l : list K),
countk j (j :: l) = S (countk j l)
    intros.H: Index
j: K
l: list K
countk j (j :: l) = S (countk j l) cbn.H: Index
j: K
l: list K
(if j == j then 1 else 0) + countk j l =
S (countk j l) compare values j and j.
  Qed.

  Lemma countk_cons_neq : forall j k l, j <> k -> countk j (k :: l) = countk j l.H: Index
forall (j k : K) (l : list K),
j <> k -> countk j (k :: l) = countk j l
  Proof.H: Index
forall (j k : K) (l : list K),
j <> k -> countk j (k :: l) = countk j l
    intros.H: Index
j, k: K
l: list K
H0: j <> k
countk j (k :: l) = countk j l cbn.H: Index
j, k: K
l: list K
H0: j <> k
(if j == k then 1 else 0) + countk j l = countk j l compare values j and k.
  Qed.

  Lemma countk_app : forall j l1 l2, countk j (l1 ++ l2) = countk j l1 + countk j l2.H: Index
forall (j : K) (l1 l2 : list K),
countk j (l1 ++ l2) = countk j l1 + countk j l2
  Proof.H: Index
forall (j : K) (l1 l2 : list K),
countk j (l1 ++ l2) = countk j l1 + countk j l2
    intros.H: Index
j: K
l1, l2: list K
countk j (l1 ++ l2) = countk j l1 + countk j l2 induction l1.H: Index
j: K
l2: list K
countk j ([] ++ l2) = countk j [] + countk j l2
H: Index
j, a: K
l1, l2: list K
IHl1: countk j (l1 ++ l2) = countk j l1 + countk j l2
countk j ((a :: l1) ++ l2) =
countk j (a :: l1) + countk j l2
    -H: Index
j: K
l2: list K
countk j ([] ++ l2) = countk j [] + countk j l2 easy.
    -H: Index
j, a: K
l1, l2: list K
IHl1: countk j (l1 ++ l2) = countk j l1 + countk j l2
countk j ((a :: l1) ++ l2) =
countk j (a :: l1) + countk j l2 cbn.H: Index
j, a: K
l1, l2: list K
IHl1: countk j (l1 ++ l2) = countk j l1 + countk j l2
(if j == a then 1 else 0) + countk j (l1 ++ l2) =
(if j == a then 1 else 0) + countk j l1 + countk j l2 compare values j and a.H: Index
a: K
l1, l2: list K
IHl1: countk a (l1 ++ l2) = countk a l1 + countk a l2
DESTR_EQs: a = a
1 + countk a (l1 ++ l2) =
1 + countk a l1 + countk a l2
      now rewrite IHl1.
  Qed.

End operations_on_context.

(** * Locally nameless variables *)
(******************************************************************************)
Inductive LN :=
| Fr : atom -> LN
| Bd : nat -> LN.

Theorem eq_dec_LN : forall l1 l2 : LN, {l1 = l2} + {l1 <> l2}.forall l1 l2 : LN, {l1 = l2} + {l1 <> l2}
Proof.forall l1 l2 : LN, {l1 = l2} + {l1 <> l2}
  decide equality.l1, l2: LN
n, n0: atom
{n = n0} + {n <> n0}
l1, l2: LN
n, n0: nat
{n = n0} + {n <> n0}
  -l1, l2: LN
n, n0: atom
{n = n0} + {n <> n0} compare values n and n0; auto.
  -l1, l2: LN
n, n0: nat
{n = n0} + {n <> n0} compare values n and n0; auto.
Qed.

#[export] Instance EqDec_LN : EquivDec.EqDec LN eq := eq_dec_LN.

Lemma compare_to_atom : forall x l (P : LN -> Prop),
    P (Fr x) ->
    (forall a : atom, a <> x -> P (Fr a)) ->
    (forall n : nat, P (Bd n)) ->
    P l.forall (x : atom) (l : LN) (P : LN -> Prop),
P (Fr x) ->
(forall a : atom, a <> x -> P (Fr a)) ->
(forall n : nat, P (Bd n)) -> P l
Proof.forall (x : atom) (l : LN) (P : LN -> Prop),
P (Fr x) ->
(forall a : atom, a <> x -> P (Fr a)) ->
(forall n : nat, P (Bd n)) -> P l
  introv case1 case2 case3.x: atom
l: LN
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
P l destruct l.x, n: atom
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
P (Fr n)
x: atom
n: nat
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
P (Bd n)
  -x, n: atom
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
P (Fr n) destruct_eq_args x n.x, n: atom
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
DESTR_NEQ: x <> n
DESTR_NEQs: n <> x
P (Fr n) auto.
  -x: atom
n: nat
P: LN -> Prop
case1: P (Fr x)
case2: forall a : atom, a <> x -> P (Fr a)
case3: forall n : nat, P (Bd n)
P (Bd n) auto.
Qed.

Tactic Notation "compare" constr(l) "to" "atom" constr(x) :=
  (induction l using (compare_to_atom x)).

(** * Syntax operations for locally nameless *)
(******************************************************************************)

(** ** Local operations *)
(******************************************************************************)
Section local_operations.

  Context
    `{Index}
    `{MReturn T}.

  Implicit Types (x : atom) (k : K).

  Definition free_loc : LN -> list atom :=
    fun l => match l with
          | Fr x => cons x List.nil
          | _ => List.nil
          end.

  Definition subst_loc k x (u : T k LN) : LN -> T k LN :=
    fun l => match l with
          | Fr y =>
            if x == y then u else mret T k (Fr y)
          | Bd n =>
            mret T k (Bd n)
          end.

  Definition open_loc k (u : T k LN) : list K * LN -> T k LN :=
    fun '(w, l) =>
      match l with
      | Fr x => mret T k (Fr x)
      | Bd n =>
        match Nat.compare n (countk k w) with
        | Gt => mret T k (Bd (n - 1))
        | Eq => u
        | Lt => mret T k (Bd n)
        end
      end.

  Definition is_opened : list K * (K * LN) -> Prop :=
    fun '(w, (k, l)) =>
      match l with
      | Fr y => False
      | Bd n => n = countk k w
      end.

  Definition close_loc k x : list K * LN -> LN :=
    fun '(w, l) =>
      match l with
      | Fr y =>
        if x == y then Bd (countk k w) else Fr y
      | Bd n =>
        match Nat.compare n (countk k w) with
        | Gt => Bd (S n)
        | Eq => Bd (S n)
        | Lt => Bd n
        end
      end.

  (** To define local closure we will take <<n = 0>>, but we can also
      consider more notions like ``local closure within a gap of 1
      binder,'' which is useful for backend reasoning. **)
  Definition lc_loc k (gap : nat) : list K * LN -> Prop :=
    fun '(w, l) =>
      match l with
      | Fr x => True
      | Bd n => n < (countk k w) + gap
      end.

End local_operations.

(** ** Lifted operations *)
(******************************************************************************)
Section LocallyNamelessOperations.

  Context
    (U : Type -> Type)
      `{MultiDecoratedTraversablePreModule (list K) T U
          (mn_op := Monoid_op_list)
          (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Definition open k (u : T k LN) : U LN -> U LN  :=
    kbindd U k (open_loc k u).

  Definition close k x : U LN -> U LN :=
    kmapd U k (close_loc k x).

  Definition subst k x (u : T k LN) : U LN -> U LN :=
    kbind U k (subst_loc k x u).

  Definition free : K -> U LN -> list atom :=
    fun k t => bind (T := list) free_loc (toklist U k t).

  Definition LCn k (gap : nat) : U LN -> Prop :=
    fun t => forall (w : list K) (l : LN),
        (w, (k, l)) ∈md t -> lc_loc k gap (w, l).

  Definition LC k : U LN -> Prop :=
    LCn k 0.

  Definition FV : K -> U LN -> AtomSet.t :=
    fun k t => AtomSet.atoms (free k t).

  Definition scoped : K -> U LN -> AtomSet.t -> Prop :=
    fun k t γ => FV k t ⊆ γ.

End LocallyNamelessOperations.

(** ** Notations for operations *)
(******************************************************************************)
Module Notations.

  Notation "t '{ k | x ~> u }" := (subst _ k x u t) (at level 35).
  Notation "t '( k | u )" := (open _ k u t) (at level 35).
  Notation "'[ k | x ] t" := (close _ k x t) (at level 35).
  Notation "( x , y , z )" := (pair x (pair y z)) : tealeaves_multi_scope.

End Notations.

Section test_notations.

  Import Notations.

  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Context
    (k j : K)
    (t1 : T k LN)
    (t2 : T j LN)
    (u : U LN)
    (x : atom)
    (n : nat).

  Check u '{ k | x ~> t1}.u '{ k | x ~> t1}
     : U LN
  Check u '(k | t1).u '( k | t1)
     : U LN
  Check '[k | x] u.'[ k | x] u
     : U LN

  Check u '{j | x ~> t2}.u '{ j | x ~> t2}
     : U LN
  Check u '(j | t2).u '( j | t2)
     : U LN
  Check '[j | x] u.'[ j | x] u
     : U LN

  Fail Check u '{j | x ~> t1}.The command has indeed failed with message:
In environment
U : Type -> Type
ix : Index
T : K -> Type -> Type
MReturn0 : MReturn T
MBind0 : MBind (list K) T U
H : forall k : K, MBind (list K) T (T k)
H0 : MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0 : MultiDecoratedTraversableMonad
                                    (list K) T
k, j : K
t1 : T k LN
t2 : T j LN
u : U LN
x : atom
n : nat
Unable to unify "T k LN" with "?T j LN"
(cannot unify "k" and "j").
  Fail Check u '(j | t1).The command has indeed failed with message:
In environment
U : Type -> Type
ix : Index
T : K -> Type -> Type
MReturn0 : MReturn T
MBind0 : MBind (list K) T U
H : forall k : K, MBind (list K) T (T k)
H0 : MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0 : MultiDecoratedTraversableMonad
                                    (list K) T
k, j : K
t1 : T k LN
t2 : T j LN
u : U LN
x : atom
n : nat
Unable to unify "T k LN" with "?T j LN"
(cannot unify "k" and "j").

End test_notations.

Import Notations.

Section operations_specifications.

  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Implicit Types (l : LN) (w : list K) (t : U LN) (x : atom).

  (** ** Identity and equality lemmas for operations *)
  (******************************************************************************)

  (** *** For [open] *)
  (******************************************************************************)
  Lemma open_eq : forall k t (u1 u2 : T k LN),
      (forall (w : list K) (l : LN), (w, (k, l)) ∈md t -> open_loc k u1 (w, l) = open_loc k u2 (w, l)) ->
      t '(k | u1) = t '(k | u2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (u1 u2 : T k LN),
(forall w l,
 (w, k, l) ∈md t ->
 open_loc k u1 (w, l) = open_loc k u2 (w, l)) ->
t '( k | u1) = t '( k | u2)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (u1 u2 : T k LN),
(forall w l,
 (w, k, l) ∈md t ->
 open_loc k u1 (w, l) = open_loc k u2 (w, l)) ->
t '( k | u1) = t '( k | u2)
    introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u1, u2: T k LN
hyp: forall w l,
(w, k, l) ∈md t ->
open_loc k u1 (w, l) = open_loc k u2 (w, l)
t '( k | u1) = t '( k | u2) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u1, u2: T k LN
hyp: forall w l,
(w, k, l) ∈md t ->
open_loc k u1 (w, l) = open_loc k u2 (w, l)
kbindd U k (open_loc k u1) t =
kbindd U k (open_loc k u2) t apply kbindd_respectful.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u1, u2: T k LN
hyp: forall w l,
(w, k, l) ∈md t ->
open_loc k u1 (w, l) = open_loc k u2 (w, l)
forall w (a : LN),
(w, k, a) ∈md t ->
open_loc k u1 (w, a) = open_loc k u2 (w, a) auto.
  Qed.

  Lemma open_id : forall k t u,
      (forall w l, (w, (k, l)) ∈md t -> open_loc k u (w, l) = mret T k l) ->
      t '(k | u) = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (u : T k LN),
(forall w l,
 (w, k, l) ∈md t -> open_loc k u (w, l) = mret T k l) ->
t '( k | u) = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (u : T k LN),
(forall w l,
 (w, k, l) ∈md t -> open_loc k u (w, l) = mret T k l) ->
t '( k | u) = t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: forall w l,
(w, k, l) ∈md t ->
open_loc k u (w, l) = mret T k l
t '( k | u) = t unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: forall w l,
(w, k, l) ∈md t ->
open_loc k u (w, l) = mret T k l
kbindd U k (open_loc k u) t = t
    now apply (kbindd_respectful_id k).
  Qed.

  (** *** For [subst] *)
  (******************************************************************************)
  Lemma subst_eq : forall k t x u1 u2,
      (forall l, (k, l) ∈m t -> subst_loc k x u1 l = subst_loc k x u2 l) ->
      t '{k | x ~> u1} = t '{k | x ~> u2}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x (u1 u2 : T k LN),
(forall l,
 (k, l) ∈m t ->
 subst_loc k x u1 l = subst_loc k x u2 l) ->
t '{ k | x ~> u1} = t '{ k | x ~> u2}
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x (u1 u2 : T k LN),
(forall l,
 (k, l) ∈m t ->
 subst_loc k x u1 l = subst_loc k x u2 l) ->
t '{ k | x ~> u1} = t '{ k | x ~> u2}
    introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
u1, u2: T k LN
hyp: forall l,
(k, l) ∈m t ->
subst_loc k x u1 l = subst_loc k x u2 l
t '{ k | x ~> u1} = t '{ k | x ~> u2} unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
u1, u2: T k LN
hyp: forall l,
(k, l) ∈m t ->
subst_loc k x u1 l = subst_loc k x u2 l
kbind U k (subst_loc k x u1) t =
kbind U k (subst_loc k x u2) t apply kbind_respectful.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
u1, u2: T k LN
hyp: forall l,
(k, l) ∈m t ->
subst_loc k x u1 l = subst_loc k x u2 l
forall a : LN,
(k, a) ∈m t -> subst_loc k x u1 a = subst_loc k x u2 a auto.
  Qed.

  Lemma subst_id : forall k t x u,
      (forall l, (k, l) ∈m t -> subst_loc k x u l = mret T k l) ->
      t '{k | x ~> u} = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x (u : T k LN),
(forall l,
 (k, l) ∈m t -> subst_loc k x u l = mret T k l) ->
t '{ k | x ~> u} = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x (u : T k LN),
(forall l,
 (k, l) ∈m t -> subst_loc k x u l = mret T k l) ->
t '{ k | x ~> u} = t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
u: T k LN
H1: forall l,
(k, l) ∈m t -> subst_loc k x u l = mret T k l
t '{ k | x ~> u} = t unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
u: T k LN
H1: forall l,
(k, l) ∈m t -> subst_loc k x u l = mret T k l
kbind U k (subst_loc k x u) t = t
    now apply kbind_respectful_id.
  Qed.

  (** *** For [close] *)
  (******************************************************************************)
  Lemma close_eq : forall k t x y,
      (forall w l, (w, (k, l)) ∈md t -> close_loc k x (w, l) = close_loc k y (w, l)) ->
      '[k | x] t = '[k | y] t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (x y : atom),
(forall w l,
 (w, k, l) ∈md t ->
 close_loc k x (w, l) = close_loc k y (w, l)) ->
'[ k | x] t = '[ k | y] t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t (x y : atom),
(forall w l,
 (w, k, l) ∈md t ->
 close_loc k x (w, l) = close_loc k y (w, l)) ->
'[ k | x] t = '[ k | y] t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
H1: forall w l,
(w, k, l) ∈md t ->
close_loc k x (w, l) = close_loc k y (w, l)
'[ k | x] t = '[ k | y] t unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
H1: forall w l,
(w, k, l) ∈md t ->
close_loc k x (w, l) = close_loc k y (w, l)
kmapd U k (close_loc k x) t =
kmapd U k (close_loc k y) t
    now apply (kmapd_respectful).
  Qed.

  Lemma close_id : forall k t x,
      (forall w l, (w, (k, l)) ∈md t -> close_loc k x (w, l) = l) ->
      '[k | x] t = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x,
(forall w l,
 (w, k, l) ∈md t -> close_loc k x (w, l) = l) ->
'[ k | x] t = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) t x,
(forall w l,
 (w, k, l) ∈md t -> close_loc k x (w, l) = l) ->
'[ k | x] t = t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: forall w l,
(w, k, l) ∈md t -> close_loc k x (w, l) = l
'[ k | x] t = t unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: forall w l,
(w, k, l) ∈md t -> close_loc k x (w, l) = l
kmapd U k (close_loc k x) t = t
    now apply (kmapd_respectful_id).
  Qed.

  (** ** Context-sensitive LN analysis of operations *)
  (******************************************************************************)

  (** *** For [open] *)
  (******************************************************************************)
  Lemma inmd_open_iff_eq : forall k l w u t,
      (w, (k, l)) ∈md t '(k | u) <-> exists w1 w2 l1,
        (w1, (k, l1)) ∈md t /\ (w2, (k, l)) ∈md open_loc k u (w1, l1) /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l w (u : T k LN) t,
(w, k, l) ∈md t '( k | u) <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\ w = w1 ● w2)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l w (u : T k LN) t,
(w, k, l) ∈md t '( k | u) <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\ w = w1 ● w2)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
u: T k LN
t: U LN
(w, k, l) ∈md t '( k | u) <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\ w = w1 ● w2) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
u: T k LN
t: U LN
(w, k, l) ∈md kbindd U k (open_loc k u) t <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\ w = w1 ● w2)
    now rewrite (inmd_kbindd_eq_iff U).
  Qed.

  Lemma inmd_open_neq_iff : forall k j l w u t,
      k <> j ->
      (w, (k, l)) ∈md t '(j | u) <-> (w, (k, l)) ∈md t \/ exists w1 w2 l1,
          (w1, (j, l1)) ∈md t /\ (w2, (k, l)) ∈md open_loc j u (w1, l1) /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l w (u : T j LN) t,
k <> j ->
(w, k, l) ∈md t '( j | u) <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md open_loc j u (w1, l1) /\ w = w1 ● w2)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l w (u : T j LN) t,
k <> j ->
(w, k, l) ∈md t '( j | u) <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md open_loc j u (w1, l1) /\ w = w1 ● w2)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
w: list K
u: T j LN
t: U LN
H1: k <> j
(w, k, l) ∈md t '( j | u) <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md open_loc j u (w1, l1) /\ w = w1 ● w2) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
w: list K
u: T j LN
t: U LN
H1: k <> j
(w, k, l) ∈md kbindd U j (open_loc j u) t <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md open_loc j u (w1, l1) /\ w = w1 ● w2)
    now rewrite (inmd_kbindd_neq_iff U); auto.
  Qed.

  (** *** For [close] *)
  (******************************************************************************)
  Lemma inmd_close_iff_eq : forall k l w x t,
      (w, (k, l)) ∈md '[k | x] t <-> exists l1,
        (w, (k, l1)) ∈md t /\ l = close_loc k x (w, l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l w x t,
(w, k, l) ∈md '[ k | x] t <->
(exists l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l w x t,
(w, k, l) ∈md '[ k | x] t <->
(exists l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
x: atom
t: U LN
(w, k, l) ∈md '[ k | x] t <->
(exists l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1)) unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
x: atom
t: U LN
(w, k, l) ∈md kmapd U k (close_loc k x) t <->
(exists l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
    rewrite (inmd_kmapd_eq_iff U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
x: atom
t: U LN
(exists a1 : LN,
   (w, k, a1) ∈md t /\ l = close_loc k x (w, a1)) <->
(exists l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
    easy.
  Qed.

  Lemma inmd_close_neq_iff : forall k j l w x t,
      j <> k ->
      (w, (k, l)) ∈md '[j | x] t <-> (w, (k, l)) ∈md t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l w x t,
j <> k ->
(w, k, l) ∈md '[ j | x] t <-> (w, k, l) ∈md t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l w x t,
j <> k ->
(w, k, l) ∈md '[ j | x] t <-> (w, k, l) ∈md t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
w: list K
x: atom
t: U LN
H1: j <> k
(w, k, l) ∈md '[ j | x] t <-> (w, k, l) ∈md t unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
w: list K
x: atom
t: U LN
H1: j <> k
(w, k, l) ∈md kmapd U j (close_loc j x) t <->
(w, k, l) ∈md t
    now rewrite (inmd_kmapd_neq_iff U).
  Qed.

  (** *** For [subst] *)
  (******************************************************************************)
  Lemma inmd_subst_iff_eq : forall k w l u t x,
      (w, (k, l)) ∈md t '{k | x ~> u} <-> exists w1 w2 l1,
        (w1, (k, l1)) ∈md t /\ (w2, (k, l)) ∈md subst_loc k x u l1 /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) w l (u : T k LN) t x,
(w, k, l) ∈md t '{ k | x ~> u} <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) w l (u : T k LN) t x,
(w, k, l) ∈md t '{ k | x ~> u} <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
l: LN
u: T k LN
t: U LN
x: atom
(w, k, l) ∈md t '{ k | x ~> u} <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
l: LN
u: T k LN
t: U LN
x: atom
(w, k, l) ∈md kbind U k (subst_loc k x u) t <->
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
    now rewrite (inmd_kbind_eq_iff U).
  Qed.

  Lemma inmd_subst_neq_iff : forall k j w l u t x,
      k <> j ->
      (w, (k, l)) ∈md t '{j | x ~> u} <-> (w, (k, l)) ∈md t \/ exists w1 w2 l1,
        (w1, (j, l1)) ∈md t /\ (w2, (k, l)) ∈md subst_loc j x u l1 /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) w l (u : T j LN) t x,
k <> j ->
(w, k, l) ∈md t '{ j | x ~> u} <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc j x u l1 /\ w = w1 ● w2)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) w l (u : T j LN) t x,
k <> j ->
(w, k, l) ∈md t '{ j | x ~> u} <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc j x u l1 /\ w = w1 ● w2)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
l: LN
u: T j LN
t: U LN
x: atom
H1: k <> j
(w, k, l) ∈md t '{ j | x ~> u} <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc j x u l1 /\ w = w1 ● w2) unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
l: LN
u: T j LN
t: U LN
x: atom
H1: k <> j
(w, k, l) ∈md kbind U j (subst_loc j x u) t <->
(w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md subst_loc j x u l1 /\ w = w1 ● w2)
    now rewrite (inmd_kbind_neq_iff U); auto.
  Qed.

  (** ** Context-agnostic LN analysis of operations *)
  (******************************************************************************)

  (** *** For [open] *)
  (******************************************************************************)
  Lemma in_open_eq_iff : forall k l u t,
      (k, l) ∈m t '(k | u) <-> exists w1 l1,
        (w1, (k, l1)) ∈md t /\ (k, l) ∈m open_loc k u (w1, l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l (u : T k LN) t,
(k, l) ∈m t '( k | u) <->
(exists w1 l1,
   (w1, k, l1) ∈md t /\
   (k, l) ∈m open_loc k u (w1, l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l (u : T k LN) t,
(k, l) ∈m t '( k | u) <->
(exists w1 l1,
   (w1, k, l1) ∈md t /\
   (k, l) ∈m open_loc k u (w1, l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
u: T k LN
t: U LN
(k, l) ∈m t '( k | u) <->
(exists w1 l1,
   (w1, k, l1) ∈md t /\
   (k, l) ∈m open_loc k u (w1, l1)) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
u: T k LN
t: U LN
(k, l) ∈m kbindd U k (open_loc k u) t <->
(exists w1 l1,
   (w1, k, l1) ∈md t /\
   (k, l) ∈m open_loc k u (w1, l1))
    now rewrite (in_kbindd_eq_iff U).
  Qed.

  Lemma in_open_neq_iff : forall k j l u t,
      k <> j ->
      (k, l) ∈m t '(j | u) <-> (k, l) ∈m t \/ exists w1 l1,
          (w1, (j, l1)) ∈md t /\ (k, l) ∈m open_loc j u (w1, l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l (u : T j LN) t,
k <> j ->
(k, l) ∈m t '( j | u) <->
(k, l) ∈m t \/
(exists w1 l1,
   (w1, j, l1) ∈md t /\
   (k, l) ∈m open_loc j u (w1, l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l (u : T j LN) t,
k <> j ->
(k, l) ∈m t '( j | u) <->
(k, l) ∈m t \/
(exists w1 l1,
   (w1, j, l1) ∈md t /\
   (k, l) ∈m open_loc j u (w1, l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
u: T j LN
t: U LN
H1: k <> j
(k, l) ∈m t '( j | u) <->
(k, l) ∈m t \/
(exists w1 l1,
   (w1, j, l1) ∈md t /\
   (k, l) ∈m open_loc j u (w1, l1)) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
u: T j LN
t: U LN
H1: k <> j
(k, l) ∈m kbindd U j (open_loc j u) t <->
(k, l) ∈m t \/
(exists w1 l1,
   (w1, j, l1) ∈md t /\
   (k, l) ∈m open_loc j u (w1, l1))
    now rewrite (in_kbindd_neq_iff U); auto.
  Qed.

  (** *** For [close] *)
  (******************************************************************************)
  Lemma in_close_eq_iff : forall k l x t,
      (k, l) ∈m '[k | x] t <-> exists w l1,
        (w, (k, l1)) ∈md t /\ l = close_loc k x (w, l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l x t,
(k, l) ∈m '[ k | x] t <->
(exists w l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l x t,
(k, l) ∈m '[ k | x] t <->
(exists w l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
x: atom
t: U LN
(k, l) ∈m '[ k | x] t <->
(exists w l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1)) unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
x: atom
t: U LN
(k, l) ∈m kmapd U k (close_loc k x) t <->
(exists w l1,
   (w, k, l1) ∈md t /\ l = close_loc k x (w, l1))
    now rewrite (in_kmapd_eq_iff U).
  Qed.

  Lemma in_close_neq_iff : forall k j l x t,
      k <> j ->
      (k, l) ∈m '[j | x] t <-> (k, l) ∈m t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l x t,
k <> j -> (k, l) ∈m '[ j | x] t <-> (k, l) ∈m t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l x t,
k <> j -> (k, l) ∈m '[ j | x] t <-> (k, l) ∈m t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
x: atom
t: U LN
H1: k <> j
(k, l) ∈m '[ j | x] t <-> (k, l) ∈m t unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
x: atom
t: U LN
H1: k <> j
(k, l) ∈m kmapd U j (close_loc j x) t <-> (k, l) ∈m t
    now rewrite (in_kmapd_neq_iff U); auto.
  Qed.

  (** *** For [subst] *)
  (******************************************************************************)
  Lemma in_subst_eq_iff : forall k l u t x,
      (k, l) ∈m t '{k | x ~> u} <-> exists l1,
        (k, l1) ∈m t /\ (k, l) ∈m subst_loc k x u l1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l (u : T k LN) t x,
(k, l) ∈m t '{ k | x ~> u} <->
(exists l1,
   (k, l1) ∈m t /\ (k, l) ∈m subst_loc k x u l1)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) l (u : T k LN) t x,
(k, l) ∈m t '{ k | x ~> u} <->
(exists l1,
   (k, l1) ∈m t /\ (k, l) ∈m subst_loc k x u l1)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
u: T k LN
t: U LN
x: atom
(k, l) ∈m t '{ k | x ~> u} <->
(exists l1,
   (k, l1) ∈m t /\ (k, l) ∈m subst_loc k x u l1) unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
u: T k LN
t: U LN
x: atom
(k, l) ∈m kbind U k (subst_loc k x u) t <->
(exists l1,
   (k, l1) ∈m t /\ (k, l) ∈m subst_loc k x u l1)
    now rewrite (in_kbind_eq_iff U).
  Qed.

  Lemma in_subst_neq_iff : forall k j l u t x,
      j <> k ->
      (k, l) ∈m t '{j | x ~> u} <-> (k, l) ∈m t \/ exists l1,
          (j, l1) ∈m t /\ (k, l) ∈m subst_loc j x u l1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l (u : T j LN) t x,
j <> k ->
(k, l) ∈m t '{ j | x ~> u} <->
(k, l) ∈m t \/
(exists l1,
   (j, l1) ∈m t /\ (k, l) ∈m subst_loc j x u l1)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) l (u : T j LN) t x,
j <> k ->
(k, l) ∈m t '{ j | x ~> u} <->
(k, l) ∈m t \/
(exists l1,
   (j, l1) ∈m t /\ (k, l) ∈m subst_loc j x u l1)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
u: T j LN
t: U LN
x: atom
H1: j <> k
(k, l) ∈m t '{ j | x ~> u} <->
(k, l) ∈m t \/
(exists l1,
   (j, l1) ∈m t /\ (k, l) ∈m subst_loc j x u l1) unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
l: LN
u: T j LN
t: U LN
x: atom
H1: j <> k
(k, l) ∈m kbind U j (subst_loc j x u) t <->
(k, l) ∈m t \/
(exists l1,
   (j, l1) ∈m t /\ (k, l) ∈m subst_loc j x u l1)
    now rewrite (in_kbind_neq_iff U).
  Qed.

End operations_specifications.


(** * Specifications for <<free>> *)
(******************************************************************************)
Section operations_specifications.

  (** ** Characterizations for [free] and [FV] *)
  (******************************************************************************)
  Section characterize_free.

    Context
      {U : Type -> Type}
      `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
      `{! MultiDecoratedTraversableMonad (list K) T}.

    #[local] Instance: Compat_ToSubset_Tolist list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
Compat_ToSubset_Tolist list
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
Compat_ToSubset_Tolist list
      unfold Compat_ToSubset_Tolist.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
@tosubset list ToSubset_list =
@tosubset list ToSubset_Tolist
      unfold tosubset, ToSubset_list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
(fun (A : Type) (l : list A) (a : A) => List.In a l) =
ToSubset_Tolist
      unfold ToSubset_Tolist, ToSubset_list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
(fun (A : Type) (l : list A) (a : A) => List.In a l) =
(fun A : Type => tosubset ∘ tolist)
      unfold compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
(fun (A : Type) (l : list A) (a : A) => List.In a l) =
(fun A : Type => tosubset ○ tolist) ext A l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
A: Type
l: list A
(fun a : A => List.In a l) = tosubset (tolist l)
      rewrite Tolist_list_id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
A: Type
l: list A
(fun a : A => List.In a l) = tosubset (id l)
      unfold tosubset.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
A: Type
l: list A
(fun a : A => List.In a l) =
(fun a : A => List.In a (id l))
      reflexivity.
    Qed.

    Theorem in_free_iff : forall (k : K) (t : U LN) (x : atom),
        x ∈ free U k t <-> (k, Fr x) ∈m t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x : atom),
x ∈ free U k t <-> (k, Fr x) ∈m t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x : atom),
x ∈ free U k t <-> (k, Fr x) ∈m t
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ free U k t <-> (k, Fr x) ∈m t unfold free.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ bind free_loc (toklist U k t) <-> (k, Fr x) ∈m t
      replace (x ∈ bind free_loc (toklist U k t)) with
        (x ∈ tolist (bind free_loc (toklist U k t))) by
        now (rewrite Tolist_list_id).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ tolist (bind free_loc (toklist U k t)) <->
(k, Fr x) ∈m t
      rewrite <- in_iff_in_tolist.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ bind free_loc (toklist U k t) <-> (k, Fr x) ∈m t
      rewrite in_bind_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
(exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a) <->
(k, Fr x) ∈m t
      split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
(exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a) ->
(k, Fr x) ∈m t
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
(k, Fr x) ∈m t ->
exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
(exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a) ->
(k, Fr x) ∈m t intros [l [lin xin]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
l: LN
lin: l ∈ toklist U k t
xin: x ∈ free_loc l
(k, Fr x) ∈m t
        rewrite in_iff_in_toklist.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
l: LN
lin: l ∈ toklist U k t
xin: x ∈ free_loc l
Fr x ∈ toklist U k t
        destruct l as [a|n].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, a: atom
lin: Fr a ∈ toklist U k t
xin: x ∈ free_loc (Fr a)
Fr x ∈ toklist U k t
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
n: nat
lin: Bd n ∈ toklist U k t
xin: x ∈ free_loc (Bd n)
Fr x ∈ toklist U k t
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, a: atom
lin: Fr a ∈ toklist U k t
xin: x ∈ free_loc (Fr a)
Fr x ∈ toklist U k t cbv in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, a: atom
lin: Fr a ∈ toklist U k t
xin: a = x \/ False
Fr x ∈ toklist U k t destruct xin as [?|[]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, a: atom
lin: Fr a ∈ toklist U k t
H1: a = x
Fr x ∈ toklist U k t
          subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lin: Fr x ∈ toklist U k t
Fr x ∈ toklist U k t cbn in lin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lin: Fr x ∈ toklist U k t
Fr x ∈ toklist U k t assumption.
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
n: nat
lin: Bd n ∈ toklist U k t
xin: x ∈ free_loc (Bd n)
Fr x ∈ toklist U k t cbv in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
n: nat
lin: Bd n ∈ toklist U k t
xin: False
Fr x ∈ toklist U k t destruct xin as [].
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
(k, Fr x) ∈m t ->
exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: (k, Fr x) ∈m t
exists a : LN, a ∈ toklist U k t /\ x ∈ free_loc a exists (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: (k, Fr x) ∈m t
Fr x ∈ toklist U k t /\ x ∈ free_loc (Fr x) rewrite <- (in_iff_in_toklist).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: (k, Fr x) ∈m t
(k, Fr x) ∈m t /\ x ∈ free_loc (Fr x)
        split; [assumption| ].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: (k, Fr x) ∈m t
x ∈ free_loc (Fr x)
        cbv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
H1: (k, Fr x) ∈m t
x = x \/ False now left.
    Qed.

    Corollary free_iff_FV : forall (k : K) (t : U LN) (x : atom),
        x ∈ free U k t <-> x `in` FV U k t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x : atom),
x ∈ free U k t <-> x `in` FV U k t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x : atom),
x ∈ free U k t <-> x `in` FV U k t
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ free U k t <-> x `in` FV U k t unfold FV.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ free U k t <-> x `in` atoms (free U k t) rewrite <- in_atoms_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x ∈ free U k t <-> x ∈ free U k t
      reflexivity.
    Qed.

  End characterize_free.

  (** ** [free] variables after <<open>>/<<close>>/<<subst>> *)
  (******************************************************************************)
  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  (** *** After [open] *)
  (******************************************************************************)
  Lemma free_open_eq_iff : forall k u t x,
      x ∈ free U k (t '(k | u)) <-> exists w l1,
        (w, (k, l1)) ∈md t /\ x ∈ free (T k) k (open_loc k u (w, l1)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (u : T k LN) (t : U LN) (x : atom),
x ∈ free U k (t '( k | u)) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) k (open_loc k u (w, l1)))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (u : T k LN) (t : U LN) (x : atom),
x ∈ free U k (t '( k | u)) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) k (open_loc k u (w, l1)))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
x ∈ free U k (t '( k | u)) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) k (open_loc k u (w, l1)))
    rewrite in_free_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
(k, Fr x) ∈m t '( k | u) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) k (open_loc k u (w, l1)))
    setoid_rewrite in_free_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
(k, Fr x) ∈m t '( k | u) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   (k, Fr x) ∈m open_loc k u (w, l1))
    now rewrite in_open_eq_iff.
  Qed.

  Lemma free_open_neq_iff : forall k j u t x,
      k <> j ->
      x ∈ free U j (t '(k | u)) <-> x ∈ free U j t \/ exists w l1,
          (w, (k, l1)) ∈md t /\ x ∈ free (T k) j (open_loc k u (w, l1)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) (u : T k LN) (t : U LN) (x : atom),
k <> j ->
x ∈ free U j (t '( k | u)) <->
x ∈ free U j t \/
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) j (open_loc k u (w, l1)))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) (u : T k LN) (t : U LN) (x : atom),
k <> j ->
x ∈ free U j (t '( k | u)) <->
x ∈ free U j t \/
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) j (open_loc k u (w, l1)))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
H1: k <> j
x ∈ free U j (t '( k | u)) <->
x ∈ free U j t \/
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) j (open_loc k u (w, l1))) rewrite 2(in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
H1: k <> j
(j, Fr x) ∈m t '( k | u) <->
(j, Fr x) ∈m t \/
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   x ∈ free (T k) j (open_loc k u (w, l1))) setoid_rewrite in_free_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
H1: k <> j
(j, Fr x) ∈m t '( k | u) <->
(j, Fr x) ∈m t \/
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\
   (j, Fr x) ∈m open_loc k u (w, l1))
    now rewrite in_open_neq_iff; auto.
  Qed.

  (** *** After [close] *)
  (******************************************************************************)
  Lemma free_close_eq_iff : forall k t x y,
      y ∈ free U k ('[k | x] t) <-> exists w l1,
        (w, (k, l1)) ∈md t /\ Fr y = close_loc k x (w, l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x y : atom),
y ∈ free U k ('[ k | x] t) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (t : U LN) (x y : atom),
y ∈ free U k ('[ k | x] t) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
y ∈ free U k ('[ k | x] t) <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)) rewrite in_free_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(k, Fr y) ∈m '[ k | x] t <->
(exists (w : list K) (l1 : LN),
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1))
    now rewrite (in_close_eq_iff U).
  Qed.

  Lemma free_close_neq_iff : forall k j t x,
      k <> j ->
      x ∈ free U j ('[k | x] t) <-> x ∈ free U j t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) (t : U LN) (x : atom),
k <> j ->
x ∈ free U j ('[ k | x] t) <-> x ∈ free U j t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k j : K) (t : U LN) (x : atom),
k <> j ->
x ∈ free U j ('[ k | x] t) <-> x ∈ free U j t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
x ∈ free U j ('[ k | x] t) <-> x ∈ free U j t rewrite 2(in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
(j, Fr x) ∈m '[ k | x] t <-> (j, Fr x) ∈m t
    now rewrite (in_close_neq_iff U); auto.
  Qed.

  (** *** For [subst] *)
  (******************************************************************************)
  Lemma free_subst_eq_iff : forall k u t x y,
      y ∈ free U k (t '{k | x ~> u}) <-> exists l1,
        (k, l1) ∈m t /\ y ∈ free (T k) k (subst_loc k x u l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (u : T k LN) (t : U LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) <->
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) k (subst_loc k x u l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (k : K) (u : T k LN) (t : U LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) <->
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) k (subst_loc k x u l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
y ∈ free U k (t '{ k | x ~> u}) <->
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) k (subst_loc k x u l1)) rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
(k, Fr y) ∈m t '{ k | x ~> u} <->
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) k (subst_loc k x u l1)) setoid_rewrite (in_free_iff (U := T k)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
(k, Fr y) ∈m t '{ k | x ~> u} <->
(exists l1 : LN,
   (k, l1) ∈m t /\ (k, Fr y) ∈m subst_loc k x u l1)
    now rewrite (in_subst_eq_iff U).
  Qed.

  Lemma free_subst_neq_iff : forall j k u t x y,
      k <> j ->
      y ∈ free U j (t '{k | x ~> u}) <-> y ∈ free U j t \/ exists l1,
        (k, l1) ∈m t /\ y ∈ free (T k) j (subst_loc k x u l1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (j k : K) (u : T k LN) (t : U LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) j (subst_loc k x u l1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (j k : K) (u : T k LN) (t : U LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) j (subst_loc k x u l1))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k: K
u: T k LN
t: U LN
x, y: atom
H1: k <> j
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) j (subst_loc k x u l1)) rewrite 2(in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k: K
u: T k LN
t: U LN
x, y: atom
H1: k <> j
(j, Fr y) ∈m t '{ k | x ~> u} <->
(j, Fr y) ∈m t \/
(exists l1 : LN,
   (k, l1) ∈m t /\
   y ∈ free (T k) j (subst_loc k x u l1)) setoid_rewrite (in_free_iff (U := T k)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k : K, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k: K
u: T k LN
t: U LN
x, y: atom
H1: k <> j
(j, Fr y) ∈m t '{ k | x ~> u} <->
(j, Fr y) ∈m t \/
(exists l1 : LN,
   (k, l1) ∈m t /\ (j, Fr y) ∈m subst_loc k x u l1)
    now rewrite (in_subst_neq_iff); auto.
  Qed.

End operations_specifications.

(** * Lemmas for local reasoning *)
(** The following lemmas govern the behavior of the <<*_loc>> operations of
      the locally nameless library. These are put into a hint database
      <<tea_local>> to use with <<autorewrite>>. Since <<autorewrite>> tries the
      first unifying lemma (even if this generates absurd side conditions), we
      use <<Hint Rewrite ... using ...>> clauses and typically prefer
      <<autorewrite*>>, which only uses hints whose side conditions are
      discharged by the associated tactic. *)
(******************************************************************************)
Create HintDb tea_local.

Section locally_nameless_local.

  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Implicit Types (l : LN) (w : list K) (t : U LN) (x : atom) (j k : K) (n : nat).

  (** ** Lemmas for proving (in)equalities between leaves *)
  (******************************************************************************)
  Lemma Fr_injective : forall (x y : atom),
      (Fr x = Fr y) <-> (x = y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, Fr x = Fr y <-> x = y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, Fr x = Fr y <-> x = y
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
Fr x = Fr y <-> x = y split; intro hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: Fr x = Fr y
x = y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: x = y
Fr x = Fr y
    now injection hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: x = y
Fr x = Fr y now subst.
  Qed.

  Lemma Fr_injective_not_iff : forall (x y : atom),
      (Fr x <> Fr y) <-> (x <> y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, Fr x <> Fr y <-> x <> y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, Fr x <> Fr y <-> x <> y
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
Fr x <> Fr y <-> x <> y split; intro hyp; contradict hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: x = y
Fr x = Fr y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: Fr x = Fr y
x = y
    now subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
hyp: Fr x = Fr y
x = y now injection hyp.
  Qed.

  Lemma Fr_injective_not : forall (x y : atom),
      x <> y -> ~ (Fr x = Fr y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, x <> y -> Fr x <> Fr y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall x y : atom, x <> y -> Fr x <> Fr y
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
x, y: atom
H1: x <> y
Fr x <> Fr y now rewrite Fr_injective.
  Qed.

  Lemma B_neq_Fr : forall n x,
      (Bd n = Fr x) = False.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall n x, (Bd n = Fr x) = False
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall n x, (Bd n = Fr x) = False
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
n: nat
x: atom
(Bd n = Fr x) = False propext.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
n: nat
x: atom
Bd n = Fr x -> False
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
n: nat
x: atom
False -> Bd n = Fr x discriminate.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
n: nat
x: atom
False -> Bd n = Fr x contradiction.
  Qed.

  Lemma prod_K_not_iff : forall (k : K) A (x y : A),
      ((k, x) <> (k, y)) <-> (x <> y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (A : Type) (x y : A),
(k, x) <> (k, y) <-> x <> y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (A : Type) (x y : A),
(k, x) <> (k, y) <-> x <> y
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
A: Type
x, y: A
(k, x) <> (k, y) <-> x <> y split; intro hyp; contradict hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
A: Type
x, y: A
hyp: x = y
(k, x) = (k, y)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
A: Type
x, y: A
hyp: (k, x) = (k, y)
x = y
    now subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
A: Type
x, y: A
hyp: (k, x) = (k, y)
x = y now injection hyp.
  Qed.

  Lemma ninf_in_neq : forall k x l (t : U LN),
      ~ x ∈ free U k t ->
      (k, l) ∈m t -> Fr x <> l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l t,
~ x ∈ free U k t -> (k, l) ∈m t -> Fr x <> l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l t,
~ x ∈ free U k t -> (k, l) ∈m t -> Fr x <> l
    introv hyp1 hyp2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
hyp1: ~ x ∈ free U k t
hyp2: (k, l) ∈m t
Fr x <> l
    rewrite in_free_iff in hyp1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, l) ∈m t
Fr x <> l destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x, n: atom
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, Fr n) ∈m t
Fr x <> Fr n
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
n: nat
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, Bd n) ∈m t
Fr x <> Bd n
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x, n: atom
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, Fr n) ∈m t
Fr x <> Fr n injection.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x, n: atom
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, Fr n) ∈m t
H1: Fr x = Fr n
x = n -> False intros; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: atom
t: U LN
hyp1: ~ (k, Fr n) ∈m t
hyp2: (k, Fr n) ∈m t
H1: Fr n = Fr n
False
      contradiction.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
n: nat
t: U LN
hyp1: ~ (k, Fr x) ∈m t
hyp2: (k, Bd n) ∈m t
Fr x <> Bd n discriminate.
  Qed.

  Lemma neq_symmetry : forall X (x y : X),
      (x <> y) = (y <> x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (X : Type) (x y : X), (x <> y) = (y <> x)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall (X : Type) (x y : X), (x <> y) = (y <> x)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
X: Type
x, y: X
(x <> y) = (y <> x) propext; intro hyp; contradict hyp; congruence.
  Qed.

  (** ** [subst_loc] *)
  (******************************************************************************)
  Lemma subst_loc_eq : forall k u x,
      subst_loc k x u (Fr x) = u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x, subst_loc k x u (Fr x) = u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x, subst_loc k x u (Fr x) = u
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
subst_loc k x u (Fr x) = u cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
(if x == x then u else mret T k (Fr x)) = u now compare values x and x.
  Qed.

  Lemma subst_loc_neq : forall k u x y,
      y <> x ->
      subst_loc k x u (Fr y) = mret T k (Fr y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) (x y : atom),
y <> x -> subst_loc k x u (Fr y) = mret T k (Fr y)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) (x y : atom),
y <> x -> subst_loc k x u (Fr y) = mret T k (Fr y)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x, y: atom
H1: y <> x
subst_loc k x u (Fr y) = mret T k (Fr y) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x, y: atom
H1: y <> x
(if x == y then u else mret T k (Fr y)) =
mret T k (Fr y) now compare values x and y.
  Qed.

  Lemma subst_loc_b : forall k u x n,
      subst_loc k x u (Bd n) = mret T k (Bd n).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x n,
subst_loc k x u (Bd n) = mret T k (Bd n)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x n,
subst_loc k x u (Bd n) = mret T k (Bd n)
    reflexivity.
  Qed.

  Lemma subst_loc_fr_neq : forall k u l x,
      Fr x <> l ->
      subst_loc k x u l = mret T k l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) l x,
Fr x <> l -> subst_loc k x u l = mret T k l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) l x,
Fr x <> l -> subst_loc k x u l = mret T k l
    introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
l: LN
x: atom
neq: Fr x <> l
subst_loc k x u l = mret T k l unfold subst_loc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
l: LN
x: atom
neq: Fr x <> l
match l with
| Fr y => if x == y then u else mret T k (Fr y)
| Bd n => mret T k (Bd n)
end = mret T k l
    destruct l as [a|?]; [compare values x and a | reflexivity].
  Qed.

  Lemma subst_in_mret_eq : forall k x l u,
      (mret T k l) '{k | x ~> u} = subst_loc k x u l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l (u : T k LN),
mret T k l '{ k | x ~> u} = subst_loc k x u l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l (u : T k LN),
mret T k l '{ k | x ~> u} = subst_loc k x u l
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
u: T k LN
mret T k l '{ k | x ~> u} = subst_loc k x u l unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
u: T k LN
kbind (T k) k (subst_loc k x u) (mret T k l) =
subst_loc k x u l
    now rewrite kbind_comp_mret_eq.
  Qed.

  Lemma subst_in_mret_neq : forall k j x l u,
      j <> k ->
      (mret T k l) '{j | x ~> u} = mret T k l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j x l (u : T j LN),
j <> k -> mret T k l '{ j | x ~> u} = mret T k l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j x l (u : T j LN),
j <> k -> mret T k l '{ j | x ~> u} = mret T k l
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
x: atom
l: LN
u: T j LN
H1: j <> k
mret T k l '{ j | x ~> u} = mret T k l unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
x: atom
l: LN
u: T j LN
H1: j <> k
kbind (T k) j (subst_loc j x u) (mret T k l) =
mret T k l
    now rewrite kbind_comp_mret_neq.
  Qed.

  (** ** [open_loc] *)
  (******************************************************************************)
  Lemma open_loc_lt : forall k u w n,
      n < countk k w ->
      open_loc k u (w, Bd n) = mret T k (Bd n).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w n,
n < countk k w ->
open_loc k u (w, Bd n) = mret T k (Bd n)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w n,
n < countk k w ->
open_loc k u (w, Bd n) = mret T k (Bd n)
    introv ineq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
n: nat
ineq: n < countk k w
open_loc k u (w, Bd n) = mret T k (Bd n) unfold open_loc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
n: nat
ineq: n < countk k w
match Nat.compare n (countk k w) with
| Eq => u
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd n) compare naturals n and (countk k w).
  Qed.

  Lemma open_loc_gt : forall k u n w,
      n > countk k w ->
      open_loc k u (w, Bd n) = mret T k (Bd (n - 1)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) n w,
n > countk k w ->
open_loc k u (w, Bd n) = mret T k (Bd (n - 1))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) n w,
n > countk k w ->
open_loc k u (w, Bd n) = mret T k (Bd (n - 1))
    introv ineq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
n: nat
w: list K
ineq: n > countk k w
open_loc k u (w, Bd n) = mret T k (Bd (n - 1)) unfold open_loc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
n: nat
w: list K
ineq: n > countk k w
match Nat.compare n (countk k w) with
| Eq => u
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd (n - 1)) compare naturals n and (countk k w).
  Qed.

  Lemma open_loc_eq : forall k w u,
      open_loc k u (w, Bd (countk k w)) = u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w (u : T k LN),
open_loc k u (w, Bd (countk k w)) = u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w (u : T k LN),
open_loc k u (w, Bd (countk k w)) = u
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
u: T k LN
open_loc k u (w, Bd (countk k w)) = u unfold open_loc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
u: T k LN
match Nat.compare (countk k w) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end = u compare naturals (countk k w) and (countk k w).
  Qed.

  Lemma open_loc_atom : forall k u w x,
      open_loc k u (w, Fr x) = mret T k (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w x,
open_loc k u (w, Fr x) = mret T k (Fr x)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w x,
open_loc k u (w, Fr x) = mret T k (Fr x)
    reflexivity.
  Qed.

End locally_nameless_local.

(** ** Tactics for local reasoning *)
(******************************************************************************)
Tactic Notation "unfold_monoid" :=
  repeat unfold monoid_op, Monoid_op_list, Monoid_op_list,
  monoid_unit, Monoid_unit_list, Monoid_unit_list in *.

#[export] Hint Rewrite @prod_K_not_iff : tea_local.

(** Rewrite rules for expressions of the form <<x ∈ mret T y>> *)
#[export] Hint Rewrite
     @in_mret_iff @in_mret_eq_iff
     @inmd_mret_iff @inmd_mret_eq_iff using typeclasses eauto : tea_local.

(** Rewrite rules for simplifying expressions involving equalities between leaves *)
#[export] Hint Rewrite
     Fr_injective Fr_injective_not_iff B_neq_Fr : tea_local.

(** Solve goals of the form <<Fr x <> Fr y>> by using <<x <> y>> *)
#[export] Hint Resolve
 Fr_injective_not : tea_local.

#[export] Hint Rewrite
     @subst_loc_eq @subst_in_mret_eq
     using typeclasses eauto : tea_local.
#[export] Hint Rewrite
     @subst_loc_neq @subst_loc_b @subst_loc_fr_neq @subst_in_mret_neq
     using first [ typeclasses eauto | auto ] : tea_local.
#[export] Hint Rewrite
     @open_loc_lt @open_loc_gt
     using first [ typeclasses eauto | auto ] : tea_local.
#[export] Hint Rewrite
     @open_loc_eq @open_loc_atom
     using typeclasses eauto : tea_local.

Tactic Notation "simpl_local" := (autorewrite* with tea_local).

(** * Metatheory for the <<subst>> operation *)
(******************************************************************************)
Section subst_metatheory.

  Section fix_dtm.

    Context
      (U : Type -> Type)
      `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
      `{! MultiDecoratedTraversableMonad (list K) T}.

    Implicit Types
             (k : K) (j : K)
             (l : LN) (p : LN)
             (x : atom) (t : U LN)
             (w : list K) (n : nat).

    (** ** LN analysis with contexts *)
    (******************************************************************************)
    Lemma inmd_subst_loc_iff : forall k l w j p u x,
        (w, (j, p)) ∈md subst_loc k x u l <->
        l <> Fr x /\ k = j /\ w = Ƶ /\ l = p \/ (* l is not replaced *)
        l = Fr x /\ (w, (j, p)) ∈md u. (* l is replaced *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w j p (u : T k LN) x,
(w, j, p) ∈md subst_loc k x u l <->
l <> Fr x /\ k = j /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, j, p) ∈md u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w j p (u : T k LN) x,
(w, j, p) ∈md subst_loc k x u l <->
l <> Fr x /\ k = j /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, j, p) ∈md u
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
j: K
p: LN
u: T k LN
x: atom
(w, j, p) ∈md subst_loc k x u l <->
l <> Fr x /\ k = j /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, j, p) ∈md u compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
(w, j, p) ∈md subst_loc k x u (Fr x) <->
Fr x <> Fr x /\ k = j /\ w = Ƶ /\ Fr x = p \/
Fr x = Fr x /\ (w, j, p) ∈md u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x, a: atom
H1: a <> x
(w, j, p) ∈md subst_loc k x u (Fr a) <->
Fr a <> Fr x /\ k = j /\ w = Ƶ /\ Fr a = p \/
Fr a = Fr x /\ (w, j, p) ∈md u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
n: nat
(w, j, p) ∈md subst_loc k x u (Bd n) <->
Bd n <> Fr x /\ k = j /\ w = Ƶ /\ Bd n = p \/
Bd n = Fr x /\ (w, j, p) ∈md u
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
(w, j, p) ∈md subst_loc k x u (Fr x) <->
Fr x <> Fr x /\ k = j /\ w = Ƶ /\ Fr x = p \/
Fr x = Fr x /\ (w, j, p) ∈md u rewrite subst_loc_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
(w, j, p) ∈md u <->
Fr x <> Fr x /\ k = j /\ w = Ƶ /\ Fr x = p \/
Fr x = Fr x /\ (w, j, p) ∈md u
        clear; firstorder.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x, a: atom
H1: a <> x
(w, j, p) ∈md subst_loc k x u (Fr a) <->
Fr a <> Fr x /\ k = j /\ w = Ƶ /\ Fr a = p \/
Fr a = Fr x /\ (w, j, p) ∈md u rewrite subst_loc_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x, a: atom
H1: a <> x
(w, j, p) ∈md mret T k (Fr a) <->
Fr a <> Fr x /\ k = j /\ w = Ƶ /\ Fr a = p \/
Fr a = Fr x /\ (w, j, p) ∈md u
        rewrite inmd_mret_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x, a: atom
H1: a <> x
w = Ƶ /\ k = j /\ Fr a = p <->
Fr a <> Fr x /\ k = j /\ w = Ƶ /\ Fr a = p \/
Fr a = Fr x /\ (w, j, p) ∈md u
        rewrite Fr_injective.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x, a: atom
H1: a <> x
w = Ƶ /\ k = j /\ Fr a = p <->
a <> x /\ k = j /\ w = Ƶ /\ Fr a = p \/
a = x /\ (w, j, p) ∈md u
        firstorder.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
n: nat
(w, j, p) ∈md subst_loc k x u (Bd n) <->
Bd n <> Fr x /\ k = j /\ w = Ƶ /\ Bd n = p \/
Bd n = Fr x /\ (w, j, p) ∈md u rewrite subst_loc_b.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
n: nat
(w, j, p) ∈md mret T k (Bd n) <->
Bd n <> Fr x /\ k = j /\ w = Ƶ /\ Bd n = p \/
Bd n = Fr x /\ (w, j, p) ∈md u
        rewrite inmd_mret_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
n: nat
w = Ƶ /\ k = j /\ Bd n = p <->
Bd n <> Fr x /\ k = j /\ w = Ƶ /\ Bd n = p \/
Bd n = Fr x /\ (w, j, p) ∈md u
        rewrite B_neq_Fr.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
j: K
p: LN
u: T k LN
x: atom
n: nat
w = Ƶ /\ k = j /\ Bd n = p <->
~ False /\ k = j /\ w = Ƶ /\ Bd n = p \/
False /\ (w, j, p) ∈md u
        firstorder.
    Qed.

    Corollary inmd_subst_loc_iff_eq : forall k l w p u x,
        (w, (k, p)) ∈md subst_loc k x u l <->
        l <> Fr x /\ w = Ƶ /\ l = p \/
        l = Fr x /\ (w, (k, p)) ∈md u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w p (u : T k LN) x,
(w, k, p) ∈md subst_loc k x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w p (u : T k LN) x,
(w, k, p) ∈md subst_loc k x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
p: LN
u: T k LN
x: atom
(w, k, p) ∈md subst_loc k x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u rewrite inmd_subst_loc_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
p: LN
u: T k LN
x: atom
l <> Fr x /\ k = k /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u <->
l <> Fr x /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u clear.ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
H: forall k, MBind (list K) T (T k)
k: K
l: LN
w: list K
p: LN
u: T k LN
x: atom
l <> Fr x /\ k = k /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u <->
l <> Fr x /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, k, p) ∈md u firstorder.
    Qed.

    Corollary inmd_subst_loc_iff_neq : forall k l w j p u x,
        k <> j ->
        (w, (j, p)) ∈md subst_loc k x u l <->
        l = Fr x /\ (w, (j, p)) ∈md u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w j p (u : T k LN) x,
k <> j ->
(w, j, p) ∈md subst_loc k x u l <->
l = Fr x /\ (w, j, p) ∈md u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l w j p (u : T k LN) x,
k <> j ->
(w, j, p) ∈md subst_loc k x u l <->
l = Fr x /\ (w, j, p) ∈md u
      introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
j: K
p: LN
u: T k LN
x: atom
neq: k <> j
(w, j, p) ∈md subst_loc k x u l <->
l = Fr x /\ (w, j, p) ∈md u rewrite inmd_subst_loc_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
w: list K
j: K
p: LN
u: T k LN
x: atom
neq: k <> j
l <> Fr x /\ k = j /\ w = Ƶ /\ l = p \/
l = Fr x /\ (w, j, p) ∈md u <->
l = Fr x /\ (w, j, p) ∈md u intuition.
    Qed.

    Theorem inmd_subst_iff : forall k j w t u l x,
        (* <<l>> occurs in the result of a <<subst>> IFF *)
        (w, j, l) ∈md t '{k | x ~> u} <->
        (* <<l>> is an occurrence in <<t>> not of the same sort as the <<subst>> OR *)
        k <> j /\ (w, j, l) ∈md t \/
        (* <<l>> is an occurrence of the same sort but not an occurrence of <<x>> OR *)
        k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
        (* <<x>> occurs and <<l>> is the any LN in a substituted <<u>> *)
        exists w1 w2 : list K, (w1, k, Fr x) ∈md t /\ (w2, j, l) ∈md u /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j w t (u : T k LN) l x,
(w, j, l) ∈md t '{ k | x ~> u} <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j w t (u : T k LN) l x,
(w, j, l) ∈md t '{ k | x ~> u} <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
(w, j, l) ∈md t '{ k | x ~> u} <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) compare values k and j.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(w, j, l) ∈md t '{ j | x ~> u} <->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w, j, l) ∈md t '{ k | x ~> u} <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(w, j, l) ∈md t '{ j | x ~> u} <->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) rewrite (inmd_subst_iff_eq U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc j x u l1 /\ w = w1 ● w2) <->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) setoid_rewrite (inmd_subst_loc_iff_eq j).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2) <->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
        split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2) ->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) ->
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ 
  w = w1 ● w2
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2) ->
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) intros [l' [n1 [n2 conditions]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
conditions: (l', j, n2) ∈md t /\
(n2 <> Fr x /\ n1 = Ƶ /\ n2 = l \/
 n2 = Fr x /\ (n1, j, l) ∈md u) /\
w = l' ● n1
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
          right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
conditions: (l', j, n2) ∈md t /\
(n2 <> Fr x /\ n1 = Ƶ /\ n2 = l \/
 n2 = Fr x /\ (n1, j, l) ∈md u) /\
w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) destruct conditions as [c1 [[c2|c2] c3]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
c3: w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 = Fr x /\ (n1, j, l) ∈md u
c3: w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
c3: w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
j = j /\ (l' ● n1, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ l' ● n1 = w1 ● w2) left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
j = j /\ (l' ● n1, j, l) ∈md t /\ l <> Fr x destructs c2; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l': list K
H1: l <> Fr x
c1: (l', j, l) ∈md t
j = j /\ (l' ● Ƶ, j, l) ∈md t /\ l <> Fr x
            rewrite monoid_id_r.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l': list K
H1: l <> Fr x
c1: (l', j, l) ∈md t
j = j /\ (l', j, l) ∈md t /\ l <> Fr x auto. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 = Fr x /\ (n1, j, l) ∈md u
c3: w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 = Fr x /\ (n1, j, l) ∈md u
c3: w = l' ● n1
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 = Fr x /\ (n1, j, l) ∈md u
j = j /\ (l' ● n1, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ l' ● n1 = w1 ● w2) right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
n2: LN
c1: (l', j, n2) ∈md t
c2: n2 = Fr x /\ (n1, j, l) ∈md u
exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ l' ● n1 = w1 ● w2 destructs c2; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
l', n1: list K
c1: (l', j, Fr x) ∈md t
H2: (n1, j, l) ∈md u
exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ l' ● n1 = w1 ● w2 eauto. }
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
j <> j /\ (w, j, l) ∈md t \/
j = j /\ (w, j, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, j, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) ->
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 intros [[contra ?] | [ [? [in_t neq]] | hyp ] ].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
contra: j <> j
H1: (w, j, l) ∈md t
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (w, j, l) ∈md t
neq: l <> Fr x
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ 
  w = w1 ● w2
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
hyp: exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ 
  w = w1 ● w2
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
contra: j <> j
H1: (w, j, l) ∈md t
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 contradiction. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (w, j, l) ∈md t
neq: l <> Fr x
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
hyp: exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ 
  w = w1 ● w2
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (w, j, l) ∈md t
neq: l <> Fr x
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 exists w (Ƶ : list K) l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (w, j, l) ∈md t
neq: l <> Fr x
(w, j, l) ∈md t /\
(l <> Fr x /\ Ƶ = Ƶ /\ l = l \/
 l = Fr x /\ (Ƶ, j, l) ∈md u) /\ w = w ● Ƶ rewrite monoid_id_r.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (w, j, l) ∈md t
neq: l <> Fr x
(w, j, l) ∈md t /\
(l <> Fr x /\ Ƶ = Ƶ /\ l = l \/
 l = Fr x /\ (Ƶ, j, l) ∈md u) /\ w = w split... }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
hyp: exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
hyp: exists w1 w2,
  (w1, j, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 destruct hyp as [w1 [w2 ?]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
w1, w2: list K
H1: (w1, j, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (l1 <> Fr x /\ w2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 exists w1 w2 (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
w: list K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
w1, w2: list K
H1: (w1, j, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w1, j, Fr x) ∈md t /\
(Fr x <> Fr x /\ w2 = Ƶ /\ Fr x = l \/
 Fr x = Fr x /\ (w2, j, l) ∈md u) /\ w = w1 ● w2 intuition. }
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w, j, l) ∈md t '{ k | x ~> u} <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) rewrite (inmd_subst_neq_iff U)...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) <->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) ->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) ->
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) ->
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) intros [? | [n1 [n2 [p [in_t in_local]]]]]...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
n1, n2: list K
p: LN
in_t: (n1, k, p) ∈md t
in_local: (n2, j, l) ∈md subst_loc k x u p /\
w = n1 ● n2
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
          rewrite (inmd_subst_loc_iff_neq k) in in_local...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
n1, n2: list K
p: LN
in_t: (n1, k, p) ∈md t
in_local: (p = Fr x /\ (n2, j, l) ∈md u) /\
w = n1 ● n2
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
          right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
n1, n2: list K
p: LN
in_t: (n1, k, p) ∈md t
in_local: (p = Fr x /\ (n2, j, l) ∈md u) /\
w = n1 ● n2
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
n1, n2: list K
p: LN
in_t: (n1, k, p) ∈md t
in_local: (p = Fr x /\ (n2, j, l) ∈md u) /\
w = n1 ● n2
exists w1 w2,
  (w1, k, Fr x) ∈md t /\
  (w2, j, l) ∈md u /\ w = w1 ● w2 exists n1 n2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
n1, n2: list K
p: LN
in_t: (n1, k, p) ∈md t
in_local: (p = Fr x /\ (n2, j, l) ∈md u) /\
w = n1 ● n2
(n1, k, Fr x) ∈md t /\ (n2, j, l) ∈md u /\ w = n1 ● n2 destruct in_local as [[? ?] ?]; subst...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) ->
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) intros [[? ?] | [[? ?] | [w1 [w2 rest]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k <> j
H2: (w, j, l) ∈md t
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (w, k, l) ∈md t /\ l <> Fr x
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k <> j
H2: (w, j, l) ∈md t
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) auto. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (w, k, l) ∈md t /\ l <> Fr x
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (w, k, l) ∈md t /\ l <> Fr x
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) contradiction. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w, j, l) ∈md t \/
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2) right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, j, l) ∈md subst_loc k x u l1 /\ w = w1 ● w2 exists w1 w2 (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
w1, w2: list K
rest: (w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md u /\ w = w1 ● w2
(w1, k, Fr x) ∈md t /\
(w2, j, l) ∈md subst_loc k x u (Fr x) /\ w = w1 ● w2 simpl_local... }
    Qed.

    Corollary inmd_subst_iff_eq' : forall k w t u l x,
        (w, k, l) ∈md t '{k | x ~> u} <->
        (w, k, l) ∈md t /\ l <> Fr x \/
        exists w1 w2 : list K, (w1, k, Fr x) ∈md t /\ (w2, k, l) ∈md u /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w t (u : T k LN) l x,
(w, k, l) ∈md t '{ k | x ~> u} <->
(w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w t (u : T k LN) l x,
(w, k, l) ∈md t '{ k | x ~> u} <->
(w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2)
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
(w, k, l) ∈md t '{ k | x ~> u} <->
(w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2) rewrite inmd_subst_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
k <> k /\ (w, k, l) ∈md t \/
k = k /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2) <->
(w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2) intuition.
    Qed.

    Corollary inmd_subst_iff_neq' : forall k j w t u l x,
        k <> j ->
        (w, j, l) ∈md t '{k | x ~> u} <->
        (w, j, l) ∈md t \/
        exists w1 w2 : list K, (w1, k, Fr x) ∈md t /\ (w2, j, l) ∈md u /\ w = w1 ● w2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j w t (u : T k LN) l x,
k <> j ->
(w, j, l) ∈md t '{ k | x ~> u} <->
(w, j, l) ∈md t \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j w t (u : T k LN) l x,
k <> j ->
(w, j, l) ∈md t '{ k | x ~> u} <->
(w, j, l) ∈md t \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
H1: k <> j
(w, j, l) ∈md t '{ k | x ~> u} <->
(w, j, l) ∈md t \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) rewrite inmd_subst_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
w: list K
t: U LN
u: T k LN
l: LN
x: atom
H1: k <> j
k <> j /\ (w, j, l) ∈md t \/
k = j /\ (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) <->
(w, j, l) ∈md t \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2) intuition.
    Qed.

    (** ** LN analysis without contexts *)
    (******************************************************************************)
    Lemma in_subst_loc_iff : forall k l j p u x,
        (j, p) ∈m subst_loc k x u l <->
        l <> Fr x /\ k = j /\ l = p \/
        l = Fr x /\ (j, p) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l j p (u : T k LN) x,
(j, p) ∈m subst_loc k x u l <->
l <> Fr x /\ k = j /\ l = p \/ l = Fr x /\ (j, p) ∈m u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l j p (u : T k LN) x,
(j, p) ∈m subst_loc k x u l <->
l <> Fr x /\ k = j /\ l = p \/ l = Fr x /\ (j, p) ∈m u
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
j: K
p: LN
u: T k LN
x: atom
(j, p) ∈m subst_loc k x u l <->
l <> Fr x /\ k = j /\ l = p \/ l = Fr x /\ (j, p) ∈m u compare l to atom x; autorewrite* with tea_local; intuition.
    Qed.

    Corollary in_subst_loc_iff_eq : forall k l p u x,
        (k, p) ∈m subst_loc k x u l <->
        l <> Fr x /\ l = p \/
        l = Fr x /\ (k, p) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l p (u : T k LN) x,
(k, p) ∈m subst_loc k x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ (k, p) ∈m u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l p (u : T k LN) x,
(k, p) ∈m subst_loc k x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ (k, p) ∈m u
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l, p: LN
u: T k LN
x: atom
(k, p) ∈m subst_loc k x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ (k, p) ∈m u rewrite in_subst_loc_iff; intuition.
    Qed.

    Corollary in_subst_loc_iff_neq : forall k l j p u x,
        k <> j ->
        (j, p) ∈m subst_loc k x u l <->
        l = Fr x /\ (j, p) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l j p (u : T k LN) x,
k <> j ->
(j, p) ∈m subst_loc k x u l <->
l = Fr x /\ (j, p) ∈m u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l j p (u : T k LN) x,
k <> j ->
(j, p) ∈m subst_loc k x u l <->
l = Fr x /\ (j, p) ∈m u
      introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
j: K
p: LN
u: T k LN
x: atom
neq: k <> j
(j, p) ∈m subst_loc k x u l <->
l = Fr x /\ (j, p) ∈m u rewrite in_subst_loc_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
j: K
p: LN
u: T k LN
x: atom
neq: k <> j
l <> Fr x /\ k = j /\ l = p \/ l = Fr x /\ (j, p) ∈m u <->
l = Fr x /\ (j, p) ∈m u intuition.
    Qed.

    Theorem in_subst_iff : forall k j t u l x,
        (* <<l>> occurs in the result of a <<subst>> IFF *)
        (j, l) ∈m t '{k | x ~> u} <->
        (* <<l>> is an occurrence in <<t>> not of the same sort as the <<subst>> OR *)
        k <> j /\ (j, l) ∈m t \/
        (* <<l>> is an occurrence of the same sort but not an occurrence of <<x>> OR *)
        k = j /\ (k, l) ∈m t /\ l <> Fr x \/
        (* <<x>> occurs and <<l>> is the any LN in a substituted <<u>> *)
        (k, Fr x) ∈m t /\ (j, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
(j, l) ∈m t '{ k | x ~> u} <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
(j, l) ∈m t '{ k | x ~> u} <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
(j, l) ∈m t '{ k | x ~> u} <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u destruct_eq_args k j.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(j, l) ∈m t '{ j | x ~> u} <->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(j, l) ∈m t '{ k | x ~> u} <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(j, l) ∈m t '{ j | x ~> u} <->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u rewrite (in_subst_eq_iff U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists l1,
   (j, l1) ∈m t /\ (j, l) ∈m subst_loc j x u l1) <->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u
        setoid_rewrite (in_subst_loc_iff_eq).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists l1,
   (j, l1) ∈m t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)) <->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists l1,
   (j, l1) ∈m t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)) ->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u ->
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
(exists l1,
   (j, l1) ∈m t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)) ->
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u intros [? [?  in_sub]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
x0: LN
H1: (j, x0) ∈m t
in_sub: x0 <> Fr x /\ x0 = l \/
x0 = Fr x /\ (j, l) ∈m u
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u
          destruct in_sub as [[? heq] | [heq ?]]; subst...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
j <> j /\ (j, l) ∈m t \/
j = j /\ (j, l) ∈m t /\ l <> Fr x \/
(j, Fr x) ∈m t /\ (j, l) ∈m u ->
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u) intros [[contra ?] | [ [? [in_t neq]] | ? ] ].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
contra: j <> j
H1: (j, l) ∈m t
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (j, l) ∈m t
neq: l <> Fr x
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
H1: (j, Fr x) ∈m t /\ (j, l) ∈m u
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
contra: j <> j
H1: (j, l) ∈m t
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u) contradiction.  }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (j, l) ∈m t
neq: l <> Fr x
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
H1: (j, Fr x) ∈m t /\ (j, l) ∈m u
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs, H1: j = j
in_t: (j, l) ∈m t
neq: l <> Fr x
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u) exists l... }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
H1: (j, Fr x) ∈m t /\ (j, l) ∈m u
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
H1: (j, Fr x) ∈m t /\ (j, l) ∈m u
exists l1,
  (j, l1) ∈m t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ (j, l) ∈m u) exists (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
DESTR_EQs: j = j
H1: (j, Fr x) ∈m t /\ (j, l) ∈m u
(j, Fr x) ∈m t /\
(Fr x <> Fr x /\ Fr x = l \/
 Fr x = Fr x /\ (j, l) ∈m u) intuition. }
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(j, l) ∈m t '{ k | x ~> u} <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u rewrite (in_subst_neq_iff U); auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) <->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) ->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u ->
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) ->
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u intros [? | [p [in_t in_local]] ]; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
p: LN
in_t: (k, p) ∈m t
in_local: (j, l) ∈m subst_loc k x u p
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
          rewrite in_subst_loc_iff_neq in in_local; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
p: LN
in_t: (k, p) ∈m t
in_local: p = Fr x /\ (j, l) ∈m u
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
          destruct in_local as [? ?]; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
in_t: (k, Fr x) ∈m t
H2: (j, l) ∈m u
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u auto.
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u ->
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) intros [[? ?] | [[? ?] | ?]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k <> j
H2: (j, l) ∈m t
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (k, l) ∈m t /\ l <> Fr x
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k <> j
H2: (j, l) ∈m t
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) auto. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (k, l) ∈m t /\ l <> Fr x
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: k = j
H2: (k, l) ∈m t /\ l <> Fr x
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) contradiction. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1)
          {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t \/
(exists l1,
   (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1) right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
exists l1,
  (k, l1) ∈m t /\ (j, l) ∈m subst_loc k x u l1 exists (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
DESTR_NEQ: k <> j
DESTR_NEQs: j <> k
H1: (k, Fr x) ∈m t /\ (j, l) ∈m u
(k, Fr x) ∈m t /\ (j, l) ∈m subst_loc k x u (Fr x) simpl_local... }
    Qed.

    Corollary in_subst_iff_eq : forall k t u l x,
        (k, l) ∈m t '{k | x ~> u} <->
        (k, l) ∈m t /\ l <> Fr x \/
        (k, Fr x) ∈m t /\ (k, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) l x,
(k, l) ∈m t '{ k | x ~> u} <->
(k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (k, l) ∈m u
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) l x,
(k, l) ∈m t '{ k | x ~> u} <->
(k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (k, l) ∈m u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
l: LN
x: atom
(k, l) ∈m t '{ k | x ~> u} <->
(k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (k, l) ∈m u rewrite in_subst_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
l: LN
x: atom
k <> k /\ (k, l) ∈m t \/
k = k /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (k, l) ∈m u <->
(k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (k, l) ∈m u intuition.
    Qed.

    Corollary in_subst_iff_neq : forall k j t u l x,
        k <> j ->
        (j, l) ∈m t '{k | x ~> u} <->
        (j, l) ∈m t \/
        (k, Fr x) ∈m t /\ (j, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
k <> j ->
(j, l) ∈m t '{ k | x ~> u} <->
(j, l) ∈m t \/ (k, Fr x) ∈m t /\ (j, l) ∈m u
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
k <> j ->
(j, l) ∈m t '{ k | x ~> u} <->
(j, l) ∈m t \/ (k, Fr x) ∈m t /\ (j, l) ∈m u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
H1: k <> j
(j, l) ∈m t '{ k | x ~> u} <->
(j, l) ∈m t \/ (k, Fr x) ∈m t /\ (j, l) ∈m u rewrite in_subst_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
H1: k <> j
k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u <->
(j, l) ∈m t \/ (k, Fr x) ∈m t /\ (j, l) ∈m u intuition.
    Qed.

    (** ** Free variables after substitution *)
    (******************************************************************************)
    Corollary in_free_subst_iff_eq : forall t k u x y,
        y ∈ free U k (t '{k | x ~> u}) <->
        y ∈ free U k t /\ y <> x \/ x ∈ free U k t /\ y ∈ free (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) <->
y ∈ free U k t /\ y <> x \/
x ∈ free U k t /\ y ∈ free (T k) k u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) <->
y ∈ free U k t /\ y <> x \/
x ∈ free U k t /\ y ∈ free (T k) k u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x, y: atom
y ∈ free U k (t '{ k | x ~> u}) <->
y ∈ free U k t /\ y <> x \/
x ∈ free U k t /\ y ∈ free (T k) k u repeat rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x, y: atom
(k, Fr y) ∈m t '{ k | x ~> u} <->
(k, Fr y) ∈m t /\ y <> x \/
(k, Fr x) ∈m t /\ (k, Fr y) ∈m u
      rewrite in_subst_iff_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x, y: atom
(k, Fr y) ∈m t /\ Fr y <> Fr x \/
(k, Fr x) ∈m t /\ (k, Fr y) ∈m u <->
(k, Fr y) ∈m t /\ y <> x \/
(k, Fr x) ∈m t /\ (k, Fr y) ∈m u now simpl_local.
    Qed.

    Corollary in_FV_subst_iff_eq : forall t k u x y,
        y `in` FV U k (t '{k | x ~> u}) <->
        y `in` FV U k t /\ y <> x \/
        x `in` FV U k t /\ y `in` FV (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x (y : AtomSet.elt),
y `in` FV U k (t '{ k | x ~> u}) <->
y `in` FV U k t /\ y <> x \/
x `in` FV U k t /\ y `in` FV (T k) k u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x (y : AtomSet.elt),
y `in` FV U k (t '{ k | x ~> u}) <->
y `in` FV U k t /\ y <> x \/
x `in` FV U k t /\ y `in` FV (T k) k u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
y: AtomSet.elt
y `in` FV U k (t '{ k | x ~> u}) <->
y `in` FV U k t /\ y <> x \/
x `in` FV U k t /\ y `in` FV (T k) k u repeat rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
y: AtomSet.elt
y ∈ free U k (t '{ k | x ~> u}) <->
y ∈ free U k t /\ y <> x \/
x ∈ free U k t /\ y ∈ free (T k) k u
      apply in_free_subst_iff_eq.
    Qed.

    Corollary in_free_subst_iff_neq : forall t k j u x y,
        k <> j ->
        y ∈ free U j (t '{k | x ~> u}) <->
        y ∈ free U j t \/ x ∈ free U k t /\ y ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/ x ∈ free U k t /\ y ∈ free (T k) j u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/ x ∈ free U k t /\ y ∈ free (T k) j u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x, y: atom
H1: k <> j
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/ x ∈ free U k t /\ y ∈ free (T k) j u repeat rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x, y: atom
H1: k <> j
(j, Fr y) ∈m t '{ k | x ~> u} <->
(j, Fr y) ∈m t \/ (k, Fr x) ∈m t /\ (j, Fr y) ∈m u
      rewrite in_subst_iff_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x, y: atom
H1: k <> j
(j, Fr y) ∈m t \/ (k, Fr x) ∈m t /\ (j, Fr y) ∈m u <->
(j, Fr y) ∈m t \/ (k, Fr x) ∈m t /\ (j, Fr y) ∈m u reflexivity.
    Qed.

    Corollary in_FV_subst_iff_neq : forall t k j u x y,
        k <> j ->
        y `in` FV U j (t '{k | x ~> u}) <->
        y `in` FV U j t \/
        x `in` FV U k t /\ y `in` FV (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x (y : AtomSet.elt),
k <> j ->
y `in` FV U j (t '{ k | x ~> u}) <->
y `in` FV U j t \/
x `in` FV U k t /\ y `in` FV (T k) j u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x (y : AtomSet.elt),
k <> j ->
y `in` FV U j (t '{ k | x ~> u}) <->
y `in` FV U j t \/
x `in` FV U k t /\ y `in` FV (T k) j u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
y: AtomSet.elt
H1: k <> j
y `in` FV U j (t '{ k | x ~> u}) <->
y `in` FV U j t \/
x `in` FV U k t /\ y `in` FV (T k) j u repeat rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
y: AtomSet.elt
H1: k <> j
y ∈ free U j (t '{ k | x ~> u}) <->
y ∈ free U j t \/ x ∈ free U k t /\ y ∈ free (T k) j u
      auto using in_free_subst_iff_neq.
    Qed.

    (** ** Upper and lower bounds for leaves after substitution *)
    (******************************************************************************)
    Corollary in_subst_upper : forall k j t u l x,
        (j, l) ∈m t '{k | x ~> u} ->
        (j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
(j, l) ∈m t '{ k | x ~> u} ->
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
(j, l) ∈m t '{ k | x ~> u} ->
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
      introv hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hin: (j, l) ∈m t '{ k | x ~> u}
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u rewrite in_subst_iff in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hin: k <> j /\ (j, l) ∈m t \/
k = j /\ (k, l) ∈m t /\ l <> Fr x \/
(k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
      destruct hin as [[hyp1 hyp2] | [hyp | hyp] ].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp1: k <> j
hyp2: (j, l) ∈m t
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: k = j /\ (k, l) ∈m t /\ l <> Fr x
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp1: k <> j
hyp2: (j, l) ∈m t
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp1: k <> j
hyp2: (j, l) ∈m t
(j, l) ∈m t /\ (j, l) <> (k, Fr x) split; [assumption |].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp1: k <> j
hyp2: (j, l) ∈m t
(j, l) <> (k, Fr x) injection...
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: k = j /\ (k, l) ∈m t /\ l <> Fr x
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u destructs hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
H1: k = j
H2: (k, l) ∈m t
H3: l <> Fr x
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
H2: (j, l) ∈m t
H3: l <> Fr x
(j, l) ∈m t /\ (j, l) <> (j, Fr x) \/ (j, l) ∈m u left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
H2: (j, l) ∈m t
H3: l <> Fr x
(j, l) ∈m t /\ (j, l) <> (j, Fr x)
        split; [assumption |].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j: K
t: U LN
u: T j LN
l: LN
x: atom
H2: (j, l) ∈m t
H3: l <> Fr x
(j, l) <> (j, Fr x) injection...
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: (k, Fr x) ∈m t /\ (j, l) ∈m u
(j, l) ∈m t /\ (j, l) <> (k, Fr x) \/ (j, l) ∈m u intuition.
    Qed.

    Corollary in_subst_upper_eq : forall k t u l x,
        (k, l) ∈m t '{k | x ~> u} ->
        (k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) l x,
(k, l) ∈m t '{ k | x ~> u} ->
(k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) l x,
(k, l) ∈m t '{ k | x ~> u} ->
(k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u
      introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: (k, l) ∈m t '{ k | x ~> u}
(k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u apply in_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: (k, l) ∈m t /\ (k, l) <> (k, Fr x) \/
(k, l) ∈m u
(k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u
      autorewrite* with tea_local in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
l: LN
x: atom
hyp: (k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u
(k, l) ∈m t /\ l <> Fr x \/ (k, l) ∈m u
      intuition.
    Qed.

    Corollary in_subst_upper_neq : forall k j t u l x,
        k <> j ->
        (j, l) ∈m t '{k | x ~> u} ->
        (j, l) ∈m t \/ (j, l) ∈m u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
k <> j ->
(j, l) ∈m t '{ k | x ~> u} ->
(j, l) ∈m t \/ (j, l) ∈m u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) l x,
k <> j ->
(j, l) ∈m t '{ k | x ~> u} ->
(j, l) ∈m t \/ (j, l) ∈m u
      introv neq hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
neq: k <> j
hyp: (j, l) ∈m t '{ k | x ~> u}
(j, l) ∈m t \/ (j, l) ∈m u apply in_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
l: LN
x: atom
neq: k <> j
hyp: (j, l) ∈m t /\ (j, l) <> (k, Fr x) \/
(j, l) ∈m u
(j, l) ∈m t \/ (j, l) ∈m u
      intuition.
    Qed.

    Corollary in_free_subst_upper : forall k j t u x y,
        y ∈ free U j (t '{k | x ~> u}) ->
        (y ∈ free U j t /\ j <> k) \/ (y ∈ free U k t /\ y <> x /\ k = j) \/ y ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) (x y : atom),
y ∈ free U j (t '{ k | x ~> u}) ->
y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) (x y : atom),
y ∈ free U j (t '{ k | x ~> u}) ->
y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
      setoid_rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) (x y : atom),
(j, Fr y) ∈m t '{ k | x ~> u} ->
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m t '{ k | x ~> u}
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u
      apply in_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m t /\ (j, Fr y) <> (k, Fr x) \/
(j, Fr y) ∈m u
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u
      destruct hyp as [hyp | hyp].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m t /\ (j, Fr y) <> (k, Fr x)
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m u
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m t /\ (j, Fr y) <> (k, Fr x)
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u destruct hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
H1: (j, Fr y) ∈m t
H2: (j, Fr y) <> (k, Fr x)
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u compare values j and k.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
H2: (k, Fr y) <> (k, Fr x)
H1: (k, Fr y) ∈m t
DESTR_EQs: k = k
(k, Fr y) ∈m t /\ k <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = k \/ (k, Fr y) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
H1: (j, Fr y) ∈m t
H2: (j, Fr y) <> (k, Fr x)
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
H2: (k, Fr y) <> (k, Fr x)
H1: (k, Fr y) ∈m t
DESTR_EQs: k = k
(k, Fr y) ∈m t /\ k <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = k \/ (k, Fr y) ∈m u right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
H2: (k, Fr y) <> (k, Fr x)
H1: (k, Fr y) ∈m t
DESTR_EQs: k = k
(k, Fr y) ∈m t /\ y <> x /\ k = k \/ (k, Fr y) ∈m u left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
H2: (k, Fr y) <> (k, Fr x)
H1: (k, Fr y) ∈m t
DESTR_EQs: k = k
(k, Fr y) ∈m t /\ y <> x /\ k = k splits; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
H2: (k, Fr y) <> (k, Fr x)
H1: (k, Fr y) ∈m t
DESTR_EQs: k = k
y <> x
          intro contra; subst; contradiction.
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
H1: (j, Fr y) ∈m t
H2: (j, Fr y) <> (k, Fr x)
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u auto.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
hyp: (j, Fr y) ∈m u
(j, Fr y) ∈m t /\ j <> k \/
(k, Fr y) ∈m t /\ y <> x /\ k = j \/ (j, Fr y) ∈m u eauto using in_subst_upper.
    Qed.

    Corollary in_free_subst_upper_eq : forall k t u x y,
        y ∈ free U k (t '{k | x ~> u}) ->
        (y ∈ free U k t /\ y <> x) \/ y ∈ free (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) ->
y ∈ free U k t /\ y <> x \/ y ∈ free (T k) k u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) (x y : atom),
y ∈ free U k (t '{ k | x ~> u}) ->
y ∈ free U k t /\ y <> x \/ y ∈ free (T k) k u
      introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
hyp: y ∈ free U k (t '{ k | x ~> u})
y ∈ free U k t /\ y <> x \/ y ∈ free (T k) k u apply in_free_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x, y: atom
hyp: y ∈ free U k t /\ k <> k \/
y ∈ free U k t /\ y <> x /\ k = k \/
y ∈ free (T k) k u
y ∈ free U k t /\ y <> x \/ y ∈ free (T k) k u
      intuition.
    Qed.

    Corollary in_free_subst_upper_neq : forall k j t u x y,
        k <> j ->
        y ∈ free U j (t '{k | x ~> u}) ->
        y ∈ free U j t \/ y ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) ->
y ∈ free U j t \/ y ∈ free (T k) j u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j (t '{ k | x ~> u}) ->
y ∈ free U j t \/ y ∈ free (T k) j u
      introv neq hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
neq: k <> j
hyp: y ∈ free U j (t '{ k | x ~> u})
y ∈ free U j t \/ y ∈ free (T k) j u apply in_free_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x, y: atom
neq: k <> j
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
y ∈ free U j t \/ y ∈ free (T k) j u
      intuition.
    Qed.

    Corollary FV_subst_upper : forall k j t u x y,
        y `in` FV U j (t '{k | x ~> u}) ->
        k = j /\ y `in` (FV U k t \\ {{x}} ∪ FV (T k) j u)%set \/
        k <> j /\ y `in` (FV U j t ∪ FV (T k) j u)%set.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x (y : AtomSet.elt),
y `in` FV U j (t '{ k | x ~> u}) ->
k = j /\ y `in` (FV U k t \\ {{x}} ∪ FV (T k) j u) \/
k <> j /\ y `in` (FV U j t ∪ FV (T k) j u)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x (y : AtomSet.elt),
y `in` FV U j (t '{ k | x ~> u}) ->
k = j /\ y `in` (FV U k t \\ {{x}} ∪ FV (T k) j u) \/
k <> j /\ y `in` (FV U j t ∪ FV (T k) j u)
      setoid_rewrite AtomSet.union_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x (y : AtomSet.elt),
y `in` FV U j (t '{ k | x ~> u}) ->
k = j /\
(y `in` (FV U k t \\ {{x}}) \/ y `in` FV (T k) j u) \/
k <> j /\ (y `in` FV U j t \/ y `in` FV (T k) j u)
      setoid_rewrite AtomSet.diff_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x (y : AtomSet.elt),
y `in` FV U j (t '{ k | x ~> u}) ->
k = j /\
(y `in` FV U k t /\ y `notin` {{x}} \/
 y `in` FV (T k) j u) \/
k <> j /\ (y `in` FV U j t \/ y `in` FV (T k) j u)
      setoid_rewrite <- free_iff_FV.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x (y : AtomSet.elt),
y ∈ free U j (t '{ k | x ~> u}) ->
k = j /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) j u) \/
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u)
      introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j (t '{ k | x ~> u})
k = j /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) j u) \/
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u) apply in_free_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
k = j /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) j u) \/
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u)
      compare values j and k.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k \/
y ∈ free U k t /\ y <> x /\ k = k \/
y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) \/
k <> k /\ (y ∈ free U k t \/ y ∈ free (T k) k u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
k = j /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) j u) \/
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u)
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k \/
y ∈ free U k t /\ y <> x /\ k = k \/
y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) \/
k <> k /\ (y ∈ free U k t \/ y ∈ free (T k) k u) left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k \/
y ∈ free U k t /\ y <> x /\ k = k \/
y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) destruct hyp as [hyp | [hyp | hyp]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) split; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ k <> k
DESTR_EQs: k = k
y ∈ free U k t /\ y `notin` {{x}} \/
y ∈ free (T k) k u intuition. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) split; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
y ∈ free U k t /\ y `notin` {{x}} \/
y ∈ free (T k) k u left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
y ∈ free U k t /\ y `notin` {{x}} split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
y ∈ free U k t
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
y `notin` {{x}} intuition.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = k
DESTR_EQs: k = k
y `notin` {{x}} fsetdec. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u)
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) k u
DESTR_EQs: k = k
k = k /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) k u) split; auto. }
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
k = j /\
(y ∈ free U k t /\ y `notin` {{x}} \/
 y ∈ free (T k) j u) \/
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u) right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
k <> j /\ (y ∈ free U j t \/ y ∈ free (T k) j u) split; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k \/
y ∈ free U k t /\ y <> x /\ k = j \/
y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u destruct hyp as [hyp | [hyp | hyp]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = j
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U j t /\ j <> k
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u intuition. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = j
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free U k t /\ y <> x /\ k = j
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u intuition. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
y: AtomSet.elt
hyp: y ∈ free (T k) j u
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
y ∈ free U j t \/ y ∈ free (T k) j u intuition. }
    Qed.

    Corollary FV_subst_upper_eq : forall k t u x,
        FV U k (t '{k | x ~> u}) ⊆
                (FV U k t \\ {{x}} ∪ FV (T k) k u).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) x,
FV U k (t '{ k | x ~> u})
⊆ FV U k t \\ {{x}} ∪ FV (T k) k u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN) x,
FV U k (t '{ k | x ~> u})
⊆ FV U k t \\ {{x}} ∪ FV (T k) k u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
FV U k (t '{ k | x ~> u})
⊆ FV U k t \\ {{x}} ∪ FV (T k) k u intros a hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
a: AtomSet.elt
hyp: a `in` FV U k (t '{ k | x ~> u})
a `in` (FV U k t \\ {{x}} ∪ FV (T k) k u) apply FV_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
x: atom
a: AtomSet.elt
hyp: k = k /\
a `in` (FV U k t \\ {{x}} ∪ FV (T k) k u) \/
k <> k /\ a `in` (FV U k t ∪ FV (T k) k u)
a `in` (FV U k t \\ {{x}} ∪ FV (T k) k u)
      intuition.
    Qed.

    Corollary scoped_subst_eq : forall t k (u : T k LN) x γ1 γ2,
        scoped U k t γ1 ->
        scoped (T k) k u γ2 ->
        scoped U k (t '{k | x ~> u}) (γ1 \\ {{x}} ∪ γ2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x (γ1 γ2 : AtomSet.t),
scoped U k t γ1 ->
scoped (T k) k u γ2 ->
scoped U k (t '{ k | x ~> u}) (γ1 \\ {{x}} ∪ γ2)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x (γ1 γ2 : AtomSet.t),
scoped U k t γ1 ->
scoped (T k) k u γ2 ->
scoped U k (t '{ k | x ~> u}) (γ1 \\ {{x}} ∪ γ2)
      introv St Su.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
γ1, γ2: AtomSet.t
St: scoped U k t γ1
Su: scoped (T k) k u γ2
scoped U k (t '{ k | x ~> u}) (γ1 \\ {{x}} ∪ γ2) unfold scoped in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
γ1, γ2: AtomSet.t
St: FV U k t ⊆ γ1
Su: FV (T k) k u ⊆ γ2
FV U k (t '{ k | x ~> u}) ⊆ γ1 \\ {{x}} ∪ γ2
      etransitivity.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
γ1, γ2: AtomSet.t
St: FV U k t ⊆ γ1
Su: FV (T k) k u ⊆ γ2
FV U k (t '{ k | x ~> u}) ⊆ ?y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
γ1, γ2: AtomSet.t
St: FV U k t ⊆ γ1
Su: FV (T k) k u ⊆ γ2
?y ⊆ γ1 \\ {{x}} ∪ γ2 apply FV_subst_upper_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
γ1, γ2: AtomSet.t
St: FV U k t ⊆ γ1
Su: FV (T k) k u ⊆ γ2
FV U k t \\ {{x}} ∪ FV (T k) k u ⊆ γ1 \\ {{x}} ∪ γ2 fsetdec.
    Qed.

    Corollary FV_subst_upper_neq : forall k j t u x,
        k <> j ->
        FV U j (t '{k | x ~> u}) ⊆
                (FV U j t ∪ FV (T k) j u).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x,
k <> j ->
FV U j (t '{ k | x ~> u}) ⊆ FV U j t ∪ FV (T k) j u
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t (u : T k LN) x,
k <> j ->
FV U j (t '{ k | x ~> u}) ⊆ FV U j t ∪ FV (T k) j u
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
H1: k <> j
FV U j (t '{ k | x ~> u}) ⊆ FV U j t ∪ FV (T k) j u intros a hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
H1: k <> j
a: AtomSet.elt
hyp: a `in` FV U j (t '{ k | x ~> u})
a `in` (FV U j t ∪ FV (T k) j u) apply FV_subst_upper in hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
u: T k LN
x: atom
H1: k <> j
a: AtomSet.elt
hyp: k = j /\
a `in` (FV U k t \\ {{x}} ∪ FV (T k) j u) \/
k <> j /\ a `in` (FV U j t ∪ FV (T k) j u)
a `in` (FV U j t ∪ FV (T k) j u)
      intuition.
    Qed.

    Corollary scoped_subst_neq : forall t j k u x γ1 γ2,
        j <> k ->
        scoped U k t γ1 ->
        scoped (T j) k u γ2 ->
        scoped U k (t '{j | x ~> u}) (γ1 ∪ γ2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t j k (u : T j LN) x (γ1 γ2 : AtomSet.t),
j <> k ->
scoped U k t γ1 ->
scoped (T j) k u γ2 ->
scoped U k (t '{ j | x ~> u}) (γ1 ∪ γ2)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t j k (u : T j LN) x (γ1 γ2 : AtomSet.t),
j <> k ->
scoped U k t γ1 ->
scoped (T j) k u γ2 ->
scoped U k (t '{ j | x ~> u}) (γ1 ∪ γ2)
      introv St Su.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
j, k: K
u: T j LN
x: atom
γ1, γ2: AtomSet.t
St: j <> k
Su: scoped U k t γ1
scoped (T j) k u γ2 ->
scoped U k (t '{ j | x ~> u}) (γ1 ∪ γ2) unfold scoped in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
j, k: K
u: T j LN
x: atom
γ1, γ2: AtomSet.t
St: j <> k
Su: FV U k t ⊆ γ1
FV (T j) k u ⊆ γ2 ->
FV U k (t '{ j | x ~> u}) ⊆ γ1 ∪ γ2
      etransitivity.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
j, k: K
u: T j LN
x: atom
γ1, γ2: AtomSet.t
St: j <> k
Su: FV U k t ⊆ γ1
H1: FV (T j) k u ⊆ γ2
FV U k (t '{ j | x ~> u}) ⊆ ?y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
j, k: K
u: T j LN
x: atom
γ1, γ2: AtomSet.t
St: j <> k
Su: FV U k t ⊆ γ1
H1: FV (T j) k u ⊆ γ2
?y ⊆ γ1 ∪ γ2 apply FV_subst_upper_neq; assumption.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
j, k: K
u: T j LN
x: atom
γ1, γ2: AtomSet.t
St: j <> k
Su: FV U k t ⊆ γ1
H1: FV (T j) k u ⊆ γ2
FV U k t ∪ FV (T j) k u ⊆ γ1 ∪ γ2
      fsetdec.
    Qed.

    Corollary in_subst_lower_eq : forall t k u l x,
        (k, l) ∈m t /\ l <> Fr x ->
        (k, l) ∈m t '{k | x ~> u}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) l x,
(k, l) ∈m t /\ l <> Fr x -> (k, l) ∈m t '{ k | x ~> u}
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) l x,
(k, l) ∈m t /\ l <> Fr x -> (k, l) ∈m t '{ k | x ~> u}
      intros; rewrite in_subst_iff...
    Qed.

    Corollary in_subst_lower_neq : forall t k j u l x,
        k <> j ->
        (j, l) ∈m t ->
        (j, l) ∈m t '{k | x ~> u}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) l x,
k <> j -> (j, l) ∈m t -> (j, l) ∈m t '{ k | x ~> u}
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) l x,
k <> j -> (j, l) ∈m t -> (j, l) ∈m t '{ k | x ~> u}
      intros; rewrite in_subst_iff...
    Qed.

    Corollary in_free_subst_lower_eq : forall t k (u : T k LN) x y,
        y ∈ free U k t /\ y <> x ->
        y ∈ free U k (t '{k | x ~> u}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) (x y : atom),
y ∈ free U k t /\ y <> x ->
y ∈ free U k (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) (x y : atom),
y ∈ free U k t /\ y <> x ->
y ∈ free U k (t '{ k | x ~> u})
      setoid_rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) (x y : atom),
(k, Fr y) ∈m t /\ y <> x ->
(k, Fr y) ∈m t '{ k | x ~> u} intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x, y: atom
H1: (k, Fr y) ∈m t /\ y <> x
(k, Fr y) ∈m t '{ k | x ~> u}
      apply in_subst_lower_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x, y: atom
H1: (k, Fr y) ∈m t /\ y <> x
(k, Fr y) ∈m t /\ Fr y <> Fr x now simpl_local.
    Qed.

    Corollary in_free_subst_lower_neq : forall t k j u x y,
        k <> j ->
        y ∈ free U j t ->
        y ∈ free U j (t '{k | x ~> u}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j t -> y ∈ free U j (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) (x y : atom),
k <> j ->
y ∈ free U j t -> y ∈ free U j (t '{ k | x ~> u})
      setoid_rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) (x y : atom),
k <> j ->
(j, Fr y) ∈m t -> (j, Fr y) ∈m t '{ k | x ~> u} intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x, y: atom
H1: k <> j
H2: (j, Fr y) ∈m t
(j, Fr y) ∈m t '{ k | x ~> u}
      now apply in_subst_lower_neq.
    Qed.

    Corollary FV_subst_lower_eq : forall t k (u : T k LN) x,
        FV U k t \\ {{ x }} ⊆ FV U k (t '{k | x ~> u}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
FV U k t \\ {{x}} ⊆ FV U k (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
FV U k t \\ {{x}} ⊆ FV U k (t '{ k | x ~> u})
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
FV U k t \\ {{x}} ⊆ FV U k (t '{ k | x ~> u}) intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a `in` (FV U k t \\ {{x}}) ->
a `in` FV U k (t '{ k | x ~> u}) rewrite AtomSet.diff_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a `in` FV U k t /\ a `notin` {{x}} ->
a `in` FV U k (t '{ k | x ~> u})
      do 2 rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a ∈ free U k t /\ a `notin` {{x}} ->
a ∈ free U k (t '{ k | x ~> u})
      intros [? ?].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
H1: a ∈ free U k t
H2: a `notin` {{x}}
a ∈ free U k (t '{ k | x ~> u}) apply in_free_subst_lower_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
H1: a ∈ free U k t
H2: a `notin` {{x}}
a ∈ free U k t /\ a <> x
      intuition.
    Qed.

    Corollary FV_subst_lower_eq_alt : forall t k (u : T k LN) x,
        FV U k t ⊆ FV U k (t '{k | x ~> u}) ∪ {{ x }}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
FV U k t ⊆ FV U k (t '{ k | x ~> u}) ∪ {{x}}
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
FV U k t ⊆ FV U k (t '{ k | x ~> u}) ∪ {{x}}
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
FV U k t ⊆ FV U k (t '{ k | x ~> u}) ∪ {{x}} intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a `in` FV U k t ->
a `in` (FV U k (t '{ k | x ~> u}) ∪ {{x}}) rewrite AtomSet.union_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a `in` FV U k t ->
a `in` FV U k (t '{ k | x ~> u}) \/ a `in` {{x}}
      do 2 rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
a ∈ free U k t ->
a ∈ free U k (t '{ k | x ~> u}) \/ a `in` {{x}}
      destruct_eq_args a x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
DESTR_EQs: x = x
x ∈ free U k t ->
x ∈ free U k (t '{ k | x ~> u}) \/ x `in` {{x}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
DESTR_NEQ: a <> x
DESTR_NEQs: x <> a
a ∈ free U k t ->
a ∈ free U k (t '{ k | x ~> u}) \/ a `in` {{x}}
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
DESTR_EQs: x = x
x ∈ free U k t ->
x ∈ free U k (t '{ k | x ~> u}) \/ x `in` {{x}} right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
DESTR_EQs: x = x
H1: x ∈ free U k t
x `in` {{x}} fsetdec.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
DESTR_NEQ: a <> x
DESTR_NEQs: x <> a
a ∈ free U k t ->
a ∈ free U k (t '{ k | x ~> u}) \/ a `in` {{x}} left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
a: AtomSet.elt
DESTR_NEQ: a <> x
DESTR_NEQs: x <> a
H1: a ∈ free U k t
a ∈ free U k (t '{ k | x ~> u}) auto using in_free_subst_lower_eq.
    Qed.

    Corollary FV_subst_lower_neq : forall t k j u x,
        k <> j ->
        FV U j t ⊆
                FV U j (t '{k | x ~> u}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
k <> j -> FV U j t ⊆ FV U j (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
k <> j -> FV U j t ⊆ FV U j (t '{ k | x ~> u})
      introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
neq: k <> j
FV U j t ⊆ FV U j (t '{ k | x ~> u}) intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
neq: k <> j
a: AtomSet.elt
a `in` FV U j t -> a `in` FV U j (t '{ k | x ~> u})
      do 2 rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
neq: k <> j
a: AtomSet.elt
a ∈ free U j t -> a ∈ free U j (t '{ k | x ~> u})
      now apply in_free_subst_lower_neq.
    Qed.

    (** ** Substitution of fresh variables *)
    (******************************************************************************)
    Theorem subst_fresh : forall t k x u,
        ~ x ∈ free U k t ->
        t '{k | x ~> u} = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x (u : T k LN),
~ x ∈ free U k t -> t '{ k | x ~> u} = t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x (u : T k LN),
~ x ∈ free U k t -> t '{ k | x ~> u} = t
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
t '{ k | x ~> u} = t apply (subst_id U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
forall l,
(k, l) ∈m t -> subst_loc k x u l = mret T k l intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
l: LN
H2: (k, l) ∈m t
subst_loc k x u l = mret T k l
      assert (Fr x <> l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
l: LN
H2: (k, l) ∈m t
Fr x <> l
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
l: LN
H2: (k, l) ∈m t
H3: Fr x <> l
subst_loc k x u l = mret T k l
      eapply (ninf_in_neq U); eauto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
u: T k LN
H1: ~ x ∈ free U k t
l: LN
H2: (k, l) ∈m t
H3: Fr x <> l
subst_loc k x u l = mret T k l
      now simpl_local.
    Qed.

    Corollary subst_fresh_set : forall t k x u,
        ~ x `in` FV U k t ->
        t '{k | x ~> u} = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x (u : T k LN),
x `notin` FV U k t -> t '{ k | x ~> u} = t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x (u : T k LN),
x `notin` FV U k t -> t '{ k | x ~> u} = t
      setoid_rewrite <- free_iff_FV.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x (u : T k LN),
~ x ∈ free U k t -> t '{ k | x ~> u} = t apply subst_fresh.
    Qed.

    Theorem free_subst_fresh : forall t k j u x,
        ~ x ∈ free U k t ->
        free U j (t '{k | x ~> u}) = free U j t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
~ x ∈ free U k t ->
free U j (t '{ k | x ~> u}) = free U j t
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
~ x ∈ free U k t ->
free U j (t '{ k | x ~> u}) = free U j t
      introv fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
fresh: ~ x ∈ free U k t
free U j (t '{ k | x ~> u}) = free U j t replace (t '{k | x ~> u}) with t...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
fresh: ~ x ∈ free U k t
t = t '{ k | x ~> u}
      rewrite subst_fresh...
    Qed.

    Corollary free_subst_fresh_eq : forall t k u x,
        ~ x ∈ free U k t ->
        free U k (t '{k | x ~> u}) = free U k t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
~ x ∈ free U k t ->
free U k (t '{ k | x ~> u}) = free U k t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
~ x ∈ free U k t ->
free U k (t '{ k | x ~> u}) = free U k t
      introv fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
fresh: ~ x ∈ free U k t
free U k (t '{ k | x ~> u}) = free U k t replace (t '{k | x ~> u}) with t; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
fresh: ~ x ∈ free U k t
t = t '{ k | x ~> u}
      now rewrite subst_fresh.
    Qed.

    Corollary FV_subst_fresh : forall t k j u x,
        ~ x `in` FV U k t ->
        FV U j (t '{k | x ~> u}) [=] FV U j t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x `notin` FV U k t ->
FV U j (t '{ k | x ~> u}) [=] FV U j t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x `notin` FV U k t ->
FV U j (t '{ k | x ~> u}) [=] FV U j t
      introv fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
fresh: x `notin` FV U k t
FV U j (t '{ k | x ~> u}) [=] FV U j t intro y.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
fresh: x `notin` FV U k t
y: AtomSet.elt
y `in` FV U j (t '{ k | x ~> u}) <-> y `in` FV U j t
      rewrite <- ?(free_iff_FV) in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
fresh: ~ x ∈ free U k t
y: AtomSet.elt
y ∈ free U j (t '{ k | x ~> u}) <-> y ∈ free U j t
      now rewrite free_subst_fresh.
    Qed.

    Corollary FV_subst_fresh_eq : forall t k u x,
        ~ x `in` FV U k t ->
        FV U k (t '{k | x ~> u}) [=] FV U k t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x `notin` FV U k t ->
FV U k (t '{ k | x ~> u}) [=] FV U k t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x `notin` FV U k t ->
FV U k (t '{ k | x ~> u}) [=] FV U k t
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
H1: x `notin` FV U k t
FV U k (t '{ k | x ~> u}) [=] FV U k t apply FV_subst_fresh; auto.
    Qed.

  End fix_dtm.

  (** ** Composing substitutions *)
  (******************************************************************************)
  Section fix_dtm.

    Context
      (U : Type -> Type)
      `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
      `{! MultiDecoratedTraversableMonad (list K) T}.

    Implicit Types
             (k : K) (j : K)
             (l : LN) (p : LN)
             (x : atom) (t : U LN)
             (w : list K) (n : nat).

    Lemma subst_subst_neq_loc : forall j k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) (x1 x2 : atom),
        k1 <> k2 ->
        ~ x1 ∈ free (T k2) k1 u2 ->
        subst (T j) k2 x2 u2 ∘ btg k1 (subst_loc k1 x1 u1) j =
        subst (T j) k1 x1 (subst (T k1) k2 x2 u2 u1) ∘ btg k2 (subst_loc k2 x2 u2) j.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall j k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
subst (T j) k2 x2 u2 ∘ btg k1 (subst_loc k1 x1 u1) j =
subst (T j) k1 x1 (u1 '{ k2 | x2 ~> u2})
∘ btg k2 (subst_loc k2 x2 u2) j
    Proof with easy.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall j k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
subst (T j) k2 x2 u2 ∘ btg k1 (subst_loc k1 x1 u1) j =
subst (T j) k1 x1 (u1 '{ k2 | x2 ~> u2})
∘ btg k2 (subst_loc k2 x2 u2) j
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
subst (T j) k2 x2 u2 ∘ btg k1 (subst_loc k1 x1 u1) j =
subst (T j) k1 x1 (u1 '{ k2 | x2 ~> u2})
∘ btg k2 (subst_loc k2 x2 u2) j ext l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
(subst (T j) k2 x2 u2 ∘ btg k1 (subst_loc k1 x1 u1) j)
  l =
(subst (T j) k1 x1 (u1 '{ k2 | x2 ~> u2})
 ∘ btg k2 (subst_loc k2 x2 u2) j) l unfold compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
btg k1 (subst_loc k1 x1 u1) j l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) j l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} compare j to both of { k1 k2 }.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
btg k1 (subst_loc k1 x1 u1) k1 l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) k1 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
btg k1 (subst_loc k1 x1 u1) k2 l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) k2 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
DESTR_NEQ1: k1 <> k2
DESTR_NEQs1: k2 <> k1
btg k1 (subst_loc k1 x1 u1) j l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) j l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
btg k1 (subst_loc k1 x1 u1) k1 l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) k1 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} do 2 simpl_tgt_fallback.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
subst_loc k1 x1 u1 l '{ k2 | x2 ~> u2} =
mret T k1 l '{ k1 | x1 ~> u1 '{ k2 | x2 ~> u2}}
        simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
subst_loc k1 x1 u1 l '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) l
        compare l to atom x1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
subst_loc k1 x1 u1 (Fr x1) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Fr x1)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
a: atom
H3: a <> x1
subst_loc k1 x1 u1 (Fr a) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
n: nat
subst_loc k1 x1 u1 (Bd n) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Bd n)
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
subst_loc k1 x1 u1 (Fr x1) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Fr x1) rewrite 2(subst_loc_eq)...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
a: atom
H3: a <> x1
subst_loc k1 x1 u1 (Fr a) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Fr a) rewrite subst_loc_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
a: atom
H3: a <> x1
mret T k1 (Fr a) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Fr a) now simpl_local.
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
n: nat
subst_loc k1 x1 u1 (Bd n) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Bd n) rewrite subst_loc_b.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_EQs: k1 = k1
DESTR_NEQ: k1 <> k2
DESTR_NEQs: k2 <> k1
n: nat
mret T k1 (Bd n) '{ k2 | x2 ~> u2} =
subst_loc k1 x1 (u1 '{ k2 | x2 ~> u2}) (Bd n) now simpl_local.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
btg k1 (subst_loc k1 x1 u1) k2 l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) k2 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_tgt.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
mret T k2 l '{ k2 | x2 ~> u2} =
subst_loc k2 x2 u2 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
subst_loc k2 x2 u2 l =
subst_loc k2 x2 u2 l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} compare l to atom x2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
subst_loc k2 x2 u2 (Fr x2) =
subst_loc k2 x2 u2 (Fr x2) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
a: atom
H3: a <> x2
subst_loc k2 x2 u2 (Fr a) =
subst_loc k2 x2 u2 (Fr a) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
n: nat
subst_loc k2 x2 u2 (Bd n) =
subst_loc k2 x2 u2 (Bd n) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
subst_loc k2 x2 u2 (Fr x2) =
subst_loc k2 x2 u2 (Fr x2) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
u2 = u2 '{ k1 | x1 ~> u1 '{ k2 | x2 ~> u2}} rewrite (subst_fresh (T k2))...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
a: atom
H3: a <> x2
subst_loc k2 x2 u2 (Fr a) =
subst_loc k2 x2 u2 (Fr a) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_local...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_NEQ0: k1 <> k2
DESTR_NEQs0: k2 <> k1
n: nat
subst_loc k2 x2 u2 (Bd n) =
subst_loc k2 x2 u2 (Bd n) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_local...
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
DESTR_NEQ1: k1 <> k2
DESTR_NEQs1: k2 <> k1
btg k1 (subst_loc k1 x1 u1) j l '{ k2 | x2 ~> u2} =
btg k2 (subst_loc k2 x2 u2) j l '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} simpl_tgt_fallback.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
j, k1, k2: K
u1: T k1 LN
u2: T k2 LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
DESTR_NEQ1: k1 <> k2
DESTR_NEQs1: k2 <> k1
mret T j l '{ k2 | x2 ~> u2} =
mret T j l '{ k1 | x1 ~> u1 '{ k2 | x2 ~> u2}} now do 2 (rewrite subst_in_mret_neq; auto).
    Qed.

    Theorem subst_subst_neq : forall k1 k2 u1 u2 t (x1 x2 : atom),
        k1 <> k2 ->
        ~ x1 ∈ free (T k2) k1 u2 ->
        t '{k1 | x1 ~> u1} '{k2 | x2 ~> u2} =
        t '{k2 | x2 ~> u2} '{k1 | x1 ~> u1 '{k2 | x2 ~> u2} }.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) t x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) t x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}}
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
kbind U k2 (subst_loc k2 x2 u2)
  (kbind U k1 (subst_loc k1 x1 u1) t) =
kbind U k1
  (subst_loc k1 x1
     (kbind (T k1) k2 (subst_loc k2 x2 u2) u1))
  (kbind U k2 (subst_loc k2 x2 u2) t)
      compose near t on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
(kbind U k2 (subst_loc k2 x2 u2)
 ∘ kbind U k1 (subst_loc k1 x1 u1)) t =
kbind U k1
  (subst_loc k1 x1
     (kbind (T k1) k2 (subst_loc k2 x2 u2) u1))
  (kbind U k2 (subst_loc k2 x2 u2) t)
      compose near t on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
(kbind U k2 (subst_loc k2 x2 u2)
 ∘ kbind U k1 (subst_loc k1 x1 u1)) t =
(kbind U k1
   (subst_loc k1 x1
      (kbind (T k1) k2 (subst_loc k2 x2 u2) u1))
 ∘ kbind U k2 (subst_loc k2 x2 u2)) t
      unfold kbind.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
(mbind U (btg k2 (subst_loc k2 x2 u2))
 ∘ mbind U (btg k1 (subst_loc k1 x1 u1))) t =
(mbind U
   (btg k1
      (subst_loc k1 x1
         (mbind (T k1) (btg k2 (subst_loc k2 x2 u2))
            u1)))
 ∘ mbind U (btg k2 (subst_loc k2 x2 u2))) t rewrite 2(mbind_mbind U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
mbind U
  (fun k =>
   mbind (T k) (btg k2 (subst_loc k2 x2 u2))
   ○ btg k1 (subst_loc k1 x1 u1) k) t =
mbind U
  (fun k =>
   mbind (T k)
     (btg k1
        (subst_loc k1 x1
           (mbind (T k1) (btg k2 (subst_loc k2 x2 u2))
              u1))) ○ btg k2 (subst_loc k2 x2 u2) k) t
      fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
(fun k =>
 mbind (T k) (btg k2 (subst_loc k2 x2 u2))
 ○ btg k1 (subst_loc k1 x1 u1) k) =
(fun k =>
 mbind (T k)
   (btg k1
      (subst_loc k1 x1
         (mbind (T k1) (btg k2 (subst_loc k2 x2 u2))
            u1))) ○ btg k2 (subst_loc k2 x2 u2) k) ext j.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
j: K
mbind (T j) (btg k2 (subst_loc k2 x2 u2))
○ btg k1 (subst_loc k1 x1 u1) j =
mbind (T j)
  (btg k1
     (subst_loc k1 x1
        (mbind (T k1) (btg k2 (subst_loc k2 x2 u2)) u1)))
○ btg k2 (subst_loc k2 x2 u2) j now apply subst_subst_neq_loc.
    Qed.

    Lemma subst_subst_eq_local : forall k u1 u2 x1 x2,
        ~ x1 ∈ free (T k) k u2 ->
        x1 <> x2 ->
        subst (T k) k x2 u2 ∘ subst_loc k x1 u1 =
        subst (T k) k x1 (subst (T k) k x2 u2 u1) ∘ subst_loc k x2 u2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) x1 x2,
~ x1 ∈ free (T k) k u2 ->
x1 <> x2 ->
subst (T k) k x2 u2 ∘ subst_loc k x1 u1 =
subst (T k) k x1 (u1 '{ k | x2 ~> u2})
∘ subst_loc k x2 u2
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) x1 x2,
~ x1 ∈ free (T k) k u2 ->
x1 <> x2 ->
subst (T k) k x2 u2 ∘ subst_loc k x1 u1 =
subst (T k) k x1 (u1 '{ k | x2 ~> u2})
∘ subst_loc k x2 u2
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
subst (T k) k x2 u2 ∘ subst_loc k x1 u1 =
subst (T k) k x1 (u1 '{ k | x2 ~> u2})
∘ subst_loc k x2 u2 ext l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
l: LN
(subst (T k) k x2 u2 ∘ subst_loc k x1 u1) l =
(subst (T k) k x1 (u1 '{ k | x2 ~> u2})
 ∘ subst_loc k x2 u2) l unfold compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
l: LN
subst_loc k x1 u1 l '{ k | x2 ~> u2} =
subst_loc k x2 u2 l '{ k | x1 ~> u1 '{ k | x2 ~> u2}} compare l to atom x1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
subst_loc k x1 u1 (Fr x1) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr x1) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
a: atom
H3: a <> x1
subst_loc k x1 u1 (Fr a) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr a) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
n: nat
subst_loc k x1 u1 (Bd n) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Bd n) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}}
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
subst_loc k x1 u1 (Fr x1) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr x1) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}} rewrite subst_loc_eq, subst_loc_neq,
        subst_in_mret_eq, subst_loc_eq...
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
a: atom
H3: a <> x1
subst_loc k x1 u1 (Fr a) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr a) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}} rewrite subst_loc_neq...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
a: atom
H3: a <> x1
mret T k (Fr a) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr a) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}}
        compare values x2 and a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1: atom
H1: ~ x1 ∈ free (T k) k u2
a: atom
H2: x1 <> a
H3: a <> x1
DESTR_EQs: a = a
mret T k (Fr a) '{ k | a ~> u2} =
subst_loc k a u2 (Fr a) '{ k | x1 ~>
u1 '{ k | a ~> u2}}
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
a: atom
H3: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
mret T k (Fr a) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr a) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}}
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1: atom
H1: ~ x1 ∈ free (T k) k u2
a: atom
H2: x1 <> a
H3: a <> x1
DESTR_EQs: a = a
mret T k (Fr a) '{ k | a ~> u2} =
subst_loc k a u2 (Fr a) '{ k | x1 ~>
u1 '{ k | a ~> u2}} rewrite subst_in_mret_eq, subst_loc_eq, (subst_fresh (T k))...
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
a: atom
H3: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
mret T k (Fr a) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Fr a) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}} rewrite subst_loc_neq, 2(subst_in_mret_eq), 2(subst_loc_neq)...
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
n: nat
subst_loc k x1 u1 (Bd n) '{ k | x2 ~> u2} =
subst_loc k x2 u2 (Bd n) '{ k | x1 ~>
u1 '{ k | x2 ~> u2}} rewrite 2(subst_loc_b), 2(subst_in_mret_eq), 2(subst_loc_b)...
    Qed.

    Theorem subst_subst_eq : forall k u1 u2 t x1 x2,
        ~ x1 ∈ free (T k) k u2 ->
        x1 <> x2 ->
        t '{k | x1 ~> u1} '{k | x2 ~> u2} =
        t '{k | x2 ~> u2} '{k | x1 ~> u1 '{k | x2 ~> u2} }.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x1 x2,
~ x1 ∈ free (T k) k u2 ->
x1 <> x2 ->
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1 '{ k | x2 ~> u2}}
    Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x1 x2,
~ x1 ∈ free (T k) k u2 ->
x1 <> x2 ->
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1 '{ k | x2 ~> u2}}
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1 '{ k | x2 ~> u2}} unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
kbind U k (subst_loc k x2 u2)
  (kbind U k (subst_loc k x1 u1) t) =
kbind U k
  (subst_loc k x1
     (kbind (T k) k (subst_loc k x2 u2) u1))
  (kbind U k (subst_loc k x2 u2) t)
      compose near t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
(kbind U k (subst_loc k x2 u2)
 ∘ kbind U k (subst_loc k x1 u1)) t =
(kbind U k
   (subst_loc k x1
      (kbind (T k) k (subst_loc k x2 u2) u1))
 ∘ kbind U k (subst_loc k x2 u2)) t
      rewrite 2(kbind_kbind U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
kbind U k (subst_loc k x2 u2 ⋆km subst_loc k x1 u1) t =
kbind U k
  (subst_loc k x1
     (kbind (T k) k (subst_loc k x2 u2) u1)
   ⋆km subst_loc k x2 u2) t
      fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: ~ x1 ∈ free (T k) k u2
H2: x1 <> x2
subst_loc k x2 u2 ⋆km subst_loc k x1 u1 =
subst_loc k x1 (kbind (T k) k (subst_loc k x2 u2) u1)
⋆km subst_loc k x2 u2 now apply subst_subst_eq_local.
    Qed.

    (** ** Commuting two substitutions *)
    (******************************************************************************)
    Corollary subst_subst_comm_eq : forall k u1 u2 t x1 x2,
        x1 <> x2 ->
        ~ x1 ∈ free (T k) k u2 ->
        ~ x2 ∈ free (T k) k u1 ->
        t '{k | x1 ~> u1} '{k | x2 ~> u2} =
        t '{k | x2 ~> u2} '{k | x1 ~> u1}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x1 x2,
x1 <> x2 ->
~ x1 ∈ free (T k) k u2 ->
~ x2 ∈ free (T k) k u1 ->
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1}
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x1 x2,
x1 <> x2 ->
~ x1 ∈ free (T k) k u2 ->
~ x2 ∈ free (T k) k u1 ->
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1}
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: x1 <> x2
H2: ~ x1 ∈ free (T k) k u2
H3: ~ x2 ∈ free (T k) k u1
(t '{ k | x1 ~> u1}) '{ k | x2 ~> u2} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1} rewrite subst_subst_eq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x1, x2: atom
H1: x1 <> x2
H2: ~ x1 ∈ free (T k) k u2
H3: ~ x2 ∈ free (T k) k u1
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1 '{ k | x2 ~> u2}} =
(t '{ k | x2 ~> u2}) '{ k | x1 ~> u1}
      rewrite (subst_fresh (T k)); auto.
    Qed.

    Corollary subst_subst_comm_neq : forall k1 k2 u1 u2 t x1 x2,
        k1 <> k2 ->
        ~ x1 ∈ free (T k2) k1 u2 ->
        ~ x2 ∈ free (T k1) k2 u1 ->
        t '{k1 | x1 ~> u1} '{k2 | x2 ~> u2} =
        t '{k2 | x2 ~> u2} '{k1 | x1 ~> u1}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) t x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
~ x2 ∈ free (T k1) k2 u1 ->
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~> u1}
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) t x1 x2,
k1 <> k2 ->
~ x1 ∈ free (T k2) k1 u2 ->
~ x2 ∈ free (T k1) k2 u1 ->
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~> u1}
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
H3: ~ x2 ∈ free (T k1) k2 u1
(t '{ k1 | x1 ~> u1}) '{ k2 | x2 ~> u2} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~> u1} rewrite subst_subst_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
t: U LN
x1, x2: atom
H1: k1 <> k2
H2: ~ x1 ∈ free (T k2) k1 u2
H3: ~ x2 ∈ free (T k1) k2 u1
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~>
u1 '{ k2 | x2 ~> u2}} =
(t '{ k2 | x2 ~> u2}) '{ k1 | x1 ~> u1}
      rewrite (subst_fresh (T k1)); auto.
    Qed.

    (** ** Local closure is preserved by substitution *)
    (******************************************************************************)
    Theorem subst_lc_eq : forall k u t x,
        LC U k t ->
        LC (T k) k u ->
        LC U k (subst U k x u t).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t x,
LC U k t -> LC (T k) k u -> LC U k (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t x,
LC U k t -> LC (T k) k u -> LC U k (t '{ k | x ~> u})
      unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t x,
LCn U k 0 t ->
LCn (T k) k 0 u -> LCn U k 0 (t '{ k | x ~> u}) introv lct lcu hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
hin: (w, k, l) ∈md t '{ k | x ~> u}
lc_loc k 0 (w, l)
      rewrite (inmd_subst_iff_eq' U) in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
hin: (w, k, l) ∈md t /\ l <> Fr x \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, k, l) ∈md u /\ w = w1 ● w2)
lc_loc k 0 (w, l)
      destruct hin as [[? ?] | [n1 [n2 [h1 [h2 h3]]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
H1: (w, k, l) ∈md t
H2: l <> Fr x
lc_loc k 0 (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, l) ∈md u
h3: w = n1 ● n2
lc_loc k 0 (w, l)
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
H1: (w, k, l) ∈md t
H2: l <> Fr x
lc_loc k 0 (w, l) auto.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
w: list K
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, l) ∈md u
h3: w = n1 ● n2
lc_loc k 0 (w, l) subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
lcu: LCn (T k) k 0 u
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, l) ∈md u
lc_loc k 0 (n1 ● n2, l) specialize (lcu n2 l h2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
l: LN
n2: list K
lcu: lc_loc k 0 (n2, l)
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, l) ∈md u
lc_loc k 0 (n1 ● n2, l)
        unfold lc_loc in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
l: LN
n2: list K
lcu: match l with
| Fr _ => True
| Bd n => n < countk k n2 + 0
end
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, l) ∈md u
match l with
| Fr _ => True
| Bd n => n < countk k (n1 ● n2) + 0
end
        destruct l; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
n: nat
n2: list K
lcu: n < countk k n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, Bd n) ∈md u
n < countk k (n1 ● n2) + 0 unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
n: nat
n2: list K
lcu: n < countk k n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, Bd n) ∈md u
n < countk k (n1 ++ n2) + 0
        rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
lct: LCn U k 0 t
n: nat
n2: list K
lcu: n < countk k n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, k, Bd n) ∈md u
n < countk k n1 + countk k n2 + 0 lia.
    Qed.

    Theorem subst_lc_neq : forall k j u t x,
        k <> j ->
        LC U j t ->
        LC (T k) j u ->
        LC U j (subst U k x u t).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j (u : T k LN) t x,
k <> j ->
LC U j t -> LC (T k) j u -> LC U j (t '{ k | x ~> u})
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j (u : T k LN) t x,
k <> j ->
LC U j t -> LC (T k) j u -> LC U j (t '{ k | x ~> u})
      unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j (u : T k LN) t x,
k <> j ->
LCn U j 0 t ->
LCn (T k) j 0 u -> LCn U j 0 (t '{ k | x ~> u}) introv neq lct lcu hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
hin: (w, j, l) ∈md t '{ k | x ~> u}
lc_loc j 0 (w, l)
      rewrite (inmd_subst_iff_neq' U) in hin; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
hin: (w, j, l) ∈md t \/
(exists w1 w2,
   (w1, k, Fr x) ∈md t /\
   (w2, j, l) ∈md u /\ w = w1 ● w2)
lc_loc j 0 (w, l)
      destruct hin as [? | [n1 [n2 [h1 [h2 h3]]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
H1: (w, j, l) ∈md t
lc_loc j 0 (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, l) ∈md u
h3: w = n1 ● n2
lc_loc j 0 (w, l)
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
H1: (w, j, l) ∈md t
lc_loc j 0 (w, l) auto.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
w: list K
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, l) ∈md u
h3: w = n1 ● n2
lc_loc j 0 (w, l) subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
lcu: LCn (T k) j 0 u
l: LN
n1, n2: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, l) ∈md u
lc_loc j 0 (n1 ● n2, l) specialize (lcu n2 l h2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
l: LN
n2: list K
lcu: lc_loc j 0 (n2, l)
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, l) ∈md u
lc_loc j 0 (n1 ● n2, l)
        unfold lc_loc in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
l: LN
n2: list K
lcu: match l with
| Fr _ => True
| Bd n => n < countk j n2 + 0
end
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, l) ∈md u
match l with
| Fr _ => True
| Bd n => n < countk j (n1 ● n2) + 0
end
        destruct l; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
n: nat
n2: list K
lcu: n < countk j n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, Bd n) ∈md u
n < countk j (n1 ● n2) + 0 unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
n: nat
n2: list K
lcu: n < countk j n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, Bd n) ∈md u
n < countk j (n1 ++ n2) + 0
        rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
u: T k LN
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
n: nat
n2: list K
lcu: n < countk j n2 + 0
n1: list K
h1: (n1, k, Fr x) ∈md t
h2: (n2, j, Bd n) ∈md u
n < countk j n1 + countk j n2 + 0 lia.
    Qed.

    (** ** Decompose substitution into closing/opening *)
    (******************************************************************************)
    Lemma subst_spec_local : forall k u w l x,
        subst_loc k x u l =
        open_loc k u (w, (close_loc k x) (w, l)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w l x,
subst_loc k x u l =
open_loc k u (w, close_loc k x (w, l))
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w l x,
subst_loc k x u l =
open_loc k u (w, close_loc k x (w, l))
      introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
l: LN
x: atom
subst_loc k x u l =
open_loc k u (w, close_loc k x (w, l)) compare l to atom x; autorewrite* with tea_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
u = open_loc k u (w, close_loc k x (w, Fr x))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x, a: atom
H1: a <> x
mret T k (Fr a) =
open_loc k u (w, close_loc k x (w, Fr a))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
mret T k (Bd n) =
open_loc k u (w, close_loc k x (w, Bd n))
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
u = open_loc k u (w, close_loc k x (w, Fr x)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
u =
match (if x == x then Bd (countk k w) else Fr x) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end compare values x and x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
DESTR_EQ, DESTR_EQs: x = x
u =
match Nat.compare (countk k w) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end unfold id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
DESTR_EQ, DESTR_EQs: x = x
u =
match Nat.compare (countk k w) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end
        compare naturals (countk k w) and (countk k w).
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x, a: atom
H1: a <> x
mret T k (Fr a) =
open_loc k u (w, close_loc k x (w, Fr a)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x, a: atom
H1: a <> x
mret T k (Fr a) =
match (if x == a then Bd (countk k w) else Fr a) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end compare values x and a. (* todo fix fragile names *)
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
mret T k (Bd n) =
open_loc k u (w, close_loc k x (w, Bd n)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
mret T k (Bd n) =
match
  match Nat.compare n (countk k w) with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end unfold id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
mret T k (Bd n) =
match
  match Nat.compare n (countk k w) with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
mret T k (Bd (countk k w)) =
match Nat.compare (S (countk k w)) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (S (countk k w)))
| Gt => mret T k (Bd (S (countk k w) - 1))
end
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd n) =
match Nat.compare (S n) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (S n))
| Gt => mret T k (Bd (S n - 1))
end
        compare naturals (Datatypes.S (countk k w)) and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
x: atom
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd n) =
match Nat.compare (S n) (countk k w) with
| Eq => u
| Lt => mret T k (Bd (S n))
| Gt => mret T k (Bd (S n - 1))
end
        now compare naturals (Datatypes.S n) and (countk k w).
    Qed.

    Theorem subst_spec : forall k x u t,
        t '{k | x ~> u} = ('[k | x] t) '(k | u).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x (u : T k LN) t,
t '{ k | x ~> u} = ('[ k | x] t) '( k | u)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x (u : T k LN) t,
t '{ k | x ~> u} = ('[ k | x] t) '( k | u)
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
t '{ k | x ~> u} = ('[ k | x] t) '( k | u) compose near t on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
t '{ k | x ~> u} = (open U k u ∘ close U k x) t
      unfold open, close, subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
kbind U k (subst_loc k x u) t =
(kbindd U k (open_loc k u) ∘ kmapd U k (close_loc k x))
  t
      rewrite (kbindd_kmapd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
kbind U k (subst_loc k x u) t =
kbindd U k
  (fun '(w, a) =>
   open_loc k u (w, close_loc k x (w, a))) t
      symmetry.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
kbindd U k
  (fun '(w, a) =>
   open_loc k u (w, close_loc k x (w, a))) t =
kbind U k (subst_loc k x u) t apply (kbindd_respectful_kbind k).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
forall w (a : LN),
(w, k, a) ∈md t ->
open_loc k u (w, close_loc k x (w, a)) =
subst_loc k x u a
      symmetry.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
u: T k LN
t: U LN
w: list K
a: LN
H1: (w, k, a) ∈md t
subst_loc k x u a =
open_loc k u (w, close_loc k x (w, a)) apply subst_spec_local.
    Qed.

    (** ** Substitution when <<u>> is a LN **)
    (******************************************************************************)
    Definition subst_loc_LN k x (u : LN) : LN -> LN :=
      fun l => match l with
            | Fr y => if x == y then u else Fr y
            | Bd n => Bd n
            end.

    Theorem subst_by_LN_spec : forall k x l,
        subst U k x (mret T k l) = kmap U k (subst_loc_LN k x l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l,
subst U k x (mret T k l) =
kmap U k (subst_loc_LN k x l)
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x l,
subst U k x (mret T k l) =
kmap U k (subst_loc_LN k x l)
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
subst U k x (mret T k l) =
kmap U k (subst_loc_LN k x l) unfold subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
kbind U k (subst_loc k x (mret T k l)) =
kmap U k (subst_loc_LN k x l) ext t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
kbind U k (subst_loc k x (mret T k l)) t =
kmap U k (subst_loc_LN k x l) t
      apply kbind_respectful_kmap.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
forall a : LN,
(k, a) ∈m t ->
subst_loc k x (mret T k l) a =
mret T k (subst_loc_LN k x l a)
      intros l' Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
l': LN
Hin: (k, l') ∈m t
subst_loc k x (mret T k l) l' =
mret T k (subst_loc_LN k x l l') destruct l'.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
n: atom
Hin: (k, Fr n) ∈m t
subst_loc k x (mret T k l) (Fr n) =
mret T k (subst_loc_LN k x l (Fr n))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
n: nat
Hin: (k, Bd n) ∈m t
subst_loc k x (mret T k l) (Bd n) =
mret T k (subst_loc_LN k x l (Bd n))
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
n: atom
Hin: (k, Fr n) ∈m t
subst_loc k x (mret T k l) (Fr n) =
mret T k (subst_loc_LN k x l (Fr n)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
n: atom
Hin: (k, Fr n) ∈m t
(if x == n then mret T k l else mret T k (Fr n)) =
mret T k (if x == n then l else Fr n) compare values x and n.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
l: LN
t: U LN
n: nat
Hin: (k, Bd n) ∈m t
subst_loc k x (mret T k l) (Bd n) =
mret T k (subst_loc_LN k x l (Bd n)) reflexivity.
    Qed.

    (** ** Substitution by the same variable is the identity *)
    (******************************************************************************)
    Theorem subst_same : forall t k x,
        t '{k | x ~> mret T k (Fr x)} = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x, t '{ k | x ~> mret T k (Fr x)} = t
    Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k x, t '{ k | x ~> mret T k (Fr x)} = t
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
t '{ k | x ~> mret T k (Fr x)} = t apply (subst_id U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
forall l,
(k, l) ∈m t ->
subst_loc k x (mret T k (Fr x)) l = mret T k l
      intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x: atom
l: LN
H1: (k, l) ∈m t
subst_loc k x (mret T k (Fr x)) l = mret T k l compare l to atom x; now simpl_local.
    Qed.

  End fix_dtm.

End subst_metatheory.

(** ** Metatheory for <<close>> *)
(******************************************************************************)
Section close_metatheory.

  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Implicit Types
           (k : K) (j : K)
           (l : LN) (p : LN)
           (x : atom) (t : U LN)
           (w : list K) (n : nat).

  (** ** Free variables after variable closing *)
  (******************************************************************************)
  Lemma in_free_close_iff_loc_1 : forall k w l t x  y,
      (w, k, l) ∈md t ->
      Fr y = close_loc k x (w, l) ->
      (k, Fr y) ∈m t /\ x <> y.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w l t (x y : atom),
(w, k, l) ∈md t ->
Fr y = close_loc k x (w, l) ->
(k, Fr y) ∈m t /\ x <> y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k w l t (x y : atom),
(w, k, l) ∈md t ->
Fr y = close_loc k x (w, l) ->
(k, Fr y) ∈m t /\ x <> y
    introv lin heq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
l: LN
t: U LN
x, y: atom
lin: (w, k, l) ∈md t
heq: Fr y = close_loc k x (w, l)
(k, Fr y) ∈m t /\ x <> y destruct l as [la | ln].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
la: atom
t: U LN
x, y: atom
lin: (w, k, Fr la) ∈md t
heq: Fr y = close_loc k x (w, Fr la)
(k, Fr y) ∈m t /\ x <> y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
ln: nat
t: U LN
x, y: atom
lin: (w, k, Bd ln) ∈md t
heq: Fr y = close_loc k x (w, Bd ln)
(k, Fr y) ∈m t /\ x <> y
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
la: atom
t: U LN
x, y: atom
lin: (w, k, Fr la) ∈md t
heq: Fr y = close_loc k x (w, Fr la)
(k, Fr y) ∈m t /\ x <> y cbn in heq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
la: atom
t: U LN
x, y: atom
lin: (w, k, Fr la) ∈md t
heq: Fr y =
(if x == la then Bd (countk k w) else Fr la)
(k, Fr y) ∈m t /\ x <> y destruct_eq_args x la.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
la: atom
t: U LN
x, y: atom
lin: (w, k, Fr la) ∈md t
DESTR_NEQ: x <> la
heq: Fr y = Fr la
DESTR_NEQs: la <> x
(k, Fr y) ∈m t /\ x <> y
      inverts heq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
la: atom
t: U LN
x: atom
lin: (w, k, Fr la) ∈md t
DESTR_NEQ: x <> la
DESTR_NEQs: la <> x
(k, Fr la) ∈m t /\ x <> la now apply (inmd_implies_in U) in lin.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
ln: nat
t: U LN
x, y: atom
lin: (w, k, Bd ln) ∈md t
heq: Fr y = close_loc k x (w, Bd ln)
(k, Fr y) ∈m t /\ x <> y cbn in heq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
w: list K
ln: nat
t: U LN
x, y: atom
lin: (w, k, Bd ln) ∈md t
heq: Fr y =
match Nat.compare ln (countk k w) with
| Lt => Bd ln
| _ => Bd (S ln)
end
(k, Fr y) ∈m t /\ x <> y compare_nats_args ln (countk k w); discriminate.
  Qed.

  Lemma in_free_close_iff_loc_2 : forall t k x y,
      x <> y ->
      (k, Fr y) ∈m t ->
      exists w l, (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (x y : atom),
x <> y ->
(k, Fr y) ∈m t ->
exists w l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (x y : atom),
x <> y ->
(k, Fr y) ∈m t ->
exists w l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l)
    introv neq yin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
yin: (k, Fr y) ∈m t
exists w l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l) apply (inmd_iff_in U) in yin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
yin: exists w, (w, k, Fr y) ∈md t
exists w l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l) destruct yin as [w yin].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
w: list K
yin: (w, k, Fr y) ∈md t
exists w l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l)
    exists w.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
w: list K
yin: (w, k, Fr y) ∈md t
exists l,
  (w, k, l) ∈md t /\ Fr y = close_loc k x (w, l) exists (Fr y).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
w: list K
yin: (w, k, Fr y) ∈md t
(w, k, Fr y) ∈md t /\ Fr y = close_loc k x (w, Fr y) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
x, y: atom
neq: x <> y
w: list K
yin: (w, k, Fr y) ∈md t
(w, k, Fr y) ∈md t /\
Fr y = (if x == y then Bd (countk k w) else Fr y) compare values x and y.
  Qed.

  Theorem in_free_close_iff : forall k t x y,
      y ∈ free U k ('[k | x] t) <-> y ∈ free U k t /\ x <> y.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (x y : atom),
y ∈ free U k ('[ k | x] t) <->
y ∈ free U k t /\ x <> y
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (x y : atom),
y ∈ free U k ('[ k | x] t) <->
y ∈ free U k t /\ x <> y
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
y ∈ free U k ('[ k | x] t) <->
y ∈ free U k t /\ x <> y rewrite (free_close_eq_iff U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(exists w l1,
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)) <->
y ∈ free U k t /\ x <> y
    rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(exists w l1,
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)) <->
(k, Fr y) ∈m t /\ x <> y split.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(exists w l1,
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)) ->
(k, Fr y) ∈m t /\ x <> y
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(k, Fr y) ∈m t /\ x <> y ->
exists w l1,
  (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(exists w l1,
   (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1)) ->
(k, Fr y) ∈m t /\ x <> y introv [? [? [? ?] ] ].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
x0: list K
x1: LN
H1: (x0, k, x1) ∈md t
H2: Fr y = close_loc k x (x0, x1)
(k, Fr y) ∈m t /\ x <> y eauto using in_free_close_iff_loc_1.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
(k, Fr y) ∈m t /\ x <> y ->
exists w l1,
  (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1) intros [? ?].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
H1: (k, Fr y) ∈m t
H2: x <> y
exists w l1,
  (w, k, l1) ∈md t /\ Fr y = close_loc k x (w, l1) eauto using in_free_close_iff_loc_2.
  Qed.

  Corollary in_free_close_iff_1 : forall k t x y,
      y <> x ->
      y ∈ free U k ('[k | x] t) <-> y ∈ free U k t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (x y : atom),
y <> x ->
y ∈ free U k ('[ k | x] t) <-> y ∈ free U k t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (x y : atom),
y <> x ->
y ∈ free U k ('[ k | x] t) <-> y ∈ free U k t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
H1: y <> x
y ∈ free U k ('[ k | x] t) <-> y ∈ free U k t rewrite in_free_close_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x, y: atom
H1: y <> x
y ∈ free U k t /\ x <> y <-> y ∈ free U k t intuition.
  Qed.

  Corollary FV_close : forall k t x,
      FV U k ('[k | x] t) [=] FV U k t \\ {{ x }}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x,
FV U k ('[ k | x] t) [=] FV U k t \\ {{x}}
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x,
FV U k ('[ k | x] t) [=] FV U k t \\ {{x}}
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
FV U k ('[ k | x] t) [=] FV U k t \\ {{x}} intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
a: AtomSet.elt
a `in` FV U k ('[ k | x] t) <->
a `in` (FV U k t \\ {{x}}) rewrite AtomSet.diff_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
a: AtomSet.elt
a `in` FV U k ('[ k | x] t) <->
a `in` FV U k t /\ a `notin` {{x}}
    rewrite <- 2(free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
a: AtomSet.elt
a ∈ free U k ('[ k | x] t) <->
a ∈ free U k t /\ a `notin` {{x}} rewrite in_free_close_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
a: AtomSet.elt
a ∈ free U k t /\ x <> a <->
a ∈ free U k t /\ a `notin` {{x}}
    fsetdec.
  Qed.

  Corollary nin_free_close : forall k t x,
      ~ (x ∈ free U k ('[k | x] t)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, ~ x ∈ free U k ('[ k | x] t)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, ~ x ∈ free U k ('[ k | x] t)
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
~ x ∈ free U k ('[ k | x] t) rewrite in_free_close_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
~ (x ∈ free U k t /\ x <> x) intuition.
  Qed.

  Corollary nin_FV_close : forall k t x,
      ~ (x `in` FV U k ('[k | x] t)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, x `notin` FV U k ('[ k | x] t)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, x `notin` FV U k ('[ k | x] t)
    introv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
x `notin` FV U k ('[ k | x] t) rewrite <- free_iff_FV.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
~ x ∈ free U k ('[ k | x] t) apply nin_free_close.
  Qed.

  (** ** Variable closing and local closure *)
  (******************************************************************************)
  Theorem close_lc_eq : forall k t x,
      LC U k t ->
      LCn U k 1 (close U k x t).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, LC U k t -> LCn U k 1 ('[ k | x] t)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, LC U k t -> LCn U k 1 ('[ k | x] t)
    unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x, LCn U k 0 t -> LCn U k 1 ('[ k | x] t) introv lct hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
l: LN
hin: (w, k, l) ∈md '[ k | x] t
lc_loc k 1 (w, l)
    rewrite (inmd_close_iff_eq U) in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
l: LN
hin: exists l1,
  (w, k, l1) ∈md t /\ l = close_loc k x (w, l1)
lc_loc k 1 (w, l)
    destruct hin as [l1 [? ?]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
l, l1: LN
H1: (w, k, l1) ∈md t
H2: l = close_loc k x (w, l1)
lc_loc k 1 (w, l) compare l1 to atom x; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
H1: (w, k, Fr x) ∈md t
lc_loc k 1 (w, close_loc k x (w, Fr x))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
a: atom
H1: (w, k, Fr a) ∈md t
H3: a <> x
lc_loc k 1 (w, close_loc k x (w, Fr a))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
n: nat
H1: (w, k, Bd n) ∈md t
lc_loc k 1 (w, close_loc k x (w, Bd n))
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
H1: (w, k, Fr x) ∈md t
lc_loc k 1 (w, close_loc k x (w, Fr x)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
H1: (w, k, Fr x) ∈md t
match (if x == x then Bd (countk k w) else Fr x) with
| Fr _ => True
| Bd n => n < countk k w + 1
end compare values x and x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
H1: (w, k, Fr x) ∈md t
DESTR_EQ, DESTR_EQs: x = x
countk k w < countk k w + 1 unfold_monoid; lia.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
a: atom
H1: (w, k, Fr a) ∈md t
H3: a <> x
lc_loc k 1 (w, close_loc k x (w, Fr a)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
a: atom
H1: (w, k, Fr a) ∈md t
H3: a <> x
match (if x == a then Bd (countk k w) else Fr a) with
| Fr _ => True
| Bd n => n < countk k w + 1
end compare values x and a.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
n: nat
H1: (w, k, Bd n) ∈md t
lc_loc k 1 (w, close_loc k x (w, Bd n)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
n: nat
H1: (w, k, Bd n) ∈md t
match
  match Nat.compare n (countk k w) with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr _ => True
| Bd n => n < countk k w + 1
end compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
H1: (w, k, Bd (countk k w)) ∈md t
S (countk k w) < countk k w + 1
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
n: nat
H1: (w, k, Bd n) ∈md t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
S n < countk k w + 1
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
H1: (w, k, Bd (countk k w)) ∈md t
S (countk k w) < countk k w + 1 specialize (lct w (Bd (countk k w)) ltac:(assumption)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
w: list K
lct: lc_loc k 0 (w, Bd (countk k w))
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
H1: (w, k, Bd (countk k w)) ∈md t
S (countk k w) < countk k w + 1
        cbn in lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
w: list K
lct: countk k w < countk k w + 0
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
H1: (w, k, Bd (countk k w)) ∈md t
S (countk k w) < countk k w + 1 now unfold_monoid; lia.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
lct: LCn U k 0 t
w: list K
n: nat
H1: (w, k, Bd n) ∈md t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
S n < countk k w + 1 specialize (lct w (Bd n) ltac:(assumption)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
w: list K
n: nat
lct: lc_loc k 0 (w, Bd n)
H1: (w, k, Bd n) ∈md t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
S n < countk k w + 1
        cbn in lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
w: list K
n: nat
lct: n < countk k w + 0
H1: (w, k, Bd n) ∈md t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
S n < countk k w + 1 now unfold_monoid; lia.
  Qed.

  Theorem close_lc_neq : forall k j t x,
      k <> j ->
      LC U j t ->
      LC U j (close U k x t).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t x,
k <> j -> LC U j t -> LC U j ('[ k | x] t)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t x,
k <> j -> LC U j t -> LC U j ('[ k | x] t)
    unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t x,
k <> j -> LCn U j 0 t -> LCn U j 0 ('[ k | x] t) introv neq lct hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
neq: k <> j
lct: LCn U j 0 t
w: list K
l: LN
hin: (w, j, l) ∈md '[ k | x] t
lc_loc j 0 (w, l)
    rewrite (inmd_close_neq_iff U) in hin; auto.
  Qed.

End close_metatheory.

(** ** Metatheory for <<open>> *)
(******************************************************************************)
Section open_metatheory.

  Context
    (U : Type -> Type)
    `{MultiDecoratedTraversablePreModule (list K) T U (mn_op := Monoid_op_list) (mn_unit := Monoid_unit_list)}
    `{! MultiDecoratedTraversableMonad (list K) T}.

  Implicit Types
           (k : K) (j : K)
           (l : LN) (p : LN)
           (x : atom) (t : U LN)
           (w : list K) (n : nat).

  (** ** Upper and lower bounds on free variables after opening *)
  (******************************************************************************)
  Lemma free_open_upper_local : forall t k j (u : T k LN) w l x,
      (k, l) ∈m t ->
      x ∈ free (T k) j (open_loc k u (w, l)) ->
      k = j /\ l = Fr x /\ x ∈ free U j t \/
      x ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) w l x,
(k, l) ∈m t ->
x ∈ free (T k) j (open_loc k u (w, l)) ->
k = j /\ l = Fr x /\ x ∈ free U j t \/
x ∈ free (T k) j u
  Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) w l x,
(k, l) ∈m t ->
x ∈ free (T k) j (open_loc k u (w, l)) ->
k = j /\ l = Fr x /\ x ∈ free U j t \/
x ∈ free (T k) j u
    introv lin xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
l: LN
x: atom
lin: (k, l) ∈m t
xin: x ∈ free (T k) j (open_loc k u (w, l))
k = j /\ l = Fr x /\ x ∈ free U j t \/
x ∈ free (T k) j u rewrite in_free_iff in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
l: LN
x: atom
lin: (k, l) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, l)
k = j /\ l = Fr x /\ x ∈ free U j t \/
x ∈ free (T k) j u
    rewrite 2(in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
l: LN
x: atom
lin: (k, l) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, l)
k = j /\ l = Fr x /\ (j, Fr x) ∈m t \/ (j, Fr x) ∈m u destruct l as [y | n].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
y, x: atom
lin: (k, Fr y) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Fr y)
k = j /\ Fr y = Fr x /\ (j, Fr x) ∈m t \/
(j, Fr x) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Bd n)
k = j /\ Bd n = Fr x /\ (j, Fr x) ∈m t \/
(j, Fr x) ∈m u
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
y, x: atom
lin: (k, Fr y) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Fr y)
k = j /\ Fr y = Fr x /\ (j, Fr x) ∈m t \/
(j, Fr x) ∈m u left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
y, x: atom
lin: (k, Fr y) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Fr y)
k = j /\ Fr y = Fr x /\ (j, Fr x) ∈m t autorewrite with tea_local in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
y, x: atom
lin: (k, Fr y) ∈m t
xin: k = j /\ y = x
k = j /\ Fr y = Fr x /\ (j, Fr x) ∈m t inverts xin...
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Bd n)
k = j /\ Bd n = Fr x /\ (j, Fr x) ∈m t \/
(j, Fr x) ∈m u right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m open_loc k u (w, Bd n)
(j, Fr x) ∈m u cbn in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x)
∈m match Nat.compare n (countk k w) with
   | Eq => u
   | Lt => mret T k (Bd n)
   | Gt => mret T k (Bd (n - 1))
   end
(j, Fr x) ∈m u compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd n)
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
(j, Fr x) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
x: atom
lin: (k, Bd (countk k w)) ∈m t
xin: (j, Fr x) ∈m u
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
(j, Fr x) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd (n - 1))
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
(j, Fr x) ∈m u
      {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd n)
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
(j, Fr x) ∈m u contradict xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
~ (j, Fr x) ∈m mret T k (Bd n) simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
~ (k = j /\ False) intuition. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
x: atom
lin: (k, Bd (countk k w)) ∈m t
xin: (j, Fr x) ∈m u
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
(j, Fr x) ∈m u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd (n - 1))
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
(j, Fr x) ∈m u
      {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
x: atom
lin: (k, Bd (countk k w)) ∈m t
xin: (j, Fr x) ∈m u
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
(j, Fr x) ∈m u assumption. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd (n - 1))
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
(j, Fr x) ∈m u
      {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
xin: (j, Fr x) ∈m mret T k (Bd (n - 1))
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
(j, Fr x) ∈m u contradict xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
~ (j, Fr x) ∈m mret T k (Bd (n - 1)) simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
n: nat
x: atom
lin: (k, Bd n) ∈m t
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
~ (k = j /\ False) intuition. }
  Qed.

  Corollary free_open_upper_local_eq : forall t k (u : T k LN) w l x,
      (k, l) ∈m t ->
      x ∈ free (T k) k (open_loc k u (w, l)) ->
      x ∈ free U k t \/ x ∈ free (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) w l x,
(k, l) ∈m t ->
x ∈ free (T k) k (open_loc k u (w, l)) ->
x ∈ free U k t \/ x ∈ free (T k) k u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) w l x,
(k, l) ∈m t ->
x ∈ free (T k) k (open_loc k u (w, l)) ->
x ∈ free U k t \/ x ∈ free (T k) k u
    introv lin xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
w: list K
l: LN
x: atom
lin: (k, l) ∈m t
xin: x ∈ free (T k) k (open_loc k u (w, l))
x ∈ free U k t \/ x ∈ free (T k) k u
    apply free_open_upper_local
      with (t:=t) in xin; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
w: list K
l: LN
x: atom
lin: (k, l) ∈m t
xin: k = k /\ l = Fr x /\ x ∈ free U k t \/
x ∈ free (T k) k u
x ∈ free U k t \/ x ∈ free (T k) k u
    intuition.
  Qed.

  Corollary free_open_upper_local_neq : forall t k j (u : T k LN) w l x,
      k <> j ->
      (k, l) ∈m t ->
      x ∈ free (T k) j (open_loc k u (w, l)) ->
      x ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) w l x,
k <> j ->
(k, l) ∈m t ->
x ∈ free (T k) j (open_loc k u (w, l)) ->
x ∈ free (T k) j u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) w l x,
k <> j ->
(k, l) ∈m t ->
x ∈ free (T k) j (open_loc k u (w, l)) ->
x ∈ free (T k) j u
    introv neq lin xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
l: LN
x: atom
neq: k <> j
lin: (k, l) ∈m t
xin: x ∈ free (T k) j (open_loc k u (w, l))
x ∈ free (T k) j u
    apply free_open_upper_local
      with (t:=t) in xin; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
w: list K
l: LN
x: atom
neq: k <> j
lin: (k, l) ∈m t
xin: k = j /\ l = Fr x /\ x ∈ free U j t \/
x ∈ free (T k) j u
x ∈ free (T k) j u
    intuition.
  Qed.

  Theorem free_open_upper : forall t k j (u : T k LN) x,
      x ∈ free U j (t '(k | u)) ->
      x ∈ free U j t \/ x ∈ free (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x ∈ free U j (t '( k | u)) ->
x ∈ free U j t \/ x ∈ free (T k) j u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x ∈ free U j (t '( k | u)) ->
x ∈ free U j t \/ x ∈ free (T k) j u
    introv xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j (t '( k | u))
x ∈ free U j t \/ x ∈ free (T k) j u compare values j and k.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: x ∈ free U k (t '( k | u))
DESTR_EQs: k = k
x ∈ free U k t \/ x ∈ free (T k) k u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j (t '( k | u))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: x ∈ free U k (t '( k | u))
DESTR_EQs: k = k
x ∈ free U k t \/ x ∈ free (T k) k u rewrite (free_open_eq_iff U) in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: exists w l1,
  (w, k, l1) ∈md t /\
  x ∈ free (T k) k (open_loc k u (w, l1))
DESTR_EQs: k = k
x ∈ free U k t \/ x ∈ free (T k) k u
      destruct xin as [w [l [hin ?]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
l: LN
hin: (w, k, l) ∈md t
H1: x ∈ free (T k) k (open_loc k u (w, l))
DESTR_EQs: k = k
x ∈ free U k t \/ x ∈ free (T k) k u
      eapply (inmd_implies_in U) in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
l: LN
hin: (k, l) ∈m t
H1: x ∈ free (T k) k (open_loc k u (w, l))
DESTR_EQs: k = k
x ∈ free U k t \/ x ∈ free (T k) k u
      eauto using free_open_upper_local_eq.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j (t '( k | u))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u rewrite (free_open_neq_iff U) in xin; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j t \/
(exists w l1,
   (w, k, l1) ∈md t /\
   x ∈ free (T k) j (open_loc k u (w, l1)))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u
      destruct xin as  [? |  [w [l [hin ?]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
H1: x ∈ free U j t
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
w: list K
l: LN
hin: (w, k, l) ∈md t
H1: x ∈ free (T k) j (open_loc k u (w, l))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u
      auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
w: list K
l: LN
hin: (w, k, l) ∈md t
H1: x ∈ free (T k) j (open_loc k u (w, l))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u apply (inmd_implies_in U) in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
w: list K
l: LN
hin: (k, l) ∈m t
H1: x ∈ free (T k) j (open_loc k u (w, l))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j t \/ x ∈ free (T k) j u
      right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
w: list K
l: LN
hin: (k, l) ∈m t
H1: x ∈ free (T k) j (open_loc k u (w, l))
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free (T k) j u eauto using free_open_upper_local_neq.
  Qed.

  Corollary FV_open_upper : forall t k j (u : T k LN),
      FV U j (t '(k | u)) ⊆ FV U j t ∪ FV (T k) j u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN),
FV U j (t '( k | u)) ⊆ FV U j t ∪ FV (T k) j u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN),
FV U j (t '( k | u)) ⊆ FV U j t ∪ FV (T k) j u
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
FV U j (t '( k | u)) ⊆ FV U j t ∪ FV (T k) j u intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
a: AtomSet.elt
a `in` FV U j (t '( k | u)) ->
a `in` (FV U j t ∪ FV (T k) j u) rewrite AtomSet.union_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
a: AtomSet.elt
a `in` FV U j (t '( k | u)) ->
a `in` FV U j t \/ a `in` FV (T k) j u
    repeat rewrite <- (free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
a: AtomSet.elt
a ∈ free U j (t '( k | u)) ->
a ∈ free U j t \/ a ∈ free (T k) j u
    auto using free_open_upper.
  Qed.

  Corollary free_open_upper_eq : forall t k (u : T k LN) x,
      x ∈ free U k (t '(k | u)) ->
      x ∈ free U k t \/ x ∈ free (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x ∈ free U k (t '( k | u)) ->
x ∈ free U k t \/ x ∈ free (T k) k u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x ∈ free U k (t '( k | u)) ->
x ∈ free U k t \/ x ∈ free (T k) k u
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
H1: x ∈ free U k (t '( k | u))
x ∈ free U k t \/ x ∈ free (T k) k u auto using free_open_upper.
  Qed.

  Corollary FV_open_upper_eq : forall t k (u : T k LN),
      FV U k (t '(k | u)) ⊆ FV U k t ∪ FV (T k) k u.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN),
FV U k (t '( k | u)) ⊆ FV U k t ∪ FV (T k) k u
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN),
FV U k (t '( k | u)) ⊆ FV U k t ∪ FV (T k) k u
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
FV U k (t '( k | u)) ⊆ FV U k t ∪ FV (T k) k u apply FV_open_upper.
  Qed.

  Theorem free_open_lower : forall t k j u x,
      x ∈ free U j t ->
      x ∈ free U j (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x ∈ free U j t -> x ∈ free U j (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN) x,
x ∈ free U j t -> x ∈ free U j (t '( k | u))
    introv xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j t
x ∈ free U j (t '( k | u)) compare values j and k.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: x ∈ free U k t
DESTR_EQs: k = k
x ∈ free U k (t '( k | u))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j t
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j (t '( k | u))
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: x ∈ free U k t
DESTR_EQs: k = k
x ∈ free U k (t '( k | u)) rewrite (in_free_iff) in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: (k, Fr x) ∈m t
DESTR_EQs: k = k
x ∈ free U k (t '( k | u))
      rewrite (inmd_iff_in U) in xin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
xin: exists w, (w, k, Fr x) ∈md t
DESTR_EQs: k = k
x ∈ free U k (t '( k | u))
      destruct xin as [w xin].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
xin: (w, k, Fr x) ∈md t
DESTR_EQs: k = k
x ∈ free U k (t '( k | u))
      rewrite (free_open_eq_iff U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
xin: (w, k, Fr x) ∈md t
DESTR_EQs: k = k
exists w l1,
  (w, k, l1) ∈md t /\
  x ∈ free (T k) k (open_loc k u (w, l1))
      setoid_rewrite (in_free_iff).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
xin: (w, k, Fr x) ∈md t
DESTR_EQs: k = k
exists w l1,
  (w, k, l1) ∈md t /\
  (k, Fr x) ∈m open_loc k u (w, l1)
      exists w (Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
w: list K
xin: (w, k, Fr x) ∈md t
DESTR_EQs: k = k
(w, k, Fr x) ∈md t /\
(k, Fr x) ∈m open_loc k u (w, Fr x) now autorewrite with tea_local.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
x: atom
xin: x ∈ free U j t
DESTR_NEQ: j <> k
DESTR_NEQs: k <> j
x ∈ free U j (t '( k | u)) rewrite (free_open_neq_iff U); auto.
  Qed.

  Theorem free_open_lower_eq : forall t k u x,
      x ∈ free U k t ->
      x ∈ free U k (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x ∈ free U k t -> x ∈ free U k (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN) x,
x ∈ free U k t -> x ∈ free U k (t '( k | u))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
x: atom
H1: x ∈ free U k t
x ∈ free U k (t '( k | u)) auto using free_open_lower.
  Qed.

  Corollary FV_open_lower : forall t k j u,
      FV U j t ⊆ FV U j (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN),
FV U j t ⊆ FV U j (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k j (u : T k LN),
FV U j t ⊆ FV U j (t '( k | u))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
FV U j t ⊆ FV U j (t '( k | u)) intro a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
a: AtomSet.elt
a `in` FV U j t -> a `in` FV U j (t '( k | u)) rewrite <- 2(free_iff_FV).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k, j: K
u: T k LN
a: AtomSet.elt
a ∈ free U j t -> a ∈ free U j (t '( k | u))
    apply free_open_lower.
  Qed.

  Corollary FV_open_lower_eq : forall t k u,
      FV U k t ⊆ FV U k (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN),
FV U k t ⊆ FV U k (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall t k (u : T k LN),
FV U k t ⊆ FV U k (t '( k | u))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
t: U LN
k: K
u: T k LN
FV U k t ⊆ FV U k (t '( k | u)) apply FV_open_lower.
  Qed.

  (** ** Opening a locally closed term is the identity *)
  (**************************************************************************)
  Lemma open_lc_local : forall k (u : T k LN) w l,
      lc_loc k 0 (w, l) ->
      open_loc k u (w, l) = mret T k l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w l,
lc_loc k 0 (w, l) -> open_loc k u (w, l) = mret T k l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) w l,
lc_loc k 0 (w, l) -> open_loc k u (w, l) = mret T k l
    introv hyp.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
l: LN
hyp: lc_loc k 0 (w, l)
open_loc k u (w, l) = mret T k l destruct l as [a | n].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
a: atom
hyp: lc_loc k 0 (w, Fr a)
open_loc k u (w, Fr a) = mret T k (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
n: nat
hyp: lc_loc k 0 (w, Bd n)
open_loc k u (w, Bd n) = mret T k (Bd n)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
a: atom
hyp: lc_loc k 0 (w, Fr a)
open_loc k u (w, Fr a) = mret T k (Fr a) reflexivity.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
n: nat
hyp: lc_loc k 0 (w, Bd n)
open_loc k u (w, Bd n) = mret T k (Bd n) cbn in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
w: list K
n: nat
hyp: n < countk k w + 0
match Nat.compare n (countk k w) with
| Eq => u
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd n) compare naturals n and (countk k w); unfold_monoid; lia.
  Qed.

  Lemma open_lc : forall k t u,
      LC U k t ->
      t '(k | u) = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN), LC U k t -> t '( k | u) = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN), LC U k t -> t '( k | u) = t
    introv lc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
lc: LC U k t
t '( k | u) = t apply (open_id U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
lc: LC U k t
forall w l,
(w, k, l) ∈md t -> open_loc k u (w, l) = mret T k l introv lin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
lc: LC U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
open_loc k u (w, l) = mret T k l
    specialize (lc _ _ lin).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
w: list K
l: LN
lc: lc_loc k 0 (w, l)
lin: (w, k, l) ∈md t
open_loc k u (w, l) = mret T k l
    destruct l; auto using open_lc_local.
  Qed.

  (** ** Opening followed by substitution *)
  (**************************************************************************)
  Lemma subst_open_eq_loc : forall k u1 u2 x,
      LC (T k) k u2 ->
      subst (T k) k x u2 ∘ open_loc k u1 =
      open_loc k (u1 '{k | x ~> u2}) ⋆kdm (subst_loc k x u2 ∘ snd).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) x,
LC (T k) k u2 ->
subst (T k) k x u2 ∘ open_loc k u1 =
open_loc k (u1 '{ k | x ~> u2})
⋆kdm (subst_loc k x u2 ∘ snd)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) x,
LC (T k) k u2 ->
subst (T k) k x u2 ∘ open_loc k u1 =
open_loc k (u1 '{ k | x ~> u2})
⋆kdm (subst_loc k x u2 ∘ snd)
    introv lcu2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
subst (T k) k x u2 ∘ open_loc k u1 =
open_loc k (u1 '{ k | x ~> u2})
⋆kdm (subst_loc k x u2 ∘ snd) ext [w l].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
l: LN
(subst (T k) k x u2 ∘ open_loc k u1) (w, l) =
(open_loc k (u1 '{ k | x ~> u2})
 ⋆kdm (subst_loc k x u2 ∘ snd)) (w, l) unfold compose_kdm.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
l: LN
(subst (T k) k x u2 ∘ open_loc k u1) (w, l) =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ∘ incr w)
  ((subst_loc k x u2 ∘ snd) (w, l))
    unfold compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
l: LN
open_loc k u1 (w, l) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, l))) compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
open_loc k u1 (w, Fr x) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Fr x)))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
a: atom
H1: a <> x
open_loc k u1 (w, Fr a) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Fr a)))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
open_loc k u1 (w, Bd n) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Bd n)))
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
open_loc k u1 (w, Fr x) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Fr x))) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
mret T k (Fr x) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (if x == x then u2 else mret T k (Fr x)) simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
u2 =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (if x == x then u2 else mret T k (Fr x)) compare values x and x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
u2 =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w) u2
      symmetry.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w) u2 = u2 apply kbindd_respectful_id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
forall w0 (a : LN),
(w0, k, a) ∈md u2 ->
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, a)) =
mret T k a intros ? [?|?] hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
w0: list K
n: atom
hin: (w0, k, Fr n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Fr n)) =
mret T k (Fr n)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
w0: list K
n: nat
hin: (w0, k, Bd n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Bd n)) =
mret T k (Bd n)
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
w0: list K
n: atom
hin: (w0, k, Fr n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Fr n)) =
mret T k (Fr n) trivial.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
DESTR_EQ, DESTR_EQs: x = x
w0: list K
n: nat
hin: (w0, k, Bd n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Bd n)) =
mret T k (Bd n) specialize (lcu2 _ _ hin).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
w0: list K
n: nat
lcu2: lc_loc k 0 (w0, Bd n)
w: list K
DESTR_EQ, DESTR_EQs: x = x
hin: (w0, k, Bd n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Bd n)) =
mret T k (Bd n) cbn in lcu2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
w0: list K
n: nat
lcu2: n < countk k w0 + 0
w: list K
DESTR_EQ, DESTR_EQs: x = x
hin: (w0, k, Bd n) ∈md u2
open_loc k (u1 '{ k | x ~> u2}) (incr w (w0, Bd n)) =
mret T k (Bd n) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
w0: list K
n: nat
lcu2: n < countk k w0 + 0
w: list K
DESTR_EQ, DESTR_EQs: x = x
hin: (w0, k, Bd n) ∈md u2
match Nat.compare n (countk k (w ● w0)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd n) unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
w0: list K
n: nat
lcu2: n < countk k w0 + 0
w: list K
DESTR_EQ, DESTR_EQs: x = x
hin: (w0, k, Bd n) ∈md u2
match Nat.compare n (countk k (w ++ w0)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd n)
        rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
w0: list K
n: nat
lcu2: n < countk k w0 + 0
w: list K
DESTR_EQ, DESTR_EQs: x = x
hin: (w0, k, Bd n) ∈md u2
match Nat.compare n (countk k w + countk k w0) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end = mret T k (Bd n)
        compare naturals n and (countk k w + countk k w0).
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
a: atom
H1: a <> x
open_loc k u1 (w, Fr a) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Fr a))) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
a: atom
H1: a <> x
mret T k (Fr a) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (if x == a then u2 else mret T k (Fr a)) simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
a: atom
H1: a <> x
mret T k (Fr a) =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (if x == a then u2 else mret T k (Fr a)) compare values x and a.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
a: atom
H1: a <> x
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
mret T k (Fr a) =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (mret T k (Fr a))
      now rewrite (kbindd_comp_mret_eq).
    (* <<< TODO standardize this lemma *)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
open_loc k u1 (w, Bd n) '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (subst_loc k x u2 (snd (w, Bd n))) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
match Nat.compare n (countk k w) with
| Eq => u1
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end '{ k | x ~> u2} =
kbindd (T k) k
  (open_loc k (u1 '{ k | x ~> u2}) ○ incr w)
  (mret T k (Bd n)) rewrite (kbindd_comp_mret_eq).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
match Nat.compare n (countk k w) with
| Eq => u1
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end '{ k | x ~> u2} =
open_loc k (u1 '{ k | x ~> u2}) (incr w (Ƶ, Bd n))
      compare naturals n and (countk k w); cbn; simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
mret T k (Bd n) =
match Nat.compare n (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
u1 '{ k | x ~> u2} =
match Nat.compare (countk k w) (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) =
match Nat.compare n (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
mret T k (Bd n) =
match Nat.compare n (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end rewrite monoid_id_r.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
mret T k (Bd n) =
match Nat.compare n (countk k w) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end compare naturals n and (countk k w).
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
u1 '{ k | x ~> u2} =
match Nat.compare (countk k w) (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end rewrite monoid_id_r.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
u1 '{ k | x ~> u2} =
match Nat.compare (countk k w) (countk k w) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
u1 '{ k | x ~> u2} =
match Nat.compare (countk k w) (countk k w) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end compare naturals (countk k w) and (countk k w).
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) =
match Nat.compare n (countk k (w ● Ƶ)) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end rewrite monoid_id_r.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) =
match Nat.compare n (countk k w) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
x: atom
lcu2: LC (T k) k u2
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) =
match Nat.compare n (countk k w) with
| Eq => u1 '{ k | x ~> u2}
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end compare naturals n and (countk k w).
  Qed.

  Theorem subst_open_eq :  forall k u1 u2 t x,
      LC (T k) k u2 ->
      t '(k | u1) '{k | x ~> u2} =
      t '{k | x ~> u2} '(k | u1 '{k | x ~> u2}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x,
LC (T k) k u2 ->
(t '( k | u1)) '{ k | x ~> u2} =
(t '{ k | x ~> u2}) '( k | u1 '{ k | x ~> u2})
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u1 u2 : T k LN) t x,
LC (T k) k u2 ->
(t '( k | u1)) '{ k | x ~> u2} =
(t '{ k | x ~> u2}) '( k | u1 '{ k | x ~> u2})
    introv lc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x: atom
lc: LC (T k) k u2
(t '( k | u1)) '{ k | x ~> u2} =
(t '{ k | x ~> u2}) '( k | u1 '{ k | x ~> u2}) compose near t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x: atom
lc: LC (T k) k u2
(subst U k x u2 ∘ open U k u1) t =
(open U k (u1 '{ k | x ~> u2}) ∘ subst U k x u2) t
    unfold open, subst at 1 3.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x: atom
lc: LC (T k) k u2
(kbind U k (subst_loc k x u2)
 ∘ kbindd U k (open_loc k u1)) t =
(kbindd U k (open_loc k (u1 '{ k | x ~> u2}))
 ∘ kbind U k (subst_loc k x u2)) t
    rewrite (kbind_kbindd U), (kbindd_kbind U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x: atom
lc: LC (T k) k u2
kbindd U k
  (kbind (T k) k (subst_loc k x u2) ∘ open_loc k u1) t =
kbindd U k
  (fun '(w, a) =>
   kbindd (T k) k
     (open_loc k (u1 '{ k | x ~> u2}) ∘ incr w)
     (subst_loc k x u2 a)) t
    fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u1, u2: T k LN
t: U LN
x: atom
lc: LC (T k) k u2
kbind (T k) k (subst_loc k x u2) ∘ open_loc k u1 =
(fun '(w, a) =>
 kbindd (T k) k
   (open_loc k (u1 '{ k | x ~> u2}) ∘ incr w)
   (subst_loc k x u2 a)) apply subst_open_eq_loc; auto.
  Qed.

  Theorem subst_open_neq_loc :  forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) (x : atom),
      k1 <> k2 ->
      LC (T k2) k1 u2 ->
      (btgd k2 (subst_loc k2 x u2 ∘ extract (W := prod (list K)))) ⋆dm btgd k1 (open_loc k1 u1) =
      (btgd k1 (open_loc k1 (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))) ⋆dm (btg k2 (subst_loc k2 x u2) ◻ const (extract (W := prod (list K)))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
btgd k2 (subst_loc k2 x u2 ∘ extract)
⋆dm btgd k1 (open_loc k1 u1) =
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
⋆dm (btg k2 (subst_loc k2 x u2) ◻ const extract)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
btgd k2 (subst_loc k2 x u2 ∘ extract)
⋆dm btgd k1 (open_loc k1 u1) =
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
⋆dm (btg k2 (subst_loc k2 x u2) ◻ const extract)
    introv Hneq HLC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
btgd k2 (subst_loc k2 x u2 ∘ extract)
⋆dm btgd k1 (open_loc k1 u1) =
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
⋆dm (btg k2 (subst_loc k2 x u2) ◻ const extract) ext j [w l].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
(btgd k2 (subst_loc k2 x u2 ∘ extract)
 ⋆dm btgd k1 (open_loc k1 u1)) j (w, l) =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
 ⋆dm (btg k2 (subst_loc k2 x u2) ◻ const extract)) j
  (w, l)
    unfold compose_dm.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
mbindd (T j)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) j (w, l)) =
mbindd (T j)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) j
     (w, l))
    compare values j and k1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) k1 (w, l)) =
mbindd (T k1)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) k1
     (w, l))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) j (w, l)) =
mbindd (T j)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) j
     (w, l))
    { (* j = k1 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) k1 (w, l)) =
mbindd (T k1)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) k1
     (w, l))
      rewrite btgd_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
mbindd (T k1)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) k1
     (w, l))
      replace ((btg k2 (subst_loc k2 x u2) ◻ const extract) k1 (w, l))
        with (btg k2 (subst_loc k2 x u2) k1 l) by (compare values k1 and k1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
mbindd (T k1)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w)) (btg k2 (subst_loc k2 x u2) k1 l)
      rewrite btg_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
mbindd (T k1)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w)) (mret T k1 l)
      compose near l on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
(mbindd (T k1)
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w)) ∘ mret T k1) l
      rewrite mbindd_comp_mret.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
((btgd k1
    (open_loc k1
       (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  ◻ allK (incr w)) k1 ∘ ret) l
      unfold vec_compose at 2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   k1 ∘ allK (incr w) k1 ∘ ret) l
      unfold allK at 2, const.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   k1 ∘ incr w ∘ ret) l
      unfold compose at 2 3.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (ret l))
      rewrite btgd_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
open_loc k1
  (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)
  (incr w (ret l))
      replace (incr w (ret l)) with (w, l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
open_loc k1
  (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)
  (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
(w, l) = incr w (ret l)
      2:{U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
(w, l) = incr w (ret l) cbn; now simpl_monoid. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (open_loc k1 u1 (w, l)) =
open_loc k1
  (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)
  (w, l)
      cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  match l with
  | Fr x => mret T k1 (Fr x)
  | Bd n =>
      match Nat.compare n (countk k1 w) with
      | Eq => u1
      | Lt => mret T k1 (Bd n)
      | Gt => mret T k1 (Bd (n - 1))
      end
  end =
match l with
| Fr x => mret T k1 (Fr x)
| Bd n =>
    match Nat.compare n (countk k1 w) with
    | Eq =>
        mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
    | Lt => mret T k1 (Bd n)
    | Gt => mret T k1 (Bd (n - 1))
    end
end destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: atom
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (mret T k1 (Fr n)) =
mret T k1 (Fr n)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  match Nat.compare n (countk k1 w) with
  | Eq => u1
  | Lt => mret T k1 (Bd n)
  | Gt => mret T k1 (Bd (n - 1))
  end =
match Nat.compare n (countk k1 w) with
| Eq => mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
| Lt => mret T k1 (Bd n)
| Gt => mret T k1 (Bd (n - 1))
end
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: atom
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (mret T k1 (Fr n)) =
mret T k1 (Fr n) compose near (Fr n) on left;
           rewrite mbindd_comp_mret;
           unfold compose; cbn;
          compare values k1 and k2.
      -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  match Nat.compare n (countk k1 w) with
  | Eq => u1
  | Lt => mret T k1 (Bd n)
  | Gt => mret T k1 (Bd (n - 1))
  end =
match Nat.compare n (countk k1 w) with
| Eq => mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
| Lt => mret T k1 (Bd n)
| Gt => mret T k1 (Bd (n - 1))
end compare naturals n and (countk k1 w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Lt
ineqp: n < countk k1 w
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (mret T k1 (Bd n)) =
mret T k1 (Bd n)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) u1 =
mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Gt
ineqp: n > countk k1 w
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) 
  (mret T k1 (Bd (n - 1))) = 
mret T k1 (Bd (n - 1))
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Lt
ineqp: n < countk k1 w
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (mret T k1 (Bd n)) =
mret T k1 (Bd n) compose near (Bd n) on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Lt
ineqp: n < countk k1 w
(mbindd (T k1)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w)) ∘ mret T k1) (Bd n) =
mret T k1 (Bd n)
          rewrite mbindd_comp_mret;
            unfold compose; cbn;
            compare values k1 and k2.
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) u1 =
mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1 unfold mbind; fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
btgd k2 (subst_loc k2 x u2 ∘ extract) ◻ allK (incr w) =
btg k2 (subst_loc k2 x u2) ◻ allK extract
          unfold vec_compose, allK, const.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
(fun k =>
 btgd k2 (subst_loc k2 x u2 ∘ extract) k ∘ incr w) =
(fun k => btg k2 (subst_loc k2 x u2) k ∘ extract)
          ext j.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
btgd k2 (subst_loc k2 x u2 ∘ extract) j ∘ incr w =
btg k2 (subst_loc k2 x u2) j ∘ extract
          destruct_eq_args j k2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
btgd k2 (subst_loc k2 x u2 ∘ extract) k2 ∘ incr w =
btg k2 (subst_loc k2 x u2) k2 ∘ extract
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
btgd k2 (subst_loc k2 x u2 ∘ extract) j ∘ incr w =
btg k2 (subst_loc k2 x u2) j ∘ extract
          *U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
btgd k2 (subst_loc k2 x u2 ∘ extract) k2 ∘ incr w =
btg k2 (subst_loc k2 x u2) k2 ∘ extract rewrite btg_eq, btgd_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
subst_loc k2 x u2 ∘ extract ∘ incr w =
subst_loc k2 x u2 ∘ extract
            reassociate ->.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
subst_loc k2 x u2 ∘ (extract ∘ incr w) =
subst_loc k2 x u2 ∘ extract
            replace (extract ∘ incr w) with (extract (A := LN)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
subst_loc k2 x u2 ∘ extract =
subst_loc k2 x u2 ∘ extract
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
extract = extract ∘ incr w
            reflexivity.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
DESTR_EQs0: k2 = k2
extract = extract ∘ incr w
            symmetry; apply (extract_incr (A := LN) w).
          *U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
btgd k2 (subst_loc k2 x u2 ∘ extract) j ∘ incr w =
btg k2 (subst_loc k2 x u2) j ∘ extract rewrite btgd_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
mret T j ∘ extract ∘ incr w =
btg k2 (subst_loc k2 x u2) j ∘ extract
            rewrite btg_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
mret T j ∘ extract ∘ incr w = mret T j ∘ extract
            reassociate ->.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
mret T j ∘ (extract ∘ incr w) = mret T j ∘ extract
            replace (extract ∘ incr w) with (extract (A := LN)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
mret T j ∘ extract = mret T j ∘ extract
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
extract = extract ∘ incr w
            reflexivity.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare 
  (countk k1 w) 
  (countk k1 w) = Eq
j: K
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
extract = extract ∘ incr w
            symmetry; apply (extract_incr (A := LN) w).
        +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Gt
ineqp: n > countk k1 w
mbindd (T k1)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) (mret T k1 (Bd (n - 1))) =
mret T k1 (Bd (n - 1)) compose near (Bd (n - 1)) on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
n: nat
DESTR_EQs: k1 = k1
ineqrw: PeanoNat.Nat.compare n (countk k1 w) = Gt
ineqp: n > countk k1 w
(mbindd (T k1)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w)) ∘ mret T k1) (Bd (n - 1)) =
mret T k1 (Bd (n - 1))
          rewrite mbindd_comp_mret;
            unfold compose; cbn;
            compare values k1 and k2.
    }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) j (w, l)) =
mbindd (T j)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) j
     (w, l))
    { (* j <> k1 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) j (w, l)) =
mbindd (T j)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) j
     (w, l))
      rewrite btgd_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w)) ((mret T j ∘ extract) (w, l)) =
mbindd (T j)
  (btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   ◻ allK (incr w))
  ((btg k2 (subst_loc k2 x u2) ◻ const extract) j
     (w, l))
      unfold vec_compose, const, compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (fun k =>
   btgd k2 (subst_loc k2 x u2 ○ extract) k
   ○ allK (incr w) k) (mret T j (extract (w, l))) =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j (extract (w, l)))
      cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
mbindd (T j)
  (fun k =>
   btgd k2 (subst_loc k2 x u2 ○ extract) k
   ○ allK (incr w) k) (mret T j l) =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l) compose near l on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(mbindd (T j)
   (fun k =>
    btgd k2 (subst_loc k2 x u2 ○ extract) k
    ○ allK (incr w) k) ∘ mret T j) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
      rewrite mbindd_comp_mret.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
(btgd k2 (subst_loc k2 x u2 ○ extract) j
 ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
      compare values j and k2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
(btgd k2 (subst_loc k2 x u2 ○ extract) k2
 ○ allK (incr w) k2 ∘ ret) l =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) k2 l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(btgd k2 (subst_loc k2 x u2 ○ extract) j
 ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
      { (* j = k2 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
(btgd k2 (subst_loc k2 x u2 ○ extract) k2
 ○ allK (incr w) k2 ∘ ret) l =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) k2 l)
        rewrite btgd_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
((fun a : list K * LN =>
  subst_loc k2 x u2 (extract (allK (incr w) k2 a)))
 ∘ ret) l =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) k2 l)
        rewrite btg_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
((fun a : list K * LN =>
  subst_loc k2 x u2 (extract (allK (incr w) k2 a)))
 ∘ ret) l =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k) (subst_loc k2 x u2 l)
        unfold compose, allK, const.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
subst_loc k2 x u2 (extract (incr w (ret l))) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 l) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
l: LN
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
subst_loc k2 x u2 l =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 l)
        compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
subst_loc k2 x u2 (Fr x) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 (Fr x))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
subst_loc k2 x u2 (Fr a) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) 
  (subst_loc k2 x u2 (Fr a))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
n: nat
subst_loc k2 x u2 (Bd n) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) 
  (subst_loc k2 x u2 (Bd n))
        - (* l = Fr x *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
subst_loc k2 x u2 (Fr x) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 (Fr x))
          cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
(if x == x then u2 else mret T k2 (Fr x)) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w)
  (if x == x then u2 else mret T k2 (Fr x)) destruct_eq_args x x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
u2 =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) u2
          symmetry.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) u2 = u2
          apply mbindd_respectful_id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
forall w0 k (a : LN),
(w0, k, a) ∈md u2 ->
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)) k
  (incr w (w0, a)) = mret T k a
          intros w' k l Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
k: K
l: LN
Hin: (w', k, l) ∈md u2
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)) k
  (incr w (w', l)) = mret T k l
          compare values k and k1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
l: LN
Hin: (w', k1, l) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', l)) = mret T k1 l
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
k: K
l: LN
Hin: (w', k, l) ∈md u2
DESTR_NEQ0: k <> k1
DESTR_NEQs0: k1 <> k
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)) k
  (incr w (w', l)) = 
mret T k l
          + (* k = k1 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
l: LN
Hin: (w', k1, l) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', l)) = mret T k1 l
            unfold LC, LCn in HLC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: forall w l,
(w, k1, l) ∈md u2 -> lc_loc k1 0 (w, l)
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
l: LN
Hin: (w', k1, l) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', l)) = mret T k1 l
            specialize (HLC _ _ Hin).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
l: LN
HLC: lc_loc k1 0 (w', l)
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, l) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', l)) = mret T k1 l
            unfold lc_loc in HLC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
l: LN
HLC: match l with
| Fr _ => True
| Bd n => n < countk k1 w' + 0
end
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, l) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', l)) = mret T k1 l
            destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: atom
HLC: True
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Fr n) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', Fr n)) = mret T k1 (Fr n)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', Bd n)) = 
mret T k1 (Bd n)
            {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: atom
HLC: True
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Fr n) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', Fr n)) = mret T k1 (Fr n)  cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: atom
HLC: True
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Fr n) ∈md u2
DESTR_EQs1: k1 = k1
match k1 == k1 with
| left x =>
    rew [fun H : K => T H LN]
        eq_ind_r (fun k => k1 = k) eq_refl x in
    mret T k1 (Fr n)
| right _ => mret T k1 (Fr n)
end = mret T k1 (Fr n) compare values k1 and k1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: atom
HLC: True
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Fr n) ∈md u2
DESTR_EQs1, DESTR_EQ0, DESTR_EQs2: k1 = k1
rew [fun H : K => T H LN]
    eq_ind_r (fun k => k1 = k) eq_refl DESTR_EQ0 in
mret T k1 (Fr n) = mret T k1 (Fr n)
               now destruct DESTR_EQ0.
            }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', Bd n)) = mret T k1 (Bd n)
            {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
  k1 (incr w (w', Bd n)) = mret T k1 (Bd n) rewrite btgd_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
open_loc k1
  (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)
  (incr w (w', Bd n)) = mret T k1 (Bd n)
              cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
match Nat.compare n (countk k1 (w ● w')) with
| Eq => mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
| Lt => mret T k1 (Bd n)
| Gt => mret T k1 (Bd (n - 1))
end = mret T k1 (Bd n) unfold_ops @Monoid_op_list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
match Nat.compare n (countk k1 (w ++ w')) with
| Eq => mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
| Lt => mret T k1 (Bd n)
| Gt => mret T k1 (Bd (n - 1))
end = mret T k1 (Bd n)
              rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
w': list K
n: nat
HLC: n < countk k1 w' + 0
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
Hin: (w', k1, Bd n) ∈md u2
DESTR_EQs1: k1 = k1
match Nat.compare n (countk k1 w + countk k1 w') with
| Eq => mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1
| Lt => mret T k1 (Bd n)
| Gt => mret T k1 (Bd (n - 1))
end = mret T k1 (Bd n)
              compare naturals n and (countk k1 w + countk k1 w'). }
          + (* k <> k1 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
DESTR_EQ, DESTR_EQs0: x = x
w': list K
k: K
l: LN
Hin: (w', k, l) ∈md u2
DESTR_NEQ0: k <> k1
DESTR_NEQs0: k1 <> k
btgd k1
  (open_loc k1
     (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)) k
  (incr w (w', l)) = mret T k l
            rewrite btgd_neq; auto.
        -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
subst_loc k2 x u2 (Fr a) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 (Fr a)) rewrite subst_loc_fr_neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
mret T k2 (Fr a) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (mret T k2 (Fr a))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
Fr x <> Fr a
          compose near (Fr a) on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
mret T k2 (Fr a) =
(mbindd (T k2)
   (fun k =>
    btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
      k ○ incr w) ∘ mret T k2) (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
Fr x <> Fr a
          rewrite mbindd_comp_mret.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
mret T k2 (Fr a) =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   k2 ○ incr w ∘ ret) (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
Fr x <> Fr a
          rewrite btgd_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
Fr x <> Fr a
          inversion 1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
a: atom
H1: a <> x
H2: Fr x = Fr a
H4: x = a
False congruence.
        -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
n: nat
subst_loc k2 x u2 (Bd n) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (subst_loc k2 x u2 (Bd n)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
n: nat
mret T k2 (Bd n) =
mbindd (T k2)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ incr w) (mret T k2 (Bd n)) compose near (Bd n) on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
n: nat
mret T k2 (Bd n) =
(mbindd (T k2)
   (fun k =>
    btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
      k ○ incr w) ∘ mret T k2) (Bd n)
          rewrite mbindd_comp_mret.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
w: list K
DESTR_NEQs: k1 <> k2
DESTR_NEQ: k2 <> k1
DESTR_EQs: k2 = k2
n: nat
mret T k2 (Bd n) =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   k2 ○ incr w ∘ ret) (Bd n)
          rewrite btgd_neq; auto.
      }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(btgd k2 (subst_loc k2 x u2 ○ extract) j
 ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
      { (* j <> k2 *)U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(btgd k2 (subst_loc k2 x u2 ○ extract) j
 ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
        rewrite btgd_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(mret T j ∘ extract ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k)
  (btg k2 (subst_loc k2 x u2) j l)
        rewrite btg_neq; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(mret T j ∘ extract ○ allK (incr w) j ∘ ret) l =
mbindd (T j)
  (fun k =>
   btgd k1
     (open_loc k1
        (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
     k ○ allK (incr w) k) (mret T j l)
        compose near l on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(mret T j ∘ extract ○ allK (incr w) j ∘ ret) l =
(mbindd (T j)
   (fun k =>
    btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
      k ○ allK (incr w) k) ∘ mret T j) l
        rewrite (mbindd_comp_mret).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
Hneq: k1 <> k2
HLC: LC (T k2) k1 u2
j: K
w: list K
l: LN
DESTR_NEQ: j <> k1
DESTR_NEQs: k1 <> j
DESTR_NEQ0: j <> k2
DESTR_NEQs0: k2 <> j
(mret T j ∘ extract ○ allK (incr w) j ∘ ret) l =
(btgd k1
   (open_loc k1
      (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
   j ○ allK (incr w) j ∘ ret) l
        rewrite btgd_neq; auto. }
    }
  Qed.

  Theorem subst_open_neq :  forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) (x : atom) (t : U LN),
      k1 <> k2 ->
      LC (T k2) k1 u2 ->
      t '(k1 | u1) '{k2 | x ~> u2} =
      t '{k2 | x ~> u2} '(k1 | u1 '{k2 | x ~> u2}).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x t,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(t '( k1 | u1)) '{ k2 | x ~> u2} =
(t '{ k2 | x ~> u2}) '( k1 | u1 '{ k2 | x ~> u2})
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x t,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(t '( k1 | u1)) '{ k2 | x ~> u2} =
(t '{ k2 | x ~> u2}) '( k1 | u1 '{ k2 | x ~> u2})
    introv neq lc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
(t '( k1 | u1)) '{ k2 | x ~> u2} =
(t '{ k2 | x ~> u2}) '( k1 | u1 '{ k2 | x ~> u2}) compose near t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
(subst U k2 x u2 ∘ open U k1 u1) t =
(open U k1 (u1 '{ k2 | x ~> u2}) ∘ subst U k2 x u2) t
    unfold subst, open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
(kbind U k2 (subst_loc k2 x u2)
 ∘ kbindd U k1 (open_loc k1 u1)) t =
(kbindd U k1
   (open_loc k1
      (kbind (T k1) k2 (subst_loc k2 x u2) u1))
 ∘ kbind U k2 (subst_loc k2 x u2)) t
    unfold kbind, kbindd.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
(mbind U (btg k2 (subst_loc k2 x u2))
 ∘ mbindd U (btgd k1 (open_loc k1 u1))) t =
(mbindd U
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)))
 ∘ mbind U (btg k2 (subst_loc k2 x u2))) t
    rewrite (mbind_to_mbindd U), (mbindd_mbindd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
mbindd U
  (fun k '(w, a) =>
   mbindd (T k)
     ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
      ◻ allK (incr w))
     (btgd k1 (open_loc k1 u1) k (w, a))) t =
(mbindd U
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1)))
 ∘ mbindd U
     (btg k2 (subst_loc k2 x u2) ◻ allK extract)) t
    rewrite (mbindd_mbindd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
mbindd U
  (fun k '(w, a) =>
   mbindd (T k)
     ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
      ◻ allK (incr w))
     (btgd k1 (open_loc k1 u1) k (w, a))) t =
mbindd U
  (fun k '(w, a) =>
   mbindd (T k)
     (btgd k1
        (open_loc k1
           (mbind (T k1) (btg k2 (subst_loc k2 x u2))
              u1)) ◻ allK (incr w))
     ((btg k2 (subst_loc k2 x u2) ◻ allK extract) k
        (w, a))) t
    fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
(fun k '(w, a) =>
 mbindd (T k)
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract) k
      (w, a)))
    pose (lemma := subst_open_neq_loc).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
btgd k2 (subst_loc k2 x u2 ∘ extract)
⋆dm btgd k1 (open_loc k1 u1) =
btgd k1
  (open_loc k1
     (mbind 
        (T k1) 
        (btg k2 (subst_loc k2 x u2)) u1))
⋆dm (btg k2 (subst_loc k2 x u2)
     ◻ const extract)
(fun k '(w, a) =>
 mbindd (T k)
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract) k
      (w, a)))
    unfold compose_dm in lemma.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
(fun k '(w, a) =>
 mbindd (T k)
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind (T k1) (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract) k
      (w, a))) setoid_rewrite <- lemma; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
(fun k '(w, a) =>
 mbindd (T k)
   ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a)))
    ext k [w a].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
mbindd (T k)
  ((btg k2 (subst_loc k2 x u2) ◻ allK extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) k (w, a)) =
mbindd (T k)
  (btgd k2 (subst_loc k2 x u2 ∘ extract)
   ◻ allK (incr w))
  (btgd k1 (open_loc k1 u1) k (w, a)) unfold vec_compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
mbindd (T k)
  (fun k =>
   btg k2 (subst_loc k2 x u2) k ∘ allK extract k
   ∘ allK (incr w) k)
  (btgd k1 (open_loc k1 u1) k (w, a)) =
mbindd (T k)
  (fun k =>
   btgd k2 (subst_loc k2 x u2 ∘ extract) k
   ∘ allK (incr w) k)
  (btgd k1 (open_loc k1 u1) k (w, a))
    fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
(fun k =>
 btg k2 (subst_loc k2 x u2) k ∘ allK extract k
 ∘ allK (incr w) k) =
(fun k =>
 btgd k2 (subst_loc k2 x u2 ∘ extract) k
 ∘ allK (incr w) k) ext j [w' a'].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
j: K
w': list K
a': LN
(btg k2 (subst_loc k2 x u2) j ∘ allK extract j
 ∘ allK (incr w) j) (w', a') =
(btgd k2 (subst_loc k2 x u2 ∘ extract) j
 ∘ allK (incr w) j) (w', a') compare values j and k2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
w': list K
a': LN
DESTR_EQs: k2 = k2
(btg k2 (subst_loc k2 x u2) k2 ∘ allK extract k2
 ∘ allK (incr w) k2) (w', a') =
(btgd k2 (subst_loc k2 x u2 ∘ extract) k2
 ∘ allK (incr w) k2) (w', a')
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
j: K
w': list K
a': LN
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
(btg k2 (subst_loc k2 x u2) j ∘ allK extract j
 ∘ allK (incr w) j) (
  w', a') =
(btgd k2 (subst_loc k2 x u2 ∘ extract) j
 ∘ allK (incr w) j) (
  w', a')
    +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
w': list K
a': LN
DESTR_EQs: k2 = k2
(btg k2 (subst_loc k2 x u2) k2 ∘ allK extract k2
 ∘ allK (incr w) k2) (w', a') =
(btgd k2 (subst_loc k2 x u2 ∘ extract) k2
 ∘ allK (incr w) k2) (w', a') rewrite btgd_eq, btg_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
w': list K
a': LN
DESTR_EQs: k2 = k2
(subst_loc k2 x u2 ∘ allK extract k2
 ∘ allK (incr w) k2) (w', a') =
(subst_loc k2 x u2 ∘ extract ∘ allK (incr w) k2)
  (w', a') cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
w': list K
a': LN
DESTR_EQs: k2 = k2
(subst_loc k2 x u2 ∘ allK extract k2
 ∘ allK (incr w) k2) (w', a') =
(subst_loc k2 x u2 ∘ extract ∘ allK (incr w) k2)
  (w', a') reflexivity.
    +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k1, k2: K
u1: T k1 LN
u2: T k2 LN
x: atom
t: U LN
neq: k1 <> k2
lc: LC (T k2) k1 u2
lemma:= subst_open_neq_loc: forall k1 k2 (u1 : T k1 LN) (u2 : T k2 LN) x,
k1 <> k2 ->
LC (T k2) k1 u2 ->
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k2 (subst_loc k2 x u2 ∘ extract)
    ◻ allK (incr w))
   (btgd k1 (open_loc k1 u1) k (w, a))) =
(fun k '(w, a) =>
 mbindd (T k)
   (btgd k1
      (open_loc k1
         (mbind 
          (T k1) 
          (btg k2 (subst_loc k2 x u2)) u1))
    ◻ allK (incr w))
   ((btg k2 (subst_loc k2 x u2)
     ◻ const extract) k (
      w, a)))
k: K
w: list K
a: LN
j: K
w': list K
a': LN
DESTR_NEQ: j <> k2
DESTR_NEQs: k2 <> j
(btg k2 (subst_loc k2 x u2) j ∘ allK extract j
 ∘ allK (incr w) j) (w', a') =
(btgd k2 (subst_loc k2 x u2 ∘ extract) j
 ∘ allK (incr w) j) (w', a') rewrite btgd_neq, btg_neq; auto.
  Qed.

  (** ** Decompose opening into variable opening followed by substitution *)
  (**************************************************************************)
  Lemma open_spec_eq_loc : forall k u x w l,
      l <> Fr x ->
      (subst (T k) k x u ∘ open_loc k (mret T k (Fr x))) (w, l) =
      open_loc k u (w, l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x w l,
l <> Fr x ->
(subst (T k) k x u ∘ open_loc k (mret T k (Fr x)))
  (w, l) = open_loc k u (w, l)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) x w l,
l <> Fr x ->
(subst (T k) k x u ∘ open_loc k (mret T k (Fr x)))
  (w, l) = open_loc k u (w, l)
    introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
l: LN
neq: l <> Fr x
(subst (T k) k x u ∘ open_loc k (mret T k (Fr x)))
  (w, l) = open_loc k u (w, l) unfold compose.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
l: LN
neq: l <> Fr x
open_loc k (mret T k (Fr x)) (w, l) '{ k | x ~> u} =
open_loc k u (w, l) compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
neq: Fr x <> Fr x
open_loc k (mret T k (Fr x)) (w, Fr x) '{ k | x ~> u} =
open_loc k u (w, Fr x)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
open_loc k (mret T k (Fr x)) (w, Fr a) '{ k | x ~> u} =
open_loc k u (w, Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
open_loc k (mret T k (Fr x)) (w, Bd n) '{ k | x ~> u} =
open_loc k u (w, Bd n)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
neq: Fr x <> Fr x
open_loc k (mret T k (Fr x)) (w, Fr x) '{ k | x ~> u} =
open_loc k u (w, Fr x) contradiction.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
open_loc k (mret T k (Fr x)) (w, Fr a) '{ k | x ~> u} =
open_loc k u (w, Fr a) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
mret T k (Fr a) '{ k | x ~> u} = mret T k (Fr a) rewrite subst_in_mret_eq, subst_loc_fr_neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
mret T k (Fr a) = mret T k (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
Fr x <> Fr a
      trivial.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
Fr x <> Fr a intuition.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
open_loc k (mret T k (Fr x)) (w, Bd n) '{ k | x ~> u} =
open_loc k u (w, Bd n) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
match Nat.compare n (countk k w) with
| Eq => mret T k (Fr x)
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end '{ k | x ~> u} =
match Nat.compare n (countk k w) with
| Eq => u
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Lt
ineqp: n < countk k w
mret T k (Bd n) '{ k | x ~> u} = mret T k (Bd n)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
neq: Bd (countk k w) <> Fr x
mret T k (Fr x) '{ k | x ~> u} = u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) '{ k | x ~> u} =
mret T k (Bd (n - 1))
      now rewrite subst_in_mret_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
neq: Bd (countk k w) <> Fr x
mret T k (Fr x) '{ k | x ~> u} = u
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) '{ k | x ~> u} =
mret T k (Bd (n - 1))
      now rewrite subst_in_mret_eq, subst_loc_eq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
mret T k (Bd (n - 1)) '{ k | x ~> u} =
mret T k (Bd (n - 1))
      now rewrite subst_in_mret_eq, subst_loc_b.
  Qed.

  (* This theorem would be easy to prove with [subst_open_eq], but
   applying that theorem would introduce a local closure hypothesis
   for <<u>> that is not actually required for our purposes. *)
  Theorem open_spec_eq : forall k u t x,
      ~ x `in` FV U k t ->
      t '(k | u) = t '(k | mret T k (Fr x)) '{k | x ~> u}.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t x,
x `notin` FV U k t ->
t '( k | u) =
(t '( k | mret T k (Fr x))) '{ k | x ~> u}
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t x,
x `notin` FV U k t ->
t '( k | u) =
(t '( k | mret T k (Fr x))) '{ k | x ~> u}
    introv fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
t '( k | u) =
(t '( k | mret T k (Fr x))) '{ k | x ~> u} compose near t on right.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
t '( k | u) =
(subst U k x u ∘ open U k (mret T k (Fr x))) t
    unfold subst, open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
kbindd U k (open_loc k u) t =
(kbind U k (subst_loc k x u)
 ∘ kbindd U k (open_loc k (mret T k (Fr x)))) t rewrite (kbind_kbindd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
kbindd U k (open_loc k u) t =
kbindd U k
  (kbind (T k) k (subst_loc k x u)
   ∘ open_loc k (mret T k (Fr x))) t
    apply (kbindd_respectful).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
forall w (a : LN),
(w, k, a) ∈md t ->
open_loc k u (w, a) =
(kbind (T k) k (subst_loc k x u)
 ∘ open_loc k (mret T k (Fr x))) (w, a) introv Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (w, k, a) ∈md t
open_loc k u (w, a) =
(kbind (T k) k (subst_loc k x u)
 ∘ open_loc k (mret T k (Fr x))) (w, a)
    assert (a <> Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (w, k, a) ∈md t
a <> Fr x
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (w, k, a) ∈md t
H1: a <> Fr x
open_loc k u (w, a) =
(kbind (T k) k (subst_loc k x u)
 ∘ open_loc k (mret T k (Fr x))) (
  w, a)
    {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (w, k, a) ∈md t
a <> Fr x apply (inmd_implies_in U) in Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (k, a) ∈m t
a <> Fr x
      rewrite <- free_iff_FV in fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: ~ x ∈ free U k t
w: list K
a: LN
Hin: (k, a) ∈m t
a <> Fr x
      eapply ninf_in_neq in fresh; eauto. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x: atom
fresh: x `notin` FV U k t
w: list K
a: LN
Hin: (w, k, a) ∈md t
H1: a <> Fr x
open_loc k u (w, a) =
(kbind (T k) k (subst_loc k x u)
 ∘ open_loc k (mret T k (Fr x))) (w, a)
    now rewrite <- (open_spec_eq_loc k u x).
  Qed.

  (** ** Opening by a variable, followed by non-equal substitution *)
  (**************************************************************************)
  Lemma subst_open_var_loc : forall k u x y,
      x <> y ->
      LC (T k) k u ->
      subst (T k) k x u ∘ open_loc k (mret T k (Fr y)) =
      open_loc k (mret T k (Fr y)) ⋆kdm (subst_loc k x u ∘ extract (W := prod (list K))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) (x y : atom),
x <> y ->
LC (T k) k u ->
subst (T k) k x u ∘ open_loc k (mret T k (Fr y)) =
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ extract)
  Proof with auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) (x y : atom),
x <> y ->
LC (T k) k u ->
subst (T k) k x u ∘ open_loc k (mret T k (Fr y)) =
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ extract)
    introv neq lc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
subst (T k) k x u ∘ open_loc k (mret T k (Fr y)) =
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ extract) rewrite subst_open_eq_loc...U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
open_loc k (mret T k (Fr y) '{ k | x ~> u})
⋆kdm (subst_loc k x u ∘ snd) =
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ extract)
    simpl_local.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ snd) =
open_loc k (mret T k (Fr y))
⋆kdm (subst_loc k x u ∘ extract) reflexivity.
  Qed.

  Theorem subst_open_var : forall k u t x y,
      x <> y ->
      LC (T k) k u ->
      t '(k | mret T k (Fr y)) '{k | x ~> u} =
      t '{k | x ~> u} '(k | mret T k (Fr y)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t (x y : atom),
x <> y ->
LC (T k) k u ->
(t '( k | mret T k (Fr y))) '{ k | x ~> u} =
(t '{ k | x ~> u}) '( k | mret T k (Fr y))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k (u : T k LN) t (x y : atom),
x <> y ->
LC (T k) k u ->
(t '( k | mret T k (Fr y))) '{ k | x ~> u} =
(t '{ k | x ~> u}) '( k | mret T k (Fr y))
    introv neq lc.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
(t '( k | mret T k (Fr y))) '{ k | x ~> u} =
(t '{ k | x ~> u}) '( k | mret T k (Fr y)) compose near t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
(subst U k x u ∘ open U k (mret T k (Fr y))) t =
(open U k (mret T k (Fr y)) ∘ subst U k x u) t
    unfold open, subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
(kbind U k (subst_loc k x u)
 ∘ kbindd U k (open_loc k (mret T k (Fr y)))) t =
(kbindd U k (open_loc k (mret T k (Fr y)))
 ∘ kbind U k (subst_loc k x u)) t
    rewrite (kbind_kbindd U), (kbindd_kbind U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
kbindd U k
  (kbind (T k) k (subst_loc k x u)
   ∘ open_loc k (mret T k (Fr y))) t =
kbindd U k
  (fun '(w, a) =>
   kbindd (T k) k
     (open_loc k (mret T k (Fr y)) ∘ incr w)
     (subst_loc k x u a)) t
    fequal.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
u: T k LN
t: U LN
x, y: atom
neq: x <> y
lc: LC (T k) k u
kbind (T k) k (subst_loc k x u)
∘ open_loc k (mret T k (Fr y)) =
(fun '(w, a) =>
 kbindd (T k) k
   (open_loc k (mret T k (Fr y)) ∘ incr w)
   (subst_loc k x u a)) apply subst_open_var_loc; auto.
  Qed.

  (** ** Closing, followed by opening *)
  (**************************************************************************)
  Lemma open_close_loc : forall k x w l,
      (open_loc k (mret T k (Fr x)) ∘ cobind (W := prod (list K))
                (close_loc k x)) (w, l) = mret T k l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x w l,
(open_loc k (mret T k (Fr x)) ∘ cobind (close_loc k x))
  (w, l) = mret T k l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x w l,
(open_loc k (mret T k (Fr x)) ∘ cobind (close_loc k x))
  (w, l) = mret T k l
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
(open_loc k (mret T k (Fr x)) ∘ cobind (close_loc k x))
  (w, l) = mret T k l cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
match
  match l with
  | Fr y => if x == y then Bd (countk k w) else Fr y
  | Bd n =>
      match Nat.compare n (countk k w) with
      | Lt => Bd n
      | _ => Bd (S n)
      end
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k l unfold id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
match
  match l with
  | Fr y => if x == y then Bd (countk k w) else Fr y
  | Bd n =>
      match Nat.compare n (countk k w) with
      | Lt => Bd n
      | _ => Bd (S n)
      end
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k l compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
match (if x == x then Bd (countk k w) else Fr x) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Fr x)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
a: atom
H1: a <> x
match (if x == a then Bd (countk k w) else Fr a) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Fr a)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
match
  match Nat.compare n (countk k w) with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Bd n)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
match (if x == x then Bd (countk k w) else Fr x) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Fr x) compare values x and x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
DESTR_EQ, DESTR_EQs: x = x
match Nat.compare (countk k w) (countk k w) with
| Eq => mret T k (Fr x)
| Lt => mret T k (Bd (countk k w))
| Gt => mret T k (Bd (countk k w - 1))
end = mret T k (Fr x) compare naturals (countk k w) and (countk k w).
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
a: atom
H1: a <> x
match (if x == a then Bd (countk k w) else Fr a) with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Fr a) compare values x and a.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
match
  match Nat.compare n (countk k w) with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => mret T k (Fr x)
| Bd n =>
    match Nat.compare n (countk k w) with
    | Eq => mret T k (Fr x)
    | Lt => mret T k (Bd n)
    | Gt => mret T k (Bd (n - 1))
    end
end = mret T k (Bd n) compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
match Nat.compare (S (countk k w)) (countk k w) with
| Eq => mret T k (Fr x)
| Lt => mret T k (Bd (S (countk k w)))
| Gt => mret T k (Bd (S (countk k w) - 1))
end = mret T k (Bd (countk k w))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
match Nat.compare (S n) (countk k w) with
| Eq => mret T k (Fr x)
| Lt => mret T k (Bd (S n))
| Gt => mret T k (Bd (S n - 1))
end = mret T k (Bd n)
      compare naturals (Datatypes.S (countk k w)) and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
match Nat.compare (S n) (countk k w) with
| Eq => mret T k (Fr x)
| Lt => mret T k (Bd (S n))
| Gt => mret T k (Bd (S n - 1))
end = mret T k (Bd n)
      compare naturals (Datatypes.S n) and (countk k w).
  Qed.

  Theorem open_close : forall k x t,
      ('[k | x] t) '(k | mret T k (Fr x)) = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x t,
('[ k | x] t) '( k | mret T k (Fr x)) = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x t,
('[ k | x] t) '( k | mret T k (Fr x)) = t
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
('[ k | x] t) '( k | mret T k (Fr x)) = t compose near t on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
(open U k (mret T k (Fr x)) ∘ close U k x) t = t
    unfold open, close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
(kbindd U k (open_loc k (mret T k (Fr x)))
 ∘ kmapd U k (close_loc k x)) t = t
    rewrite (kbindd_kmapd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
kbindd U k
  (fun '(w, a) =>
   open_loc k (mret T k (Fr x))
     (w, close_loc k x (w, a))) t = t
    apply (kbindd_respectful_id); intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
w: list K
a: LN
H1: (w, k, a) ∈md t
open_loc k (mret T k (Fr x)) (w, close_loc k x (w, a)) =
mret T k a
    apply open_close_loc.
  Qed.

  (** ** Opening by a LN reduces to an [kmapd] *)
  (**************************************************************************)
  Definition open_LN_loc k (u : LN) : list K * LN -> LN :=
    fun wl =>
      match wl with
      | (w, l) =>
        match l with
        | Fr x => Fr x
        | Bd n =>
          match Nat.compare n (countk k w) with
          | Gt => Bd (n - 1)
          | Eq => u
          | Lt => Bd n
          end
        end
      end.

  Lemma open_by_LN_spec : forall k l,
      open U k (mret T k l) = kmapd U k (open_LN_loc k l).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l,
open U k (mret T k l) = kmapd U k (open_LN_loc k l)
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k l,
open U k (mret T k l) = kmapd U k (open_LN_loc k l)
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
open U k (mret T k l) = kmapd U k (open_LN_loc k l) unfold open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
kbindd U k (open_loc k (mret T k l)) =
kmapd U k (open_LN_loc k l) ext t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
kbindd U k (open_loc k (mret T k l)) t =
kmapd U k (open_LN_loc k l) t
    apply (kbindd_respectful_kmapd).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
forall w (a : LN),
(w, k, a) ∈md t ->
open_loc k (mret T k l) (w, a) =
mret T k (open_LN_loc k l (w, a))
    intros w l' l'in.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
l': LN
l'in: (w, k, l') ∈md t
open_loc k (mret T k l) (w, l') =
mret T k (open_LN_loc k l (w, l')) destruct l'.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
n: atom
l'in: (w, k, Fr n) ∈md t
open_loc k (mret T k l) (w, Fr n) =
mret T k (open_LN_loc k l (w, Fr n))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
n: nat
l'in: (w, k, Bd n) ∈md t
open_loc k (mret T k l) (w, Bd n) =
mret T k (open_LN_loc k l (w, Bd n))
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
n: atom
l'in: (w, k, Fr n) ∈md t
open_loc k (mret T k l) (w, Fr n) =
mret T k (open_LN_loc k l (w, Fr n)) reflexivity.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
n: nat
l'in: (w, k, Bd n) ∈md t
open_loc k (mret T k l) (w, Bd n) =
mret T k (open_LN_loc k l (w, Bd n)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
l: LN
t: U LN
w: list K
n: nat
l'in: (w, k, Bd n) ∈md t
match Nat.compare n (countk k w) with
| Eq => mret T k l
| Lt => mret T k (Bd n)
| Gt => mret T k (Bd (n - 1))
end =
mret T k
  match Nat.compare n (countk k w) with
  | Eq => l
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end compare naturals n and (countk k w).
  Qed.

  (** ** Opening, followed by closing *)
  (**************************************************************************)
  Lemma close_open_local : forall k x w l,
      l <> Fr x ->
      (close_loc k x ∘ cobind (W := prod (list K))
                 (open_LN_loc k (Fr x))) (w, l) = l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x w l,
l <> Fr x ->
(close_loc k x ∘ cobind (open_LN_loc k (Fr x))) (w, l) =
l
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x w l,
l <> Fr x ->
(close_loc k x ∘ cobind (open_LN_loc k (Fr x))) (w, l) =
l
    introv neq.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
neq: l <> Fr x
(close_loc k x ∘ cobind (open_LN_loc k (Fr x))) (w, l) =
l cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
neq: l <> Fr x
match
  match l with
  | Fr x => Fr x
  | Bd n =>
      match Nat.compare n (countk k w) with
      | Eq => Fr x
      | Lt => Bd n
      | Gt => Bd (n - 1)
      end
  end
with
| Fr y => if x == y then Bd (countk k w) else Fr y
| Bd n =>
    match Nat.compare n (countk k w) with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = l unfold id.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
l: LN
neq: l <> Fr x
match
  match l with
  | Fr x => Fr x
  | Bd n =>
      match Nat.compare n (countk k w) with
      | Eq => Fr x
      | Lt => Bd n
      | Gt => Bd (n - 1)
      end
  end
with
| Fr y => if x == y then Bd (countk k w) else Fr y
| Bd n =>
    match Nat.compare n (countk k w) with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = l compare l to atom x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
neq: Fr x <> Fr x
(if x == x then Bd (countk k w) else Fr x) = Fr x
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
(if x == a then Bd (countk k w) else Fr a) = Fr a
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
match
  match Nat.compare n (countk k w) with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd (countk k w) else Fr y
| Bd n =>
    match Nat.compare n (countk k w) with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
neq: Fr x <> Fr x
(if x == x then Bd (countk k w) else Fr x) = Fr x contradiction.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
(if x == a then Bd (countk k w) else Fr a) = Fr a unfold compose; cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
a: atom
neq: Fr a <> Fr x
H1: a <> x
(if x == a then Bd (countk k w) else Fr a) = Fr a now compare values x and a.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
match
  match Nat.compare n (countk k w) with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd (countk k w) else Fr y
| Bd n =>
    match Nat.compare n (countk k w) with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n unfold compose; cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
match
  match Nat.compare n (countk k w) with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd (countk k w) else Fr y
| Bd n =>
    match Nat.compare n (countk k w) with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n
      compare naturals n and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
neq: Bd (countk k w) <> Fr x
(if x == x then Bd (countk k w) else Fr x) =
Bd (countk k w)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
match Nat.compare (n - 1) (countk k w) with
| Lt => Bd (n - 1)
| _ => Bd (S (n - 1))
end = Bd n compare values x and x.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
w: list K
n: nat
neq: Bd n <> Fr x
ineqrw: PeanoNat.Nat.compare n (countk k w) = Gt
ineqp: n > countk k w
match Nat.compare (n - 1) (countk k w) with
| Lt => Bd (n - 1)
| _ => Bd (S (n - 1))
end = Bd n
      compare naturals (n - 1) and (countk k w).
  Qed.

  Theorem close_open : forall k x t,
      ~ x ∈ free U k t ->
      '[k | x] (t '(k | mret T k (Fr x))) = t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x t,
~ x ∈ free U k t ->
'[ k | x] (t '( k | mret T k (Fr x))) = t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k x t,
~ x ∈ free U k t ->
'[ k | x] (t '( k | mret T k (Fr x))) = t
    introv fresh.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
'[ k | x] (t '( k | mret T k (Fr x))) = t compose near t on left.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
(close U k x ∘ open U k (mret T k (Fr x))) t = t
    rewrite open_by_LN_spec.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
(close U k x ∘ kmapd U k (open_LN_loc k (Fr x))) t = t unfold close.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
(kmapd U k (close_loc k x)
 ∘ kmapd U k (open_LN_loc k (Fr x))) t = t
    rewrite (kmapd_kmapd U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
kmapd U k
  (fun '(w, a) =>
   close_loc k x (w, open_LN_loc k (Fr x) (w, a))) t =
t
    apply (kmapd_respectful_id).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
forall w (a : LN),
(w, k, a) ∈md t ->
close_loc k x (w, open_LN_loc k (Fr x) (w, a)) = a
    intros w l lin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
close_loc k x (w, open_LN_loc k (Fr x) (w, l)) = l
    assert (l <> Fr x).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
l <> Fr x
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
H1: l <> Fr x
close_loc k x (w, open_LN_loc k (Fr x) (w, l)) = l
    {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
l <> Fr x rewrite neq_symmetry.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
Fr x <> l apply (inmd_implies_in) in lin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (k, l) ∈m t
Fr x <> l
      eapply (ninf_in_neq U).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (k, l) ∈m t
~ x ∈ free U ?k ?t
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (k, l) ∈m t
(?k, l) ∈m ?t eassumption.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (k, l) ∈m t
(k, l) ∈m t auto. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
x: atom
t: U LN
fresh: ~ x ∈ free U k t
w: list K
l: LN
lin: (w, k, l) ∈md t
H1: l <> Fr x
close_loc k x (w, open_LN_loc k (Fr x) (w, l)) = l
    now apply close_open_local.
  Qed.

  (** ** Opening and local closure *)
  (**************************************************************************)
  Lemma open_lc_gap_eq_1 : forall k n t u,
      LC (T k) k u ->
      LCn U k n t ->
      LCn U k (n - 1) (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
LC (T k) k u ->
LCn U k n t -> LCn U k (n - 1) (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
LC (T k) k u ->
LCn U k n t -> LCn U k (n - 1) (t '( k | u))
    unfold LCn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
LC (T k) k u ->
(forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)) ->
forall w l,
(w, k, l) ∈md t '( k | u) -> lc_loc k (n - 1) (w, l)
    introv lcu lct Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t '( k | u)
lc_loc k (n - 1) (w, l) rewrite (inmd_open_iff_eq U) in Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, l) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2
lc_loc k (n - 1) (w, l)
    destruct Hin as [w1 [w2 [l1 [h1 [h2 h3]]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
l1: LN
h1: (w1, k, l1) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, l1)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l)
    destruct l1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, Fr n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: nat
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, Bd n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, Fr n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l) cbn in h2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
h2: (w2, k, l) ∈md mret T k (Fr n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l) rewrite inmd_mret_eq_iff in h2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
h2: w2 = Ƶ /\ Fr n0 = l
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l)
      destruct h2; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
lc_loc k (n - 1) (w1 ● Ƶ, Fr n0) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
n0: atom
h1: (w1, k, Fr n0) ∈md t
True trivial.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
n0: nat
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, Bd n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l) specialize (lct _ _ h1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md open_loc k u (w1, Bd n0)
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l) cbn in h2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l)
∈md match Nat.compare n0 (countk k w1) with
    | Eq => u
    | Lt => mret T k (Bd n0)
    | Gt => mret T k (Bd (n0 - 1))
    end
h3: w = w1 ● w2
lc_loc k (n - 1) (w, l) compare naturals n0 and (countk k w1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md mret T k (Bd n0)
h3: w = w1 ● w2
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
lc_loc k (n - 1) (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
lct: lc_loc k n (w1, Bd (countk k w1))
l: LN
w2: list K
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, l) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
lc_loc k (n - 1) (w1 ● w2, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md mret T k (Bd (n0 - 1))
h3: w = w1 ● w2
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
lc_loc k (n - 1) (w, l)
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md mret T k (Bd n0)
h3: w = w1 ● w2
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
lc_loc k (n - 1) (w, l) rewrite inmd_mret_eq_iff in h2; destruct h2; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
lc_loc k (n - 1) (w1 ● Ƶ, Bd n0)
        cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
n0 < countk k (w1 ● Ƶ) + (n - 1) unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
n0 < countk k (w1 ++ []) + (n - 1) List.simpl_list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Lt
ineqp: n0 < countk k w1
n0 < countk k w1 + (n - 1) lia.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
lct: lc_loc k n (w1, Bd (countk k w1))
l: LN
w2: list K
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, l) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
lc_loc k (n - 1) (w1 ● w2, l) specialize (lcu w2 l h2).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
l: LN
w2: list K
lcu: lc_loc k 0 (w2, l)
w1: list K
lct: lc_loc k n (w1, Bd (countk k w1))
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, l) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
lc_loc k (n - 1) (w1 ● w2, l) unfold lc_loc in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
l: LN
w2: list K
lcu: match l with
| Fr _ => True
| Bd n => n < countk k w2 + 0
end
w1: list K
lct: countk k w1 < countk k w1 + n
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, l) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
match l with
| Fr _ => True
| Bd n0 => n0 < countk k (w1 ● w2) + (n - 1)
end
        destruct l; [trivial|].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
n0: nat
w2: list K
lcu: n0 < countk k w2 + 0
w1: list K
lct: countk k w1 < countk k w1 + n
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, Bd n0) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
n0 < countk k (w1 ● w2) + (n - 1) unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
n0: nat
w2: list K
lcu: n0 < countk k w2 + 0
w1: list K
lct: countk k w1 < countk k w1 + n
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, Bd n0) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
n0 < countk k (w1 ++ w2) + (n - 1) rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
n0: nat
w2: list K
lcu: n0 < countk k w2 + 0
w1: list K
lct: countk k w1 < countk k w1 + n
h1: (w1, k, Bd (countk k w1)) ∈md t
h2: (w2, k, Bd n0) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk k w1) 
  (countk k w1) = Eq
n0 < countk k w1 + countk k w2 + (n - 1) lia.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: (w2, k, l) ∈md mret T k (Bd (n0 - 1))
h3: w = w1 ● w2
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
lc_loc k (n - 1) (w, l) rewrite inmd_mret_eq_iff in h2.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
w: list K
l: LN
w2: list K
h1: (w1, k, Bd n0) ∈md t
h2: w2 = Ƶ /\ Bd (n0 - 1) = l
h3: w = w1 ● w2
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
lc_loc k (n - 1) (w, l) destruct h2; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: lc_loc k n (w1, Bd n0)
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
lc_loc k (n - 1) (w1 ● Ƶ, Bd (n0 - 1))
        unfold lc_loc in *.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: n0 < countk k w1 + n
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
n0 - 1 < countk k (w1 ● Ƶ) + (n - 1) unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: n0 < countk k w1 + n
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
n0 - 1 < countk k (w1 ++ []) + (n - 1) List.simpl_list.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
lcu: LC (T k) k u
w1: list K
n0: nat
lct: n0 < countk k w1 + n
h1: (w1, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w1) = Gt
ineqp: n0 > countk k w1
n0 - 1 < countk k w1 + (n - 1) lia.
  Qed.

  Lemma open_lc_gap_eq_2 : forall k n t u,
      n > 0 ->
      LC (T k) k u ->
      LCn U k (n - 1) (t '(k | u)) ->
      LCn U k n t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
n > 0 ->
LC (T k) k u ->
LCn U k (n - 1) (t '( k | u)) -> LCn U k n t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
n > 0 ->
LC (T k) k u ->
LCn U k (n - 1) (t '( k | u)) -> LCn U k n t
    unfold LCn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
n > 0 ->
LC (T k) k u ->
(forall w l,
 (w, k, l) ∈md t '( k | u) -> lc_loc k (n - 1) (w, l)) ->
forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
    introv ngt lcu lct Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(w, k, l) ∈md t '( k | u) ->
lc_loc k (n - 1) (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t
lc_loc k n (w, l) setoid_rewrite (inmd_open_iff_eq U) in lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t
lc_loc k n (w, l)
    destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
lc_loc k n (w, Fr n0)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
lc_loc k n (w, Bd n0)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
lc_loc k n (w, Fr n0) cbv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
True trivial.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
lc_loc k n (w, Bd n0) compare naturals n0 and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
lc_loc k n (w, Bd n0)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
Hin: (w, k, Bd (countk k w)) ∈md t
lc_loc k n (w, Bd (countk k w))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
lc_loc k n (w, Bd n0)
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
lc_loc k n (w, Bd n0) specialize (lct w (Bd n0)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
lc_loc k n (w, Bd n0)
        lapply lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
lc_loc k (n - 1) (w, Bd n0) -> lc_loc k n (w, Bd n0)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
lc_loc k (n - 1) (w, Bd n0) -> lc_loc k n (w, Bd n0) cbn; unfold_monoid; lia. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2 exists w (Ƶ : list K) (Bd n0).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
(w, k, Bd n0) ∈md t /\
(Ƶ, k, Bd n0) ∈md open_loc k u (w, Bd n0) /\ w = w ● Ƶ
          split; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
(Ƶ, k, Bd n0) ∈md open_loc k u (w, Bd n0) /\ w = w ● Ƶ cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp: n0 < countk k w
(Ƶ, k, Bd n0)
∈md match Nat.compare n0 (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n0)
    | Gt => mret T k (Bd (n0 - 1))
    end /\ w = w ● Ƶ compare naturals n0 and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: Lt = Lt
ineqp: n0 < countk k w
ineqrw0: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp0: n0 < countk k w
(Ƶ, k, Bd n0) ∈md mret T k (Bd n0) /\ w = w ● Ƶ
          rewrite inmd_mret_eq_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd n0) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
ineqrw: Lt = Lt
ineqp: n0 < countk k w
ineqrw0: PeanoNat.Nat.compare n0 (countk k w) = Lt
ineqp0: n0 < countk k w
(Ƶ = Ƶ /\ Bd n0 = Bd n0) /\ w = w ● Ƶ now simpl_monoid. }
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
Hin: (w, k, Bd (countk k w)) ∈md t
lc_loc k n (w, Bd (countk k w)) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
ineqrw: PeanoNat.Nat.compare 
  (countk k w) 
  (countk k w) = Eq
Hin: (w, k, Bd (countk k w)) ∈md t
countk k w < countk k w + n unfold_monoid; lia.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
lc_loc k n (w, Bd n0) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
lct: forall w l,
(exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, l) ∈md open_loc k u (w1, l1) /\
   w = w1 ● w2) -> 
lc_loc k (n - 1) (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
n0 < countk k w + n specialize (lct w (Bd (n0 - 1))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
n0 < countk k w + n
        lapply lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
lc_loc k (n - 1) (w, Bd (n0 - 1)) ->
n0 < countk k w + n
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd (n0 - 1)) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
lc_loc k (n - 1) (w, Bd (n0 - 1)) ->
n0 < countk k w + n cbn; unfold_monoid; lia. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd (n0 - 1)) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
exists w1 w2 l1,
  (w1, k, l1) ∈md t /\
  (w2, k, Bd (n0 - 1)) ∈md open_loc k u (w1, l1) /\
  w = w1 ● w2 exists w (Ƶ : list K) (Bd n0).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
(w, k, Bd n0) ∈md t /\
(Ƶ, k, Bd (n0 - 1)) ∈md open_loc k u (w, Bd n0) /\
w = w ● Ƶ
          split; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
(Ƶ, k, Bd (n0 - 1)) ∈md open_loc k u (w, Bd n0) /\
w = w ● Ƶ cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp: n0 > countk k w
(Ƶ, k, Bd (n0 - 1))
∈md match Nat.compare n0 (countk k w) with
    | Eq => u
    | Lt => mret T k (Bd n0)
    | Gt => mret T k (Bd (n0 - 1))
    end /\ w = w ● Ƶ compare naturals n0 and (countk k w).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: Gt = Gt
ineqp: n0 > countk k w
ineqrw0: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp0: n0 > countk k w
(Ƶ, k, Bd (n0 - 1)) ∈md mret T k (Bd (n0 - 1)) /\
w = w ● Ƶ
          rewrite inmd_mret_eq_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
u: T k LN
ngt: n > 0
lcu: LC (T k) k u
w: list K
n0: nat
lct: (exists w1 w2 l1,
   (w1, k, l1) ∈md t /\
   (w2, k, Bd (n0 - 1))
   ∈md open_loc k u (w1, l1) /\ 
   w = w1 ● w2) ->
lc_loc k (n - 1) (w, Bd (n0 - 1))
Hin: (w, k, Bd n0) ∈md t
ineqrw: Gt = Gt
ineqp: n0 > countk k w
ineqrw0: PeanoNat.Nat.compare n0 (countk k w) = Gt
ineqp0: n0 > countk k w
(Ƶ = Ƶ /\ Bd (n0 - 1) = Bd (n0 - 1)) /\ w = w ● Ƶ now simpl_monoid. }
  Qed.

  Theorem open_lc_gap_eq_iff : forall k n t u,
      n > 0 ->
      LC (T k) k u ->
      LCn U k n t <->
      LCn U k (n - 1) (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
n > 0 ->
LC (T k) k u ->
LCn U k n t <-> LCn U k (n - 1) (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t (u : T k LN),
n > 0 ->
LC (T k) k u ->
LCn U k n t <-> LCn U k (n - 1) (t '( k | u))
    intros; intuition (eauto using open_lc_gap_eq_1, open_lc_gap_eq_2).
  Qed.

  Corollary open_lc_gap_eq_var : forall k n t x,
      n > 0 ->
      LCn U k n t <->
      LCn U k (n - 1) (t '(k | mret T k (Fr x))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t x,
n > 0 ->
LCn U k n t <->
LCn U k (n - 1) (t '( k | mret T k (Fr x)))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k n t x,
n > 0 ->
LCn U k n t <->
LCn U k (n - 1) (t '( k | mret T k (Fr x)))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
LCn U k n t <->
LCn U k (n - 1) (t '( k | mret T k (Fr x))) apply open_lc_gap_eq_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
n > 0
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
LC (T k) k (mret T k (Fr x)) auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
LC (T k) k (mret T k (Fr x))
    intros w l hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
w: list K
l: LN
hin: (w, k, l) ∈md mret T k (Fr x)
lc_loc k 0 (w, l) rewrite inmd_mret_eq_iff in hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
w: list K
l: LN
hin: w = Ƶ /\ Fr x = l
lc_loc k 0 (w, l)
    destruct hin; subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
lc_loc k 0 (Ƶ, Fr x) cbv.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
n: nat
t: U LN
x: atom
H1: n > 0
True trivial.
  Qed.

  Corollary open_lc_gap_eq_iff_1 : forall k t u,
      LC (T k) k u ->
      LCn U k 1 t <->
      LC U k (t '(k | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN),
LC (T k) k u -> LCn U k 1 t <-> LC U k (t '( k | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t (u : T k LN),
LC (T k) k u -> LCn U k 1 t <-> LC U k (t '( k | u))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: LC (T k) k u
LCn U k 1 t <-> LC U k (t '( k | u)) unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: LC (T k) k u
LCn U k 1 t <-> LCn U k 0 (t '( k | u))
    change 0 with (1 - 1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: LC (T k) k u
LCn U k (S (1 - 1)) t <->
LCn U k (1 - 1) (t '( k | u))
    rewrite <- open_lc_gap_eq_iff; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
u: T k LN
H1: LC (T k) k u
LCn U k (S (1 - 1)) t <-> LCn U k 1 t
    reflexivity.
  Qed.

  Corollary open_lc_gap_eq_var_1 : forall k t x,
      LCn U k 1 t <->
      LC U k (t '(k | mret T k (Fr x))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x,
LCn U k 1 t <-> LC U k (t '( k | mret T k (Fr x)))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k t x,
LCn U k 1 t <-> LC U k (t '( k | mret T k (Fr x)))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
LCn U k 1 t <-> LC U k (t '( k | mret T k (Fr x))) unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
LCn U k 1 t <-> LCn U k 0 (t '( k | mret T k (Fr x)))
    change 0 with (1 - 1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
LCn U k (S (1 - 1)) t <->
LCn U k (1 - 1) (t '( k | mret T k (Fr x)))
    rewrite <- open_lc_gap_eq_var; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k: K
t: U LN
x: atom
LCn U k (S (1 - 1)) t <-> LCn U k 1 t
    reflexivity.
  Qed.

  Lemma open_lc_gap_neq_1 : forall k j n t u,
      k <> j ->
      LC (T j) k u ->
      LCn U k n t ->
      LCn U k n (t '(j | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u -> LCn U k n t -> LCn U k n (t '( j | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u -> LCn U k n t -> LCn U k n (t '( j | u))
    unfold LCn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u ->
(forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)) ->
forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
    introv neq lcu lct Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t '( j | u)
lc_loc k n (w, l)
    rewrite (inmd_open_neq_iff U) in Hin; auto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t \/
(exists w1 w2 l1,
   (w1, j, l1) ∈md t /\
   (w2, k, l) ∈md open_loc j u (w1, l1) /\
   w = w1 ● w2)
lc_loc k n (w, l)
    destruct Hin as [Hin | Hin].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t
lc_loc k n (w, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (w2, k, l) ∈md open_loc j u (w1, l1) /\
  w = w1 ● w2
lc_loc k n (w, l)
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t
lc_loc k n (w, l) auto.
    -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
Hin: exists w1 w2 l1,
  (w1, j, l1) ∈md t /\
  (w2, k, l) ∈md open_loc j u (w1, l1) /\
  w = w1 ● w2
lc_loc k n (w, l) destruct Hin as [w1 [w2 [l1 [in_t [in_open ?]]]]].U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w: list K
l: LN
w1, w2: list K
l1: LN
in_t: (w1, j, l1) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, l1)
H1: w = w1 ● w2
lc_loc k n (w, l)
      subst.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
l1: LN
in_t: (w1, j, l1) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, l1)
lc_loc k n (w1 ● w2, l) destruct l1.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: atom
in_t: (w1, j, Fr n0) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, Fr n0)
lc_loc k n (w1 ● w2, l)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, l)
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: atom
in_t: (w1, j, Fr n0) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, Fr n0)
lc_loc k n (w1 ● w2, l) cbn in in_open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: atom
in_t: (w1, j, Fr n0) ∈md t
in_open: (w2, k, l) ∈md mret T j (Fr n0)
lc_loc k n (w1 ● w2, l) rewrite inmd_mret_iff in in_open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: atom
in_t: (w1, j, Fr n0) ∈md t
in_open: w2 = Ƶ /\ j = k /\ Fr n0 = l
lc_loc k n (w1 ● w2, l)
        false.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: atom
in_t: (w1, j, Fr n0) ∈md t
in_open: w2 = Ƶ /\ j = k /\ Fr n0 = l
False intuition.
      +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
l: LN
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, l) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, l) destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: atom
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Fr n1) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, Fr n1)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, Bd n1)
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: atom
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Fr n1) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, Fr n1) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: atom
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Fr n1) ∈md open_loc j u (w1, Bd n0)
True trivial. }U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, Bd n1)
        {U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md open_loc j u (w1, Bd n0)
lc_loc k n (w1 ● w2, Bd n1) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md open_loc j u (w1, Bd n0)
n1 < countk k (w1 ● w2) + n cbn in in_open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1)
∈md match Nat.compare n0 (countk j w1) with
    | Eq => u
    | Lt => mret T j (Bd n0)
    | Gt => mret T j (Bd (n0 - 1))
    end
n1 < countk k (w1 ● w2) + n
          compare naturals n0 and (countk j w1).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md mret T j (Bd n0)
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Lt
ineqp: n0 < countk j w1
n1 < countk k (w1 ● w2) + n
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k (w1 ● w2) + n
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md mret T j (Bd (n0 - 1))
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Gt
ineqp: n0 > countk j w1
n1 < countk k (w1 ● w2) + n
          -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md mret T j (Bd n0)
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Lt
ineqp: n0 < countk j w1
n1 < countk k (w1 ● w2) + n rewrite inmd_mret_iff in in_open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: w2 = Ƶ /\ j = k /\ Bd n0 = Bd n1
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Lt
ineqp: n0 < countk j w1
n1 < countk k (w1 ● w2) + n false; intuition.
          -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k (w1 ● w2) + n specialize (lcu _ _ in_open).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
n1: nat
w2: list K
lcu: lc_loc k 0 (w2, Bd n1)
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k (w1 ● w2) + n
            unfold lc_loc in lcu.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
n1: nat
w2: list K
lcu: n1 < countk k w2 + 0
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k (w1 ● w2) + n unfold_monoid.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
n1: nat
w2: list K
lcu: n1 < countk k w2 + 0
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k (w1 ++ w2) + n
            rewrite countk_app.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
n1: nat
w2: list K
lcu: n1 < countk k w2 + 0
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
w1: list K
in_t: (w1, j, Bd (countk j w1)) ∈md t
in_open: (w2, k, Bd n1) ∈md u
ineqrw: PeanoNat.Nat.compare 
  (countk j w1) 
  (countk j w1) = Eq
n1 < countk k w1 + countk k w2 + n lia.
          -U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: (w2, k, Bd n1) ∈md mret T j (Bd (n0 - 1))
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Gt
ineqp: n0 > countk j w1
n1 < countk k (w1 ● w2) + n rewrite inmd_mret_iff in in_open.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
n1: nat
w1, w2: list K
n0: nat
in_t: (w1, j, Bd n0) ∈md t
in_open: w2 = Ƶ /\ j = k /\ Bd (n0 - 1) = Bd n1
ineqrw: PeanoNat.Nat.compare n0 (countk j w1) = Gt
ineqp: n0 > countk j w1
n1 < countk k (w1 ● w2) + n false; intuition.
        }
  Qed.

  Lemma open_lc_gap_neq_2 : forall k j n t u,
      k <> j ->
      LC (T j) k u ->
      LCn U k n (t '(j | u)) ->
      LCn U k n t.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u -> LCn U k n (t '( j | u)) -> LCn U k n t
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u -> LCn U k n (t '( j | u)) -> LCn U k n t
    unfold LCn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u ->
(forall w l,
 (w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)) ->
forall w l, (w, k, l) ∈md t -> lc_loc k n (w, l)
    introv neq lcu lct Hin.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
l: LN
Hin: (w, k, l) ∈md t
lc_loc k n (w, l)
    destruct l.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
lc_loc k n (w, Fr n0)
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
lc_loc k n (w, Bd n0)
    +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
lc_loc k n (w, Fr n0) cbn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
n0: atom
Hin: (w, k, Fr n0) ∈md t
True auto.
    +U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
lct: forall w l,
(w, k, l) ∈md t '( j | u) -> lc_loc k n (w, l)
w: list K
n0: nat
Hin: (w, k, Bd n0) ∈md t
lc_loc k n (w, Bd n0) specialize (lct w (Bd n0)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
w: list K
n0: nat
lct: (w, k, Bd n0) ∈md t '( j | u) ->
lc_loc k n (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
lc_loc k n (w, Bd n0)
      apply lct.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
neq: k <> j
lcu: LC (T j) k u
w: list K
n0: nat
lct: (w, k, Bd n0) ∈md t '( j | u) ->
lc_loc k n (w, Bd n0)
Hin: (w, k, Bd n0) ∈md t
(w, k, Bd n0) ∈md t '( j | u) rewrite (inmd_open_neq_iff U); auto.
  Qed.

  Theorem open_lc_gap_neq_iff : forall k j n t u,
      k <> j ->
      LC (T j) k u ->
      LCn U k n t <->
      LCn U k n (t '(j | u)).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u ->
LCn U k n t <-> LCn U k n (t '( j | u))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j n t (u : T j LN),
k <> j ->
LC (T j) k u ->
LCn U k n t <-> LCn U k n (t '( j | u))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
n: nat
t: U LN
u: T j LN
H1: k <> j
H2: LC (T j) k u
LCn U k n t <-> LCn U k n (t '( j | u)) intuition (eauto using open_lc_gap_neq_1, open_lc_gap_neq_2).
  Qed.

  Corollary open_lc_gap_neq_var : forall k j t x,
      k <> j ->
      LC U k t <->
      LC U k (t '(j | mret T j (Fr x))).U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t x,
k <> j ->
LC U k t <-> LC U k (t '( j | mret T j (Fr x)))
  Proof.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
forall k j t x,
k <> j ->
LC U k t <-> LC U k (t '( j | mret T j (Fr x)))
    intros.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
LC U k t <-> LC U k (t '( j | mret T j (Fr x))) unfold LC.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
LCn U k 0 t <-> LCn U k 0 (t '( j | mret T j (Fr x)))
    rewrite open_lc_gap_neq_iff; eauto.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
LCn U k 0 (t '( j | ?Goal0)) <->
LCn U k 0 (t '( j | mret T j (Fr x)))
U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
LC (T j) k ?Goal0
    reflexivity.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
LC (T j) k (mret T j (Fr x)) unfold LC, LCn.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
forall w l,
(w, k, l) ∈md mret T j (Fr x) -> lc_loc k 0 (w, l)
    setoid_rewrite inmd_mret_iff.U: Type -> Type
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind (list K) T U
H: forall k, MBind (list K) T (T k)
H0: MultiDecoratedTraversablePreModule (list K) T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  (list K) T
k, j: K
t: U LN
x: atom
H1: k <> j
forall w l,
w = Ƶ /\ j = k /\ Fr x = l -> lc_loc k 0 (w, l) intuition.
  Qed.

End open_metatheory.