(** This files contains metatheorems for the locally nameless variables
 that closely parallel those of LNgen. *)
From Tealeaves.Backends.Common Require Export
  Names AtomSet.
From Tealeaves.Misc Require Export
  NaturalNumbers.
From Tealeaves.Classes Require Import
  Kleisli.Theory.DecoratedTraversableMonad.
From Tealeaves.Theory Require Import
  DecoratedTraversableMonad.

Import Coq.Init.Nat. (* Notation <? *)

Import
  List.ListNotations
  Product.Notations
  Monoid.Notations
  Batch.Notations
  Subset.Notations
  Kleisli.Monad.Notations
  Kleisli.Comonad.Notations
  ContainerFunctor.Notations
  DecoratedMonad.Notations
  DecoratedContainerFunctor.Notations
  DecoratedTraversableMonad.Notations
  AtomSet.Notations.

#[local] Generalizable Variables T U.

(** * Deriving All Other Typeclass Instances from a DTM *)
(******************************************************************************)
Import UsefulInstances.

(** * 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.

(** [compare_to_atom] is an induction principle for leaves that splits
      the <<Fr x>> case into subcases <<x = y>> and <<x <> y>> for
      some user-specified atom <<y>>. *)
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)).

(** * Locally nameless operations *)
(******************************************************************************)

(** ** Localized operations *)
(******************************************************************************)
Section locally_nameless_local_operations.

  Context
    `{Return T}.

  #[local] Arguments ret (T)%function_scope {Return} {A}%type_scope _.

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

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

  Definition open_loc (u: T LN): nat * LN -> T LN :=
    fun p => match p with
          | (w, l) =>
              match l with
              | Fr x => ret T (Fr x)
              | Bd n =>
                  match Nat.compare n w with
                  | Gt => ret T (Bd (n - 1))
                  | Eq => u
                  | Lt => ret T (Bd n)
                  end
              end
          end.

  Definition is_opened: nat * LN -> Prop :=
    fun p =>
      match p with
      | (ctx, l) =>
          match l with
          | Fr y => False
          | Bd n => n = ctx
          end
      end.

  Definition close_loc (x: atom ): nat * LN -> LN :=
    fun p => match p with
          | (w, l) =>
              match l with
              | Fr y => if x == y then Bd w else Fr y
              | Bd n =>
                  match n ?= w with
                  | Gt => Bd (S n)
                  | Eq => Bd (S n)
                  | Lt => Bd n
                  end
              end
          end.

  (** The argument <<n>> is appended the context---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 (gap: nat): nat * LN -> Prop :=
    fun p => match p with
          | (w, l) =>
              match l with
              | Fr x => True
              | Bd n => n < w + gap
              end
          end.

  Definition lcb_loc (gap: nat): nat * LN -> bool :=
    fun p => match p with
          | (w, l) =>
              match l with
              | Fr x => true
              | Bd n => n <? w + gap
              end
          end.

  Definition level_loc: nat * LN -> nat :=
    fun p => match p with
          | (w, l) =>
            match l with
            | Fr x => 0
            | Bd n => n + 1 - w
            end
          end.

End locally_nameless_local_operations.

(** ** Lifted Operations *)
(******************************************************************************)
Section locally_nameless_operations.

  Context
    `{Return_T: Return T}
    `{Map_U: Map U}
    `{Bind_TU: Bind T U}
    `{Traverse_U: Traverse U}
    `{Mapd_U: Mapd nat U}
    `{Bindt_TU: Bindt T U}
    `{Bindd_U: Bindd nat T U}
    `{Mapdt_U: Mapdt nat U}
    `{Binddt_TU: Binddt nat T U}.

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

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

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

  Definition free: U LN -> list atom :=
    mapReduce free_loc.

  Definition FV: U LN -> AtomSet.t :=
    AtomSet.atoms ○ free.

  Definition LCnb (gap: nat): U LN -> bool :=
    mapdReduce (lcb_loc gap).

  Definition LCb: U LN -> bool :=
    LCnb 0.

  Definition LCn (gap: nat): U LN -> Prop :=
    Forall_ctx (lc_loc gap).

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

 Definition level: U LN -> nat :=
    mapdReduce (op := Monoid_op_max) (unit := Monoid_unit_max)
      level_loc.

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

End locally_nameless_operations.

(** ** Operation Notations *)
(******************************************************************************)
Module OpNotations.
  Notation "t '{ x ~> u }" := (subst x u t) (at level 35).
  Notation "t ' ( u )" := (open u t) (at level 35, format "t  ' ( u )" ).
  Notation "' [ x ] t" := (close x t) (at level 35, format "' [ x ]  t" ).
End OpNotations.

Import OpNotations.

Section test_notations.

  Context
    `{DecoratedTraversableMonad nat T}.

  Context
    (t: T LN)
    (u: T LN)
    (x: atom)
    (n: nat).

  Check u '{x ~> t}.
u '{ x ~> t}
     : T LN
  Check u '(t).
u '(t)
     : T LN
  Check '[x] u.
'[x] u
     : T LN

End test_notations.

(** * Some Tactics *)
(******************************************************************************)
Create HintDb tea_local.

Tactic Notation "unfold_monoid" :=
  repeat unfold monoid_op, Monoid_op_plus,
  monoid_unit, Monoid_unit_zero in *.

Tactic Notation "unfold_lia" :=
  unfold_monoid; lia.

(** * Basic principles and lemmas *)
(******************************************************************************)
Section locally_nameless_basic_principles.

  Import Notations.

  (*
  Context
    `{Return_T: Return T}
    `{Binddt_TT: Binddt nat T T}
    `{Binddt_TU: Binddt nat T U}
    `{Monad_inst: ! DecoratedTraversableMonad nat T}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
                        (unit := Monoid_unit_zero)
                        (op := Monoid_op_plus)}.
   *)

  Context
    `{Return_T: Return T}
    `{Map_T: Map T}
    `{Bind_TT: Bind T T}
    `{Traverse_T: Traverse T}
    `{Mapd_T: Mapd nat T}
    `{Bindt_TT: Bindt T T}
    `{Bindd_T: Bindd nat T}
    `{Mapdt_T: Mapdt nat T}
    `{Binddt_TT: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}.

  Context
    `{Map_U: Map U}
    `{Bind_TU: Bind T U}
    `{Traverse_U: Traverse U}
    `{Mapd_U: Mapd nat U}
    `{Bindt_TU: Bindt T U}
    `{Bindd_TU: Bindd nat T U}
    `{Mapdt_U: Mapdt nat U}
    `{Binddt_TU: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad nat T}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
                        (unit := Monoid_unit_zero)
                        (op := Monoid_op_plus)}.

  Implicit Types (l: LN) (n: nat) (t: U LN) (x: atom).

  (** ** Local closure spec *)
  (******************************************************************************)
  Lemma LCn_spec: forall gap,
      LCn (U := U) gap = fun t => forall w l, (w, l) ∈d t -> lc_loc gap (w, l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall gap : nat,
LCn gap =
(fun t =>
 forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall gap : nat,
LCn gap =
(fun t =>
 forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
    intro.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
LCn gap =
(fun t =>
 forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l)) ext t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn gap t =
(forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
    apply propositional_extensionality.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn gap t <->
(forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
    unfold LCn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
Forall_ctx (lc_loc gap) t <->
(forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
    rewrite forall_ctx_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
(forall (e : nat) (a : LN),
 (e, a) ∈d t -> lc_loc gap (e, a)) <->
(forall (w : nat) l, (w, l) ∈d t -> lc_loc gap (w, l))
    reflexivity.
  Qed.

  Definition LC_spec:
    LC (U := U) = fun t => forall w l, (w, l) ∈d t -> lc_loc 0 (w, l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
LC =
(fun t =>
 forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
LC =
(fun t =>
 forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l))
    ext t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
LC t =
(forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l))
    unfold LC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
LCn 0 t =
(forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l))
    rewrite LCn_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
(forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l)) =
(forall (w : nat) l, (w, l) ∈d t -> lc_loc 0 (w, l))
    reflexivity.
  Qed.

  Lemma lc_loc_mono: forall gap gap' w l,
      gap <= gap' ->
      lc_loc gap (w, l) ->
      lc_loc gap' (w, l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' w : nat) l,
gap <= gap' -> lc_loc gap (w, l) -> lc_loc gap' (w, l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' w : nat) l,
gap <= gap' -> lc_loc gap (w, l) -> lc_loc gap' (w, l)
    unfold lc_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' w : nat) l,
gap <= gap' ->
match l with
| Fr _ => True
| Bd n => n < w + gap
end ->
match l with
| Fr _ => True
| Bd n => n < w + gap'
end
    unfold transparent tcs.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' w : nat) l,
gap <= gap' ->
match l with
| Fr _ => True
| Bd n => n < w + gap
end ->
match l with
| Fr _ => True
| Bd n => n < w + gap'
end
    destruct l; lia.
  Qed.

  Theorem local_closure_mono: forall gap gap' t,
      gap <= gap' ->
      LCn gap t ->
      LCn gap' t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' : nat) t,
gap <= gap' -> LCn gap t -> LCn gap' t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap gap' : nat) t,
gap <= gap' -> LCn gap t -> LCn gap' t
    introv Hineq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, gap': nat
t: U LN
Hineq: gap <= gap'
LCn gap t -> LCn gap' t
    do 2 rewrite LCn_spec; intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, gap': nat
t: U LN
Hineq: gap <= gap'
H: forall (w : nat) l,
(w, l) ∈d t -> lc_loc gap (w, l)
w: nat
l: LN
H0: (w, l) ∈d t
lc_loc gap' (w, l)
    apply (lc_loc_mono gap); auto.
  Qed.

  Lemma level1: forall (t: U LN),
    forall w n, (w, Bd n) ∈d t -> n < (w + level t)%nat.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (w n : nat),
(w, Bd n) ∈d t -> n < w + level t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (w n : nat),
(w, Bd n) ∈d t -> n < w + level t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n < w + level t
    enough (n + 1 - w <= level t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
H0: n + 1 - w <= level t
n < w + level t
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n + 1 - w <= level t
    lia.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n + 1 - w <= level t
    change (n + 1 - w) with
      (level_loc (w, Bd n)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
level_loc (w, Bd n) <= level t
    unfold level.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
level_loc (w, Bd n) <= mapdReduce level_loc t
    apply (mapdReduce_pompos (R := le)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
(w, Bd n) ∈d t
    assumption.
  Qed.

  Lemma level2: forall (t: U LN),
      LCn (level t) t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t, LCn (level t) t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t, LCn (level t) t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
LCn (level t) t
    rewrite LCn_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
forall (w : nat) l,
(w, l) ∈d t -> lc_loc (level t) (w, l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w: nat
l: LN
H: (w, l) ∈d t
lc_loc (level t) (w, l)
    unfold lc_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w: nat
l: LN
H: (w, l) ∈d t
match l with
| Fr _ => True
| Bd n => n < w + level t
end
    destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w: nat
n: atom
H: (w, Fr n) ∈d t
True
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n < w + level t
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w: nat
n: atom
H: (w, Fr n) ∈d t
True trivial.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n < w + level t unfold transparent tcs.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
n < w + level t
      apply level1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
w, n: nat
H: (w, Bd n) ∈d t
(w, Bd n) ∈d t
      auto.
  Qed.

  Lemma level3: forall (gap: nat) (t: U LN),
      level t <= gap ->
      LCn gap t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, level t <= gap -> LCn gap t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, level t <= gap -> LCn gap t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
H: level t <= gap
LCn gap t apply (local_closure_mono (level t)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
H: level t <= gap
level t <= gap
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
H: level t <= gap
LCn (level t) t
    assumption.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
H: level t <= gap
LCn (level t) t
    apply level2.
  Qed.

  Lemma level4: forall (gap: nat) (t: U LN),
      LCn gap t ->
      level t <= gap.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, LCn gap t -> level t <= gap
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, LCn gap t -> level t <= gap
    introv hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: LCn gap t
level t <= gap
    unfold LCn in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: Forall_ctx (lc_loc gap) t
level t <= gap
    unfold Forall_ctx in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: mapdReduce (lc_loc gap) t
level t <= gap
    rewrite mapdReduce_through_toctxlist in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: (mapReduce (lc_loc gap) ∘ toctxlist) t
level t <= gap
    unfold level.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: (mapReduce (lc_loc gap) ∘ toctxlist) t
mapdReduce level_loc t <= gap
    rewrite mapdReduce_through_toctxlist.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: (mapReduce (lc_loc gap) ∘ toctxlist) t
(mapReduce level_loc ∘ toctxlist) t <= gap
    unfold compose in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: mapReduce (lc_loc gap) (toctxlist t)
mapReduce level_loc (toctxlist t) <= gap
    induction (toctxlist t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: mapReduce (lc_loc gap) []
mapReduce level_loc [] <= gap
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: mapReduce (lc_loc gap) (a :: e)
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce level_loc (a :: e) <= gap
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: mapReduce (lc_loc gap) []
mapReduce level_loc [] <= gap cbv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
hyp: mapReduce (lc_loc gap) []
0 <= gap lia.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: mapReduce (lc_loc gap) (a :: e)
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce level_loc (a :: e) <= gap cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: mapReduce (lc_loc gap) (a :: e)
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a ● mapReduce level_loc e <= gap
      rewrite mapReduce_eq_mapReduce_list in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: mapReduce_list (lc_loc gap) (a :: e)
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a ● mapReduce level_loc e <= gap
      rewrite mapReduce_list_cons in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: lc_loc gap a ● mapReduce_list (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a ● mapReduce level_loc e <= gap
      rewrite <- mapReduce_eq_mapReduce_list in hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
hyp: lc_loc gap a ● mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a ● mapReduce level_loc e <= gap
      destruct hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
H: lc_loc gap a
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a ● mapReduce level_loc e <= gap
      apply PeanoNat.Nat.max_lub.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
H: lc_loc gap a
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a <= gap
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
H: lc_loc gap a
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce level_loc e <= gap
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
H: lc_loc gap a
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc a <= gap destruct a as [n l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n: nat
l: LN
e: list (nat * LN)
H: lc_loc gap (n, l)
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
level_loc (n, l) <= gap
        unfold level_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n: nat
l: LN
e: list (nat * LN)
H: lc_loc gap (n, l)
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
match l with
| Fr _ => 0
| Bd n0 => n0 + 1 - n
end <= gap
        unfold lc_loc in H.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n: nat
l: LN
e: list (nat * LN)
H: match l with
| Fr _ => True
| Bd n0 => n0 < n + gap
end
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
match l with
| Fr _ => 0
| Bd n0 => n0 + 1 - n
end <= gap
        destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n: nat
n0: atom
e: list (nat * LN)
H: True
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
0 <= gap
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n, n0: nat
e: list (nat * LN)
H: n0 < n + gap
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
n0 + 1 - n <= gap
        *
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n: nat
n0: atom
e: list (nat * LN)
H: True
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
0 <= gap lia.
        *
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
n, n0: nat
e: list (nat * LN)
H: n0 < n + gap
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
n0 + 1 - n <= gap lia.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
a: nat * LN
e: list (nat * LN)
H: lc_loc gap a
H0: mapReduce (lc_loc gap) e
IHe: mapReduce (lc_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce level_loc e <= gap auto.
  Qed.

  Lemma level_iff: forall (gap: nat) (t: U LN),
      LCn gap t <-> level t <= gap.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, LCn gap t <-> level t <= gap
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (gap : nat) t, LCn gap t <-> level t <= gap
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn gap t <-> level t <= gap split.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn gap t -> level t <= gap
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
level t <= gap -> LCn gap t
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn gap t -> level t <= gap apply level4.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
level t <= gap -> LCn gap t apply level3.
  Qed.

  Lemma local_closure_iff: forall (gap: nat) (t: U LN),
      LC t <-> level t = 0.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
nat -> forall t, LC t <-> level t = 0
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
nat -> forall t, LC t <-> level t = 0
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LC t <-> level t = 0 unfold LC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
LCn 0 t <-> level t = 0
    rewrite level_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U LN
level t <= 0 <-> level t = 0
    lia.
  Qed.

  (** ** Reasoning principles for proving equalities *)
  (******************************************************************************)

  (** *** <<open>> *)
  (******************************************************************************)
  Lemma open_eq: forall t u1 u2,
      (forall w l, (w, l) ∈d t -> open_loc u1 (w, l) = open_loc u2 (w, l)) ->
      t '(u1) = t '(u2).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (u1 u2 : T LN),
(forall (w : nat) l,
 (w, l) ∈d t ->
 open_loc u1 (w, l) = open_loc u2 (w, l)) ->
t '(u1) = t '(u2)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (u1 u2 : T LN),
(forall (w : nat) l,
 (w, l) ∈d t ->
 open_loc u1 (w, l) = open_loc u2 (w, l)) ->
t '(u1) = t '(u2)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u1, u2: T LN
H: forall (w : nat) l,
(w, l) ∈d t ->
open_loc u1 (w, l) = open_loc u2 (w, l)
t '(u1) = t '(u2) unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u1, u2: T LN
H: forall (w : nat) l,
(w, l) ∈d t ->
open_loc u1 (w, l) = open_loc u2 (w, l)
bindd (open_loc u1) t = bindd (open_loc u2) t
    now apply bindd_respectful.
  Qed.

  Lemma open_id: forall t u,
      (forall w l, (w, l) ∈d t -> open_loc u (w, l) = ret l) ->
      t '(u) = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (u : T LN),
(forall (w : nat) l,
 (w, l) ∈d t -> open_loc u (w, l) = ret l) ->
t '(u) = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (u : T LN),
(forall (w : nat) l,
 (w, l) ∈d t -> open_loc u (w, l) = ret l) ->
t '(u) = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: forall (w : nat) l,
(w, l) ∈d t -> open_loc u (w, l) = ret l
t '(u) = t unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: forall (w : nat) l,
(w, l) ∈d t -> open_loc u (w, l) = ret l
bindd (open_loc u) t = t
    now apply bindd_respectful_id.
  Qed.

  Lemma open_ret: forall l (u: T LN),
      (ret l) '(u) = open_loc u (0, l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN), ret l '(u) = open_loc u (0, l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN), ret l '(u) = open_loc u (0, l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
ret l '(u) = open_loc u (0, l) unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
bindd (open_loc u) (ret l) = open_loc u (0, l)
    compose near l on left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
(bindd (open_loc u) ∘ ret) l = open_loc u (0, l)
    rewrite kdm_bindd0.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
(open_loc u ∘ ret) l = open_loc u (0, l)
    reflexivity.
  Qed.

  (** *** <<close>> *)
  (******************************************************************************)
  Lemma close_eq: forall t x y,
      (forall w l, (w, l) ∈d t -> close_loc x (w, l) = close_loc y (w, l)) ->
      '[x] t = '[y] t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
(forall (w : nat) l,
 (w, l) ∈d t ->
 close_loc x (w, l) = close_loc y (w, l)) ->
'[x] t = '[y] t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
(forall (w : nat) l,
 (w, l) ∈d t ->
 close_loc x (w, l) = close_loc y (w, l)) ->
'[x] t = '[y] t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
H: forall (w : nat) l,
(w, l) ∈d t ->
close_loc x (w, l) = close_loc y (w, l)
'[x] t = '[y] t unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
H: forall (w : nat) l,
(w, l) ∈d t ->
close_loc x (w, l) = close_loc y (w, l)
mapd (close_loc x) t = mapd (close_loc y) t
    now apply mapd_respectful.
  Qed.

  Lemma close_id: forall t x,
      (forall w l, (w, l) ∈d t -> close_loc x (w, l) = l) ->
      '[x] t = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x,
(forall (w : nat) l,
 (w, l) ∈d t -> close_loc x (w, l) = l) -> '[x] t = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x,
(forall (w : nat) l,
 (w, l) ∈d t -> close_loc x (w, l) = l) -> '[x] t = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
H: forall (w : nat) l,
(w, l) ∈d t -> close_loc x (w, l) = l
'[x] t = t unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
H: forall (w : nat) l,
(w, l) ∈d t -> close_loc x (w, l) = l
mapd (close_loc x) t = t
    now apply mapd_respectful_id.
  Qed.

  Lemma close_ret: forall l x,
      '[x] (ret l) = ret (close_loc x (0, l)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l x, '[x] ret l = ret (close_loc x (0, l))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l x, '[x] ret l = ret (close_loc x (0, l))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
'[x] ret l = ret (close_loc x (0, l)) unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
mapd (close_loc x) (ret l) = ret (close_loc x (0, l))
    compose near l on left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
(mapd (close_loc x) ∘ ret) l =
ret (close_loc x (0, l))
    rewrite mapd_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
(ret ∘ close_loc x ∘ ret) l = ret (close_loc x (0, l))
    reflexivity.
  Qed.

  (** *** <<subst>> *)
  (******************************************************************************)
  Lemma subst_eq: forall t x y (u1 u2: T LN),
      (forall l, l ∈ t -> subst_loc x u1 l = subst_loc y u2 l) ->
      t '{x ~> u1} = t '{y ~> u2}.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom) (u1 u2 : T LN),
(forall l,
 l ∈ t -> subst_loc x u1 l = subst_loc y u2 l) ->
t '{ x ~> u1} = t '{ y ~> u2}
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom) (u1 u2 : T LN),
(forall l,
 l ∈ t -> subst_loc x u1 l = subst_loc y u2 l) ->
t '{ x ~> u1} = t '{ y ~> u2}
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
u1, u2: T LN
H: forall l,
l ∈ t -> subst_loc x u1 l = subst_loc y u2 l
t '{ x ~> u1} = t '{ y ~> u2} unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
u1, u2: T LN
H: forall l,
l ∈ t -> subst_loc x u1 l = subst_loc y u2 l
bind (subst_loc x u1) t = bind (subst_loc y u2) t
    now apply bind_respectful.
  Qed.

  Lemma subst_id: forall t x u,
      (forall l, l ∈ t -> subst_loc x u l = ret l) ->
      t '{x ~> u} = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x (u : T LN),
(forall l, l ∈ t -> subst_loc x u l = ret l) ->
t '{ x ~> u} = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x (u : T LN),
(forall l, l ∈ t -> subst_loc x u l = ret l) ->
t '{ x ~> u} = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: forall l, l ∈ t -> subst_loc x u l = ret l
t '{ x ~> u} = t unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: forall l, l ∈ t -> subst_loc x u l = ret l
bind (subst_loc x u) t = t
    now apply mod_bind_respectful_id.
  Qed.

  Lemma subst_ret: forall x l (u: T LN),
      (ret l) '{x ~> u} = subst_loc x u l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l (u : T LN),
ret l '{ x ~> u} = subst_loc x u l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l (u : T LN),
ret l '{ x ~> u} = subst_loc x u l
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
u: T LN
ret l '{ x ~> u} = subst_loc x u l unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
u: T LN
bind (subst_loc x u) (ret l) = subst_loc x u l
    compose near l on left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
u: T LN
(bind (subst_loc x u) ∘ ret) l = subst_loc x u l
    rewrite kmon_bind0.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
u: T LN
subst_loc x u l = subst_loc x u l
    reflexivity.
  Qed.

  (** ** Occurrence analysis lemmas *)
  (******************************************************************************)

  (** *** <<open>> *)
  (******************************************************************************)
  Lemma ind_open_iff: forall l n u t,
      (n, l) ∈d t '(u) <-> exists n1 n2 l1,
        (n1, l1) ∈d t /\ (n2, l) ∈d open_loc u (n1, l1) /\ n = n1 ● n2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l n (u : T LN) t,
(n, l) ∈d t '(u) <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ n = n1 ● n2)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l n (u : T LN) t,
(n, l) ∈d t '(u) <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ n = n1 ● n2)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
n: nat
u: T LN
t: U LN
(n, l) ∈d t '(u) <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ n = n1 ● n2) unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
n: nat
u: T LN
t: U LN
(n, l) ∈d bindd (open_loc u) t <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ n = n1 ● n2)
    now rewrite ind_bindd_iff'.
  Qed.

  Lemma in_open_iff: forall l u t,
      l ∈ t '(u) <-> exists n1 l1,
        (n1, l1) ∈d t /\ l ∈ open_loc u (n1, l1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN) t,
l ∈ t '(u) <->
(exists n1 l1,
   (n1, l1) ∈d t /\ l ∈ open_loc u (n1, l1))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN) t,
l ∈ t '(u) <->
(exists n1 l1,
   (n1, l1) ∈d t /\ l ∈ open_loc u (n1, l1))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
t: U LN
l ∈ t '(u) <->
(exists n1 l1,
   (n1, l1) ∈d t /\ l ∈ open_loc u (n1, l1)) unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
t: U LN
l ∈ bindd (open_loc u) t <->
(exists n1 l1,
   (n1, l1) ∈d t /\ l ∈ open_loc u (n1, l1))
    now rewrite in_bindd_iff.
  Qed.

  (** *** <<close>> *)
  (******************************************************************************)
  Lemma ind_close_iff: forall l n x t,
      (n, l) ∈d '[x] t <-> exists l1,
        (n, l1) ∈d t /\ close_loc x (n, l1) = l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l n x t,
(n, l) ∈d '[x] t <->
(exists l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l n x t,
(n, l) ∈d '[x] t <->
(exists l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
n: nat
x: atom
t: U LN
(n, l) ∈d '[x] t <->
(exists l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l) unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
n: nat
x: atom
t: U LN
(n, l) ∈d mapd (close_loc x) t <->
(exists l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
    rewrite ind_mapd_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
n: nat
x: atom
t: U LN
(exists a : LN, (n, a) ∈d t /\ close_loc x (n, a) = l) <->
(exists l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
    easy.
  Qed.

  Lemma in_close_iff: forall l x t,
      l ∈ '[x] t <-> exists n l1,
        (n, l1) ∈d t /\ close_loc x (n, l1) = l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l x t,
l ∈ '[x] t <->
(exists n l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l x t,
l ∈ '[x] t <->
(exists n l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
t: U LN
l ∈ '[x] t <->
(exists n l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l) unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
x: atom
t: U LN
l ∈ mapd (close_loc x) t <->
(exists n l1, (n, l1) ∈d t /\ close_loc x (n, l1) = l)
    now rewrite in_mapd_iff.
  Qed.

  (** *** <<subst>> *)
  (******************************************************************************)
  Lemma ind_subst_iff: forall n l u t x,
      (n, l) ∈d t '{x ~> u} <-> exists n1 n2 l1,
        (n1, l1) ∈d t /\ (n2, l) ∈d subst_loc x u l1 /\ n = n1 ● n2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n l (u : T LN) t x,
(n, l) ∈d t '{ x ~> u} <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d subst_loc x u l1 /\ n = n1 ● n2)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n l (u : T LN) t x,
(n, l) ∈d t '{ x ~> u} <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d subst_loc x u l1 /\ n = n1 ● n2)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
l: LN
u: T LN
t: U LN
x: atom
(n, l) ∈d t '{ x ~> u} <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d subst_loc x u l1 /\ n = n1 ● n2) unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
l: LN
u: T LN
t: U LN
x: atom
(n, l) ∈d bind (subst_loc x u) t <->
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d subst_loc x u l1 /\ n = n1 ● n2)
    now rewrite ind_bind_iff'.
  Qed.

  Lemma in_subst_iff: forall l u t x,
      l ∈ t '{x ~> u} <-> exists l1,
       l1 ∈ t /\ l ∈ subst_loc x u l1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN) t x,
l ∈ t '{ x ~> u} <->
(exists l1, l1 ∈ t /\ l ∈ subst_loc x u l1)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l (u : T LN) t x,
l ∈ t '{ x ~> u} <->
(exists l1, l1 ∈ t /\ l ∈ subst_loc x u l1)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
t: U LN
x: atom
l ∈ t '{ x ~> u} <->
(exists l1, l1 ∈ t /\ l ∈ subst_loc x u l1) unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
u: T LN
t: U LN
x: atom
l ∈ bind (subst_loc x u) t <->
(exists l1, l1 ∈ t /\ l ∈ subst_loc x u l1)
    now rewrite mod_in_bind_iff.
  Qed.

End locally_nameless_basic_principles.

(** * Utilities for reasoning at the leaves *)
(******************************************************************************)
Section locally_nameless_utilities.

  Context
    `{Return_T: Return T}
    `{Map_T: Map T}
    `{Bind_TT: Bind T T}
    `{Traverse_T: Traverse T}
    `{Mapd_T: Mapd nat T}
    `{Bindt_TT: Bindt T T}
    `{Bindd_T: Bindd nat T}
    `{Mapdt_T: Mapdt nat T}
    `{Binddt_TT: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}.

  Context
    `{Map_U: Map U}
    `{Bind_TU: Bind T U}
    `{Traverse_U: Traverse U}
    `{Mapd_U: Mapd nat U}
    `{Bindt_TU: Bindt T U}
    `{Bindd_TU: Bindd nat T U}
    `{Mapdt_U: Mapdt nat U}
    `{Binddt_TU: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad nat T}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
                        (unit := Monoid_unit_zero)
                        (op := Monoid_op_plus)}.

  Import Notations.

  (** ** (In)equalities between leaves *)
  (******************************************************************************)
  Lemma Fr_injective: forall (x y: atom),
      (Fr x = Fr y) <-> (x = y).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, Fr x = Fr y <-> x = y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, Fr x = Fr y <-> x = y
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
Fr x = Fr y <-> x = y split; intro hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
hyp: Fr x = Fr y
x = y
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
hyp: x = y
Fr x = Fr y
    now injection hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
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).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, Fr x <> Fr y <-> x <> y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, Fr x <> Fr y <-> x <> y
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
Fr x <> Fr y <-> x <> y split; intro hyp; contradict hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
hyp: x = y
Fr x = Fr y
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
hyp: Fr x = Fr y
x = y
    now subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
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).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, x <> y -> Fr x <> Fr y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x y : atom, x <> y -> Fr x <> Fr y
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
H: x <> y
Fr x <> Fr y now rewrite Fr_injective.
  Qed.

  Lemma Bd_neq_Fr: forall n x,
      (Bd n = Fr x) = False.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) (x : atom), (Bd n = Fr x) = False
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) (x : atom), (Bd n = Fr x) = False
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
x: atom
(Bd n = Fr x) = False propext.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
x: atom
Bd n = Fr x -> False
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
x: atom
False -> Bd n = Fr x discriminate.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
x: atom
False -> Bd n = Fr x contradiction.
  Qed.

  (** ** [subst_loc] *)
  (******************************************************************************)
  Lemma subst_loc_eq: forall (u: T LN) x,
      subst_loc x u (Fr x) = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom), subst_loc x u (Fr x) = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom), subst_loc x u (Fr x) = u intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
subst_loc x u (Fr x) = u cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
(if x == x then u else ret (Fr x)) = u now compare values x and x. Qed.

  Lemma subst_loc_eq_impl: forall (u: T LN) x y,
      Fr x = y -> subst_loc x u y = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom) (y : LN),
Fr x = y -> subst_loc x u y = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom) (y : LN),
Fr x = y -> subst_loc x u y = u intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
y: LN
H: Fr x = y
subst_loc x u y = u subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
subst_loc x u (Fr x) = u now rewrite subst_loc_eq. Qed.

  Lemma subst_loc_fr_eq_impl: forall (u: T LN) x y,
      x = y -> subst_loc x u (Fr y) = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x y : atom),
x = y -> subst_loc x u (Fr y) = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x y : atom),
x = y -> subst_loc x u (Fr y) = u intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
H: x = y
subst_loc x u (Fr y) = u subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
y: atom
subst_loc y u (Fr y) = u now rewrite subst_loc_eq. Qed.

  Lemma subst_loc_neq: forall (u: T LN) x y,
      y <> x -> subst_loc x u (Fr y) = ret (Fr y).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x y : atom),
y <> x -> subst_loc x u (Fr y) = ret (Fr y)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x y : atom),
y <> x -> subst_loc x u (Fr y) = ret (Fr y) intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
H: y <> x
subst_loc x u (Fr y) = ret (Fr y) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
H: y <> x
(if x == y then u else ret (Fr y)) = ret (Fr y) now compare values x and y. Qed.

  Lemma subst_loc_b: forall u x n,
      subst_loc x u (Bd n) = ret (Bd n).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom) (n : nat),
subst_loc x u (Bd n) = ret (Bd n)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (x : atom) (n : nat),
subst_loc x u (Bd n) = ret (Bd n) reflexivity. Qed.

  Lemma subst_loc_fr_neq: forall (u: T LN) l x,
      Fr x <> l -> subst_loc x u l = ret l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (l : LN) (x : atom),
Fr x <> l -> subst_loc x u l = ret l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (l : LN) (x : atom),
Fr x <> l -> subst_loc x u l = ret l
    introv neq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
l: LN
x: atom
neq: Fr x <> l
subst_loc x u l = ret l unfold subst_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
l: LN
x: atom
neq: Fr x <> l
match l with
| Fr y => if x == y then u else ret (Fr y)
| Bd n => ret (Bd n)
end = ret l
    destruct l as [a|?]; [compare values x and a | reflexivity ].
  Qed.

  (** ** [open_loc] *)
  (******************************************************************************)
  Lemma open_loc_lt: forall (u: T LN) w n,
      n < w -> open_loc u (w, Bd n) = ret (Bd n).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (w n : nat),
n < w -> open_loc u (w, Bd n) = ret (Bd n)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (w n : nat),
n < w -> open_loc u (w, Bd n) = ret (Bd n)
    introv ineq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w, n: nat
ineq: n < w
open_loc u (w, Bd n) = ret (Bd n) unfold open_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w, n: nat
ineq: n < w
match n ?= w with
| Eq => u
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end = ret (Bd n) compare naturals n and w.
  Qed.

  Lemma open_loc_gt: forall (u: T LN) n w,
      n > w -> open_loc u (w, Bd n) = ret (Bd (n - 1)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (n w : nat),
n > w -> open_loc u (w, Bd n) = ret (Bd (n - 1))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (n w : nat),
n > w -> open_loc u (w, Bd n) = ret (Bd (n - 1))
    introv ineq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
n, w: nat
ineq: n > w
open_loc u (w, Bd n) = ret (Bd (n - 1)) unfold open_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
n, w: nat
ineq: n > w
match n ?= w with
| Eq => u
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end = ret (Bd (n - 1)) compare naturals n and w.
  Qed.

  Lemma open_loc_eq: forall w (u: T LN),
      open_loc u (w, Bd w) = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (u : T LN), open_loc u (w, Bd w) = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (u : T LN), open_loc u (w, Bd w) = u
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
u: T LN
open_loc u (w, Bd w) = u unfold open_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
u: T LN
match w ?= w with
| Eq => u
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end = u compare naturals w and w.
  Qed.

  Lemma open_loc_atom: forall (u: T LN) w x,
      open_loc u (w, Fr x) = ret (Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (w : nat) (x : atom),
open_loc u (w, Fr x) = ret (Fr x)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) (w : nat) (x : atom),
open_loc u (w, Fr x) = ret (Fr x)
    reflexivity.
  Qed.

  (** ** [close_loc] *)
  (******************************************************************************)
  Lemma close_loc_neq: forall w x y,
      x <> y -> close_loc x (w, Fr y) = Fr y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (x y : atom),
x <> y -> close_loc x (w, Fr y) = Fr y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (x y : atom),
x <> y -> close_loc x (w, Fr y) = Fr y
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
x, y: atom
H: x <> y
close_loc x (w, Fr y) = Fr y cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
x, y: atom
H: x <> y
(if x == y then Bd w else Fr y) = Fr y compare values x and y.
  Qed.

  Lemma close_loc_eq: forall w x,
      close_loc x (w, Fr x) = Bd w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (x : atom),
close_loc x (w, Fr x) = Bd w
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (w : nat) (x : atom),
close_loc x (w, Fr x) = Bd w
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
x: atom
close_loc x (w, Fr x) = Bd w cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
x: atom
(if x == x then Bd w else Fr x) = Bd w compare values x and x.
  Qed.

  Lemma close_loc_lt: forall x w n,
      n < w ->
      close_loc x (w, Bd n) = Bd n.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x : atom) (w n : nat),
n < w -> close_loc x (w, Bd n) = Bd n
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x : atom) (w n : nat),
n < w -> close_loc x (w, Bd n) = Bd n
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
H: n < w
close_loc x (w, Bd n) = Bd n cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
H: n < w
match n ?= w with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n compare naturals n and w.
  Qed.

  Lemma close_loc_ge: forall x w n,
      n >= w ->
      close_loc x (w, Bd n) = Bd (S n).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x : atom) (w n : nat),
n >= w -> close_loc x (w, Bd n) = Bd (S n)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x : atom) (w n : nat),
n >= w -> close_loc x (w, Bd n) = Bd (S n)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
H: n >= w
close_loc x (w, Bd n) = Bd (S n) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
H: n >= w
match n ?= w with
| Lt => Bd n
| _ => Bd (S n)
end = Bd (S n) compare naturals n and w.
  Qed.

  (** ** Local reasoning principles for LC *)
  (******************************************************************************)
  Lemma lc_loc_preincr: forall n m,
      lc_loc n ⦿ m =
        lc_loc (n + m).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n m : nat, lc_loc n ⦿ m = lc_loc (n + m)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n m : nat, lc_loc n ⦿ m = lc_loc (n + m)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
lc_loc n ⦿ m = lc_loc (n + m)
    ext [w l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w: nat
l: LN
(lc_loc n ⦿ m) (w, l) = lc_loc (n + m) (w, l)
    destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w: nat
n0: atom
(lc_loc n ⦿ m) (w, Fr n0) = lc_loc (n + m) (w, Fr n0)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w, n0: nat
(lc_loc n ⦿ m) (w, Bd n0) = lc_loc (n + m) (w, Bd n0)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w: nat
n0: atom
(lc_loc n ⦿ m) (w, Fr n0) = lc_loc (n + m) (w, Fr n0) reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w, n0: nat
(lc_loc n ⦿ m) (w, Bd n0) = lc_loc (n + m) (w, Bd n0) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w, n0: nat
(n0 < (m ● w) + n) = (n0 < w + (n + m)) unfold_monoid.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m, w, n0: nat
(n0 < m + w + n) = (n0 < w + (n + m)) propext; lia.
  Qed.

  Lemma lc_loc_S: forall n,
      lc_loc n ⦿ 1 =
        lc_loc (S n).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, lc_loc n ⦿ 1 = lc_loc (S n)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, lc_loc n ⦿ 1 = lc_loc (S n)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
lc_loc n ⦿ 1 = lc_loc (S n)
    rewrite lc_loc_preincr.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
lc_loc (n + 1) = lc_loc (S n)
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
(n + 1)%nat = S n lia.
  Qed.

  Lemma lc_loc_nBd: forall n w b,
      lc_loc n (w, Bd b) = (b < w + n).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n w b : nat, lc_loc n (w, Bd b) = (b < w + n)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n w b : nat, lc_loc n (w, Bd b) = (b < w + n)
    reflexivity.
  Qed.

  Lemma lc_loc_0Bd: forall w b,
      lc_loc 0 (w, Bd b) = (b < w).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w b : nat, lc_loc 0 (w, Bd b) = (b < w)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w b : nat, lc_loc 0 (w, Bd b) = (b < w)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, b: nat
lc_loc 0 (w, Bd b) = (b < w)
    cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, b: nat
(b < w + 0) = (b < w)
    unfold_monoid.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, b: nat
(b < w + 0) = (b < w)
    propext; lia.
  Qed.

  Lemma lc_loc_00Bd: forall b,
      lc_loc 0 (0, Bd b) = False.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall b : nat, lc_loc 0 (0, Bd b) = False
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall b : nat, lc_loc 0 (0, Bd b) = False
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
b: nat
lc_loc 0 (0, Bd b) = False
    rewrite lc_loc_0Bd.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
b: nat
(b < 0) = False
    propext; lia.
  Qed.

  Lemma lc_loc_Fr: forall n w x,
      lc_loc n (w, Fr x) = True.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n w : nat) (x : atom),
lc_loc n (w, Fr x) = True
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n w : nat) (x : atom),
lc_loc n (w, Fr x) = True
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, w: nat
x: atom
lc_loc n (w, Fr x) = True
    reflexivity.
  Qed.

  Lemma neq_symmetry: forall X (x y: X),
      (x <> y) = (y <> x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (X : Type) (x y : X), (x <> y) = (y <> x)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (X : Type) (x y : X), (x <> y) = (y <> x)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
X: Type
x, y: X
(x <> y) = (y <> x) propext; intro hyp; contradict hyp; congruence.
  Qed.

End locally_nameless_utilities.


Create HintDb subst_local.

#[export] Hint Rewrite
  @subst_loc_eq: subst_local.

#[export] Hint Rewrite
  @subst_loc_eq_impl
  @subst_loc_fr_eq_impl
  @subst_loc_b
  @subst_loc_neq
  @subst_loc_fr_neq
  using solve [auto]: subst_local.

Ltac simplify_subst_local :=
  autorewrite with subst_local.


#[export] Hint Rewrite @in_ret_iff @ind_ret_iff
     using typeclasses eauto: tea_local.

#[export] Hint Rewrite Fr_injective Fr_injective_not_iff Bd_neq_Fr: tea_local.
#[export] Hint Resolve Fr_injective_not: tea_local.

#[export] Hint Rewrite
     @subst_loc_eq (*@subst_in_ret*)
     using typeclasses eauto: tea_local.
#[export] Hint Rewrite
     @subst_loc_neq @subst_loc_b @subst_loc_fr_neq (*@subst_in_ret_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).

Ltac simplify_lc_loc :=
  match goal with
  | |- context[lc_loc ?n ⦿ 1] =>
      rewrite lc_loc_S
  | |- context[lc_loc ?n ⦿ ?m] =>
      rewrite lc_loc_preincr
  | |- context[lc_loc ?n (?w, Fr ?x)] =>
      rewrite lc_loc_Fr
  | |- context[lc_loc 0 (0, Bd ?b)] =>
      rewrite lc_loc_00Bd
  | |- context[lc_loc 0 (?w, Bd ?b)] =>
      rewrite lc_loc_0Bd
  | |- context[lc_loc ?n (?w, Bd ?b)] =>
      rewrite lc_loc_nBd
  end.

(** * Free variables *)
(******************************************************************************)
Section locally_nameless_free_variables.

  Import Notations.

  Context
    `{Return_T: Return T}
    `{Map_T: Map T}
    `{Bind_TT: Bind T T}
    `{Traverse_T: Traverse T}
    `{Mapd_T: Mapd nat T}
    `{Bindt_TT: Bindt T T}
    `{Bindd_T: Bindd nat T}
    `{Mapdt_T: Mapdt nat T}
    `{Binddt_TT: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}.

  Context
    `{Map_U: Map U}
    `{Bind_TU: Bind T U}
    `{Traverse_U: Traverse U}
    `{Mapd_U: Mapd nat U}
    `{Bindt_TU: Bindt T U}
    `{Bindd_TU: Bindd nat T U}
    `{Mapdt_U: Mapdt nat U}
    `{Binddt_TU: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad nat T}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
                        (unit := Monoid_unit_zero)
                        (op := Monoid_op_plus)}.

  Implicit Types (l: LN) (n: nat) (t: U LN) (x: atom).

  (** ** Free variables *)
  (******************************************************************************)

  (** *** Specifications for [free] and [FV] *)
  (******************************************************************************)
  Lemma in_free_iff_local: forall x l,
      x ∈ free_loc l = (Fr x = l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l, x ∈ free_loc l = (Fr x = l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l, x ∈ free_loc l = (Fr x = l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
x ∈ free_loc l = (Fr x = l)
    destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
x ∈ free_loc (Fr n) = (Fr x = Fr n)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
x ∈ free_loc (Bd n) = (Fr x = Bd n)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
x ∈ free_loc (Fr n) = (Fr x = Fr n) cbv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
(n = x \/ False) = (Fr x = Fr n) propext.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
n = x \/ False -> Fr x = Fr n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
Fr x = Fr n -> n = x \/ False
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
n = x \/ False -> Fr x = Fr n intros [hyp|hyp].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
hyp: n = x
Fr x = Fr n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
hyp: False
Fr x = Fr n
        *
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
hyp: n = x
Fr x = Fr n now subst.
        *
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
hyp: False
Fr x = Fr n contradiction.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
Fr x = Fr n -> n = x \/ False inversion 1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
H: Fr x = Fr n
H1: x = n
n = n \/ False now left.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
x ∈ free_loc (Bd n) = (Fr x = Bd n) cbv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
False = (Fr x = Bd n) propext.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
False -> Fr x = Bd n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
Fr x = Bd n -> False
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
False -> Fr x = Bd n inversion 1.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
Fr x = Bd n -> False inversion 1.
  Qed.

  Theorem in_free_iff: forall t x,
      x ∈ free (U := U) t = Fr x ∈ t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x ∈ free t = Fr x ∈ t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x ∈ free t = Fr x ∈ t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x ∈ free t = Fr x ∈ t unfold free.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x ∈ mapReduce free_loc t = Fr x ∈ t
    change_left ((element_of x ∘ mapReduce free_loc) t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
(element_of x ∘ mapReduce free_loc) t = Fr x ∈ t
    rewrite (mapReduce_morphism
               (morphism := Monoid_Morphism_element_list atom x)
               (list atom) Prop (ϕ := element_of x)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
mapReduce (element_of x ∘ free_loc) t = Fr x ∈ t
    rewrite (element_of_to_mapReduce LN (Fr x)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
mapReduce (element_of x ∘ free_loc) t =
mapReduce {{Fr x}} t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
element_of x ∘ free_loc = {{Fr x}} ext l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
l: LN
(element_of x ∘ free_loc) l = {{Fr x}} l
    apply in_free_iff_local.
  Qed.

  Theorem free_iff_FV: forall t x,
      x ∈ free t <-> x `in` FV t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x ∈ free t <-> x `in` FV t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x ∈ free t <-> x `in` FV t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x ∈ free t <-> x `in` FV t
    unfold FV.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x ∈ free t <-> x `in` atoms (free t)
    rewrite <- in_atoms_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x ∈ free t <-> x ∈ free t
    reflexivity.
  Qed.

  Theorem in_free_iff_T_internal: forall (t: T LN) x,
      x ∈ free (U := T) t = Fr x ∈ t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) x, x ∈ free t = Fr x ∈ t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) x, x ∈ free t = Fr x ∈ t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
x ∈ free t = Fr x ∈ t unfold free.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
x ∈ mapReduce free_loc t = Fr x ∈ t
    change_left ((element_of x ∘ mapReduce free_loc) t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
(element_of x ∘ mapReduce free_loc) t = Fr x ∈ t
    rewrite (mapReduce_morphism
               (morphism := Monoid_Morphism_element_list atom x)
               (list atom) Prop (ϕ := element_of x)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
mapReduce (element_of x ∘ free_loc) t = Fr x ∈ t
    rewrite (element_of_to_mapReduce LN (Fr x)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
mapReduce (element_of x ∘ free_loc) t =
mapReduce {{Fr x}} t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
element_of x ∘ free_loc = {{Fr x}} ext l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
l: LN
(element_of x ∘ free_loc) l = {{Fr x}} l
    apply in_free_iff_local.
  Qed.

  (** *** Opening *)
  (******************************************************************************)
  Lemma free_open_iff: forall u t x,
      x ∈ free (t '(u)) <-> exists w l1,
        (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) t x,
x ∈ free (t '(u)) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1)))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) t x,
x ∈ free (t '(u)) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1)))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
x ∈ free (t '(u)) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1)))
    rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
Fr x ∈ t '(u) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1)))
    setoid_rewrite (in_free_iff_T_internal).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
Fr x ∈ t '(u) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ Fr x ∈ open_loc u (w, l1))
    now rewrite in_open_iff.
  Qed.

  (** *** Closing *)
  (******************************************************************************)
  Lemma free_close_iff: forall t x y,
      y ∈ free ('[x] t) <-> exists w l1,
        (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y ∈ free ('[x] t) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y ∈ free ('[x] t) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
y ∈ free ('[x] t) <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y)
    rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
Fr y ∈ '[x] t <->
(exists (w : nat) l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y)
    now rewrite in_close_iff.
  Qed.

  (** *** Substitution *)
  (******************************************************************************)
  Lemma free_subst_iff: forall u t x y,
      y ∈ free (t '{x ~> u}) <-> exists l1,
        l1 ∈ t /\ y ∈ free (subst_loc x u l1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) t (x y : atom),
y ∈ free (t '{ x ~> u}) <->
(exists l1, l1 ∈ t /\ y ∈ free (subst_loc x u l1))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (u : T LN) t (x y : atom),
y ∈ free (t '{ x ~> u}) <->
(exists l1, l1 ∈ t /\ y ∈ free (subst_loc x u l1))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
y ∈ free (t '{ x ~> u}) <->
(exists l1, l1 ∈ t /\ y ∈ free (subst_loc x u l1))
    rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Fr y ∈ t '{ x ~> u} <->
(exists l1, l1 ∈ t /\ y ∈ free (subst_loc x u l1))
    setoid_rewrite in_free_iff_T_internal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Fr y ∈ t '{ x ~> u} <->
(exists l1, l1 ∈ t /\ Fr y ∈ subst_loc x u l1)
    now rewrite in_subst_iff.
  Qed.

  (** ** [Miscellaneous utilities] *)
  (******************************************************************************)
  Lemma ninf_in_neq: forall x l (t: U LN),
      ~ x ∈ free t ->
      l ∈ t ->
      Fr x <> l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l t, ~ x ∈ free t -> l ∈ t -> Fr x <> l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l t, ~ x ∈ free t -> l ∈ t -> Fr x <> l
    introv hyp1 hyp2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
hyp1: ~ x ∈ free t
hyp2: l ∈ t
Fr x <> l
    rewrite in_free_iff in hyp1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: l ∈ t
Fr x <> l destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: Fr n ∈ t
Fr x <> Fr n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: Bd n ∈ t
Fr x <> Bd n
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: Fr n ∈ t
Fr x <> Fr n injection.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, n: atom
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: Fr n ∈ t
H: Fr x = Fr n
x = n -> False intros; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: atom
t: U LN
hyp1: ~ Fr n ∈ t
hyp2: Fr n ∈ t
H: Fr n = Fr n
False
      contradiction.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
n: nat
t: U LN
hyp1: ~ Fr x ∈ t
hyp2: Bd n ∈ t
Fr x <> Bd n discriminate.
  Qed.

End locally_nameless_free_variables.

(** * Locally nameless metatheory *)
(******************************************************************************)
Section locally_nameless_metatheory.

  Context
    `{Return_T: Return T}
    `{Map_T: Map T}
    `{Bind_TT: Bind T T}
    `{Traverse_T: Traverse T}
    `{Mapd_T: Mapd nat T}
    `{Bindt_TT: Bindt T T}
    `{Bindd_T: Bindd nat T}
    `{Mapdt_T: Mapdt nat T}
    `{Binddt_TT: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}.

  Context
    `{Map_U: Map U}
    `{Bind_TU: Bind T U}
    `{Traverse_U: Traverse U}
    `{Mapd_U: Mapd nat U}
    `{Bindt_TU: Bindt T U}
    `{Bindd_TU: Bindd nat T U}
    `{Mapdt_U: Mapdt nat U}
    `{Binddt_TU: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}.

  Context
    `{Monad_inst: ! DecoratedTraversableMonad nat T}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
                        (unit := Monoid_unit_zero)
                        (op := Monoid_op_plus)}.

  Open Scope set_scope.

  Implicit Types
    (l: LN) (p: LN)
    (w: nat) (n: nat)
    (t: U LN) (u: T LN)
    (x: atom).

  (** ** Occurrence analysis: substitution with contexts *)
  (******************************************************************************)
  Lemma ind_subst_loc_iff: forall l w p (u: T LN) x,
      (w, p) ∈d subst_loc x u l <->
      l <> Fr x /\ w = Ƶ /\ l = p \/ (* l is not replaced *)
      l = Fr x /\ (w, p) ∈d u. (* l is replaced *)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l w p u x,
(w, p) ∈d subst_loc x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/ l = Fr x /\ (w, p) ∈d u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l w p u x,
(w, p) ∈d subst_loc x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/ l = Fr x /\ (w, p) ∈d u
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
w: nat
p: LN
u: T LN
x: atom
(w, p) ∈d subst_loc x u l <->
l <> Fr x /\ w = Ƶ /\ l = p \/ l = Fr x /\ (w, p) ∈d u compare l to atom x; simpl_local; intuition.
  Qed.

  Theorem ind_subst_iff2: forall w (t: U LN) (u: T LN) l x,
      (w, l) ∈d t '{x ~> u} <->
      (w, l) ∈d t /\ l <> Fr x \/
      exists w1 w2: nat, (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w t u l x,
(w, l) ∈d t '{ x ~> u} <->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w t u l x,
(w, l) ∈d t '{ x ~> u} <->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(w, l) ∈d t '{ x ~> u} <->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
    rewrite ind_subst_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d subst_loc x u l1 /\ w = n1 ● n2) <->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
    setoid_rewrite ind_subst_loc_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2) <->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2) split.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2) ->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2) ->
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ 
  w = n1 ● n2
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
    l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2) ->
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2) intros [l' [n1 [n2 conditions]]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
conditions: (l', n2) ∈d t /\
(n2 <> Fr x /\ n1 = Ƶ /\ n2 = l \/
 n2 = Fr x /\ (n1, l) ∈d u) /\
w = l' ● n1
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2)
      destruct conditions as [c1 [[c2|c2] c3]]; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
(l' ● n1, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\
   (w2, l) ∈d u /\ l' ● n1 = w1 ● w2)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 = Fr x /\ (n1, l) ∈d u
(l' ● n1, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\
   (w2, l) ∈d u /\ l' ● n1 = w1 ● w2)
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
(l' ● n1, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\
   (w2, l) ∈d u /\ l' ● n1 = w1 ● w2) left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 <> Fr x /\ n1 = Ƶ /\ n2 = l
(l' ● n1, l) ∈d t /\ l <> Fr x destructs c2; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l': nat
H: l <> Fr x
c1: (l', l) ∈d t
(l' ● Ƶ, l) ∈d t /\ l <> Fr x now rewrite monoid_id_r.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 = Fr x /\ (n1, l) ∈d u
(l' ● n1, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\
   (w2, l) ∈d u /\ l' ● n1 = w1 ● w2) right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
n2: LN
c1: (l', n2) ∈d t
c2: n2 = Fr x /\ (n1, l) ∈d u
exists w1 w2,
  (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ l' ● n1 = w1 ● w2 destructs c2; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l', n1: nat
c1: (l', Fr x) ∈d t
H0: (n1, l) ∈d u
exists w1 w2,
  (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ l' ● n1 = w1 ● w2 eauto.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
(w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\ (w2, l) ∈d u /\ w = w1 ● w2) ->
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2 intros [[? ?] | [n1 [n2 [? [? ?]]]]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
H: (w, l) ∈d t
H0: l <> Fr x
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
n1, n2: nat
H: (n1, Fr x) ∈d t
H0: (n2, l) ∈d u
H1: w = n1 ● n2
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ 
  w = n1 ● n2
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
H: (w, l) ∈d t
H0: l <> Fr x
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2 exists w (Ƶ: nat) l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
H: (w, l) ∈d t
H0: l <> Fr x
(w, l) ∈d t /\
(l <> Fr x /\ Ƶ = Ƶ /\ l = l \/
 l = Fr x /\ (Ƶ, l) ∈d u) /\ w = w ● Ƶ rewrite monoid_id_r.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
H: (w, l) ∈d t
H0: l <> Fr x
(w, l) ∈d t /\
(l <> Fr x /\ Ƶ = Ƶ /\ l = l \/
 l = Fr x /\ (Ƶ, l) ∈d u) /\ w = w splits; auto.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
n1, n2: nat
H: (n1, Fr x) ∈d t
H0: (n2, l) ∈d u
H1: w = n1 ● n2
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (l1 <> Fr x /\ n2 = Ƶ /\ l1 = l \/
   l1 = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2 exists n1 n2 (Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
u: T LN
l: LN
x: atom
n1, n2: nat
H: (n1, Fr x) ∈d t
H0: (n2, l) ∈d u
H1: w = n1 ● n2
(n1, Fr x) ∈d t /\
(Fr x <> Fr x /\ n2 = Ƶ /\ Fr x = l \/
 Fr x = Fr x /\ (n2, l) ∈d u) /\ w = n1 ● n2 splits; auto.
  Qed.

  (** ** Occurrence analysis: substitution without contexts *)
  (******************************************************************************)
  Lemma in_subst_loc_iff: forall l p (u: T LN) x,
      p ∈ subst_loc x u l <->
      l <> Fr x /\ l = p \/
      l = Fr x /\ p ∈ u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l p u x,
p ∈ subst_loc x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ p ∈ u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l p u x,
p ∈ subst_loc x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ p ∈ u
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, p: LN
u: T LN
x: atom
p ∈ subst_loc x u l <->
l <> Fr x /\ l = p \/ l = Fr x /\ p ∈ u compare l to atom x; autorewrite* with tea_local; intuition.
  Qed.

  Theorem in_subst_iff': forall t u l x,
      l ∈ t '{x ~> u} <->
      l ∈ t /\ l <> Fr x \/
      Fr x ∈ t /\ l ∈ u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x,
l ∈ t '{ x ~> u} <->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
  Proof with auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x,
l ∈ t '{ x ~> u} <->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l ∈ t '{ x ~> u} <->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u rewrite in_subst_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
(exists l1, l1 ∈ t /\ l ∈ subst_loc x u l1) <->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
    setoid_rewrite in_subst_loc_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
(exists l1,
   l1 ∈ t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ l ∈ u)) <->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u split.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
(exists l1,
   l1 ∈ t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ l ∈ u)) ->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u ->
exists l1,
  l1 ∈ t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ l ∈ u)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
(exists l1,
   l1 ∈ t /\
   (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ l ∈ u)) ->
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u intros [? [? in_sub]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
x0: LN
H: x0 ∈ t
in_sub: x0 <> Fr x /\ x0 = l \/ x0 = Fr x /\ l ∈ u
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
      destruct in_sub as [[? heq] | [heq ?]]; subst...
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u ->
exists l1,
  l1 ∈ t /\
  (l1 <> Fr x /\ l1 = l \/ l1 = Fr x /\ l ∈ u) intros [[? ?]|[? ?]]; eauto.
  Qed.

  (** ** Free variables after substitution *)
  (******************************************************************************)
  Corollary in_free_subst_iff: forall t u x y,
      y ∈ free (t '{x ~> u}) <->
      y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free (t '{ x ~> u}) <->
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free (t '{ x ~> u}) <->
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
y ∈ free (t '{ x ~> u}) <->
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u
    do 4 rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
Fr y ∈ t '{ x ~> u} <->
Fr y ∈ t /\ y <> x \/ Fr x ∈ t /\ Fr y ∈ u
    rewrite in_subst_iff'.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
Fr y ∈ t /\ Fr y <> Fr x \/ Fr x ∈ t /\ Fr y ∈ u <->
Fr y ∈ t /\ y <> x \/ Fr x ∈ t /\ Fr y ∈ u
    now simpl_local.
  Qed.

  Corollary in_FV_subst_iff: forall t u x y,
      y `in` FV (t '{x ~> u}) <->
      y `in` FV t /\ y <> x \/
      x `in` FV t /\ y `in` FV u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x (y : AtomSet.elt),
y `in` FV (t '{ x ~> u}) <->
y `in` FV t /\ y <> x \/ x `in` FV t /\ y `in` FV u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x (y : AtomSet.elt),
y `in` FV (t '{ x ~> u}) <->
y `in` FV t /\ y <> x \/ x `in` FV t /\ y `in` FV u
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
y: AtomSet.elt
y `in` FV (t '{ x ~> u}) <->
y `in` FV t /\ y <> x \/ x `in` FV t /\ y `in` FV u
    do 4 rewrite <- free_iff_FV.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
y: AtomSet.elt
y ∈ free (t '{ x ~> u}) <->
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u
    rewrite in_free_subst_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
y: AtomSet.elt
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u <->
y ∈ free t /\ y <> x \/ x ∈ free t /\ y ∈ free u
    reflexivity.
  Qed.

  (** ** Upper and lower bounds for leaves after substitution *)
  (******************************************************************************)
  Corollary in_subst_upper: forall t u l x,
      l ∈ t '{x ~> u} ->
      (l ∈ t /\ l <> Fr x) \/ l ∈ u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x,
l ∈ t '{ x ~> u} -> l ∈ t /\ l <> Fr x \/ l ∈ u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x,
l ∈ t '{ x ~> u} -> l ∈ t /\ l <> Fr x \/ l ∈ u
    introv hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
hin: l ∈ t '{ x ~> u}
l ∈ t /\ l <> Fr x \/ l ∈ u rewrite in_subst_iff' in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
l: LN
x: atom
hin: l ∈ t /\ l <> Fr x \/ Fr x ∈ t /\ l ∈ u
l ∈ t /\ l <> Fr x \/ l ∈ u
    intuition.
  Qed.

  Corollary in_free_subst_upper: forall t u x y,
      y ∈ free (t '{x ~> u}) ->
      (y ∈ free t /\ y <> x) \/ y ∈ free u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free (t '{ x ~> u}) ->
y ∈ free t /\ y <> x \/ y ∈ free u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free (t '{ x ~> u}) ->
y ∈ free t /\ y <> x \/ y ∈ free u
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
y ∈ free (t '{ x ~> u}) ->
y ∈ free t /\ y <> x \/ y ∈ free u
    do 3 rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
Fr y ∈ t '{ x ~> u} -> Fr y ∈ t /\ y <> x \/ Fr y ∈ u
    rewrite in_subst_iff'.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
Fr y ∈ t /\ Fr y <> Fr x \/ Fr x ∈ t /\ Fr y ∈ u ->
Fr y ∈ t /\ y <> x \/ Fr y ∈ u
    rewrite Fr_injective_not_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
Fr y ∈ t /\ y <> x \/ Fr x ∈ t /\ Fr y ∈ u ->
Fr y ∈ t /\ y <> x \/ Fr y ∈ u
    tauto.
  Qed.

  (* LNgen 4: fv-subst-upper *)
  Corollary FV_subst_upper: forall t u x,
      FV (t '{x ~> u}) ⊆
              (FV t \\ {{x}} ∪ FV u)%set.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x, FV (t '{ x ~> u}) ⊆ FV t \\ {{x}} ∪ FV u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x, FV (t '{ x ~> u}) ⊆ FV t \\ {{x}} ∪ FV u
    intros t y x a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
y: T LN
x: atom
a: AtomSet.elt
a `in` FV (t '{ x ~> y}) ->
a `in` (FV t \\ {{x}} ∪ FV y)
    rewrite AtomSet.union_spec, AtomSet.diff_spec,
    AtomSet.singleton_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
y: T LN
x: atom
a: AtomSet.elt
a `in` FV (t '{ x ~> y}) ->
a `in` FV t /\ a <> x \/ a `in` FV y
    rewrite <- ?(free_iff_FV).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
y: T LN
x: atom
a: AtomSet.elt
a ∈ free (t '{ x ~> y}) ->
a ∈ free t /\ a <> x \/ a ∈ free y
    apply in_free_subst_upper.
  Qed.

  Corollary in_subst_lower: forall t u l x,
      l ∈ t /\ l <> Fr x ->
      l ∈ t '{x ~> u}.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x, l ∈ t /\ l <> Fr x -> l ∈ t '{ x ~> u}
  Proof with auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u l x, l ∈ t /\ l <> Fr x -> l ∈ t '{ x ~> u}
    intros; rewrite in_subst_iff'...
  Qed.

  Corollary in_free_subst_lower: forall t (u: T LN) x y,
      y ∈ free t -> y <> x ->
      y ∈ free (t '{x ~> u}).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free t -> y <> x -> y ∈ free (t '{ x ~> u})
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
y ∈ free t -> y <> x -> y ∈ free (t '{ x ~> u})
    setoid_rewrite in_free_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u (x y : atom),
Fr y ∈ t -> y <> x -> Fr y ∈ t '{ x ~> u} intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x, y: atom
H: Fr y ∈ t
H0: y <> x
Fr y ∈ t '{ x ~> u}
    apply in_subst_lower; now simpl_local.
  Qed.

  (* LNgen 5: fv-subst-lower *)
  Corollary FV_subst_lower: forall (t: T LN) (u: U LN) x,
      FV u \\ {{ x }} ⊆ FV (u '{x ~> t}).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) (u : U LN) x,
FV u \\ {{x}} ⊆ FV (u '{ x ~> t})
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) (u : U LN) x,
FV u \\ {{x}} ⊆ FV (u '{ x ~> t})
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
u: U LN
x: atom
FV u \\ {{x}} ⊆ FV (u '{ x ~> t}) intro a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
u: U LN
x: atom
a: AtomSet.elt
a `in` (FV u \\ {{x}}) -> a `in` FV (u '{ x ~> t})
    rewrite AtomSet.diff_spec, AtomSet.singleton_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
u: U LN
x: atom
a: AtomSet.elt
a `in` FV u /\ a <> x -> a `in` FV (u '{ x ~> t})
    do 2 rewrite <- free_iff_FV.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
u: U LN
x: atom
a: AtomSet.elt
a ∈ free u /\ a <> x -> a ∈ free (u '{ x ~> t})
    intros [? ?].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
u: U LN
x: atom
a: AtomSet.elt
H: a ∈ free u
H0: a <> x
a ∈ free (u '{ x ~> t})
    now apply in_free_subst_lower.
  Qed.

  Corollary scoped_subst: forall t (u: T LN) x γ1 γ2,
      scoped t γ1 ->
      scoped u γ2 ->
      scoped (t '{x ~> u}) (γ1 \\ {{x}} ∪ γ2).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x (γ1 γ2 : AtomSet.t),
scoped t γ1 ->
scoped u γ2 ->
scoped (t '{ x ~> u}) (γ1 \\ {{x}} ∪ γ2)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x (γ1 γ2 : AtomSet.t),
scoped t γ1 ->
scoped u γ2 ->
scoped (t '{ x ~> u}) (γ1 \\ {{x}} ∪ γ2)
    introv St Su.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
γ1, γ2: AtomSet.t
St: scoped t γ1
Su: scoped u γ2
scoped (t '{ x ~> u}) (γ1 \\ {{x}} ∪ γ2) unfold scoped in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
γ1, γ2: AtomSet.t
St: FV t ⊆ γ1
Su: FV u ⊆ γ2
FV (t '{ x ~> u}) ⊆ γ1 \\ {{x}} ∪ γ2
    etransitivity.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
γ1, γ2: AtomSet.t
St: FV t ⊆ γ1
Su: FV u ⊆ γ2
FV (t '{ x ~> u}) ⊆ ?y
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
γ1, γ2: AtomSet.t
St: FV t ⊆ γ1
Su: FV u ⊆ γ2
?y ⊆ γ1 \\ {{x}} ∪ γ2 apply FV_subst_upper.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
γ1, γ2: AtomSet.t
St: FV t ⊆ γ1
Su: FV u ⊆ γ2
FV t \\ {{x}} ∪ FV u ⊆ γ1 \\ {{x}} ∪ γ2 fsetdec.
  Qed.

  (** ** Substitution of fresh variables *)
  (******************************************************************************)
  Theorem subst_fresh: forall t x u,
      ~ x ∈ free t ->
      t '{x ~> u} = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u, ~ x ∈ free t -> t '{ x ~> u} = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u, ~ x ∈ free t -> t '{ x ~> u} = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: ~ x ∈ free t
t '{ x ~> u} = t apply subst_id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: ~ x ∈ free t
forall l, l ∈ t -> subst_loc x u l = ret l intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: ~ x ∈ free t
l: LN
H0: l ∈ t
subst_loc x u l = ret l
    assert (Fr x <> l) by (eapply ninf_in_neq; eauto).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: ~ x ∈ free t
l: LN
H0: l ∈ t
H1: Fr x <> l
subst_loc x u l = ret l
    now simpl_local.
  Qed.

  Theorem subst_fresh_T: forall (t: T LN) x u,
      ~ x ∈ free t ->
      t '{x ~> u} = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) x u,
~ x ∈ free t -> t '{ x ~> u} = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T LN) x u,
~ x ∈ free t -> t '{ x ~> u} = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
u: T LN
H: ~ x ∈ free t
t '{ x ~> u} = t
    apply subst_id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
u: T LN
H: ~ x ∈ free t
forall l, l ∈ t -> subst_loc x u l = ret l intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
u: T LN
H: ~ x ∈ free t
l: LN
H0: l ∈ t
subst_loc x u l = ret l
    assert (Fr x <> l) by (apply (ninf_in_neq x l t); eauto).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T LN
x: atom
u: T LN
H: ~ x ∈ free t
l: LN
H0: l ∈ t
H1: Fr x <> l
subst_loc x u l = ret l
    now simpl_local.
  Qed.

  (* LNgen 7: subst-fresh-eq *)
  Corollary subst_fresh_set: forall t x u,
      ~ x `in` FV t ->
      t '{x ~> u} = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u, x `notin` FV t -> t '{ x ~> u} = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u, x `notin` FV t -> t '{ x ~> u} = t
    setoid_rewrite <- free_iff_FV.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u, ~ x ∈ free t -> t '{ x ~> u} = t apply subst_fresh.
  Qed.

  (* LNgen 6: fv-subst-fresh *)
  Corollary FV_subst_fresh: forall t x u,
      ~ x `in` FV t ->
             FV (t '{x ~> u}) = FV t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u,
x `notin` FV t -> FV (t '{ x ~> u}) = FV t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x u,
x `notin` FV t -> FV (t '{ x ~> u}) = FV t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
u: T LN
H: x `notin` FV t
FV (t '{ x ~> u}) = FV t rewrite subst_fresh_set; auto.
  Qed.

  (** ** Composing substitutions *)
  (******************************************************************************)
  Lemma subst_subst_local: forall (u1: T LN) (u2: T LN) x1 x2,
      ~ x1 ∈ free u2 ->
      x1 <> x2 ->
      subst_loc x2 u2 ⋆ subst_loc x1 u1 =
    subst_loc x1 (subst x2 u2 u1) ⋆ subst_loc x2 u2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 x1 x2,
~ x1 ∈ free u2 ->
x1 <> x2 ->
subst_loc x2 u2 ⋆ subst_loc x1 u1 =
subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2
  Proof with try assumption.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 x1 x2,
~ x1 ∈ free u2 ->
x1 <> x2 ->
subst_loc x2 u2 ⋆ subst_loc x1 u1 =
subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
subst_loc x2 u2 ⋆ subst_loc x1 u1 =
subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2 ext l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
l: LN
(subst_loc x2 u2 ⋆ subst_loc x1 u1) l =
(subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2) l unfold kc1, compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
l: LN
(subst_loc x2 u2 ⋆ subst_loc x1 u1) l =
(subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2) l
    change_left (subst x2 u2 (subst_loc x1 u1 l)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
l: LN
subst_loc x1 u1 l '{ x2 ~> u2} =
(subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2) l
    change_right (subst x1 (subst x2 u2 u1) (subst_loc x2 u2 l)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
l: LN
subst_loc x1 u1 l '{ x2 ~> u2} =
subst_loc x2 u2 l '{ x1 ~> u1 '{ x2 ~> u2}}
    compare l to atom x1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
subst_loc x1 u1 (Fr x1) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr x1) '{ x1 ~> u1 '{ x2 ~> u2}}
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
subst_loc x1 u1 (Fr a) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}}
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
subst_loc x1 u1 (Bd n) '{ x2 ~> u2} =
subst_loc x2 u2 (Bd n) '{ x1 ~> u1 '{ x2 ~> u2}}
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
subst_loc x1 u1 (Fr x1) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr x1) '{ x1 ~> u1 '{ x2 ~> u2}} rewrite subst_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
u1 '{ x2 ~> u2} =
subst_loc x2 u2 (Fr x1) '{ x1 ~> u1 '{ x2 ~> u2}}
      rewrite subst_loc_neq...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
u1 '{ x2 ~> u2} =
ret (Fr x1) '{ x1 ~> u1 '{ x2 ~> u2}}
      rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
u1 '{ x2 ~> u2} =
subst_loc x1 (u1 '{ x2 ~> u2}) (Fr x1)
      rewrite subst_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
u1 '{ x2 ~> u2} = u1 '{ x2 ~> u2}
      reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
subst_loc x1 u1 (Fr a) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}} rewrite subst_loc_neq...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
ret (Fr a) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}}
      compare values x2 and a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1: atom
H: ~ x1 ∈ free u2
a: atom
H0: x1 <> a
H1: a <> x1
DESTR_EQs: a = a
ret (Fr a) '{ a ~> u2} =
subst_loc a u2 (Fr a) '{ x1 ~> u1 '{ a ~> u2}}
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
ret (Fr a) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}}
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1: atom
H: ~ x1 ∈ free u2
a: atom
H0: x1 <> a
H1: a <> x1
DESTR_EQs: a = a
ret (Fr a) '{ a ~> u2} =
subst_loc a u2 (Fr a) '{ x1 ~> u1 '{ a ~> u2}} rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1: atom
H: ~ x1 ∈ free u2
a: atom
H0: x1 <> a
H1: a <> x1
DESTR_EQs: a = a
subst_loc a u2 (Fr a) =
subst_loc a u2 (Fr a) '{ x1 ~> u1 '{ a ~> u2}}
        rewrite subst_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1: atom
H: ~ x1 ∈ free u2
a: atom
H0: x1 <> a
H1: a <> x1
DESTR_EQs: a = a
u2 = u2 '{ x1 ~> u1 '{ a ~> u2}}
        rewrite subst_fresh_T...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1: atom
H: ~ x1 ∈ free u2
a: atom
H0: x1 <> a
H1: a <> x1
DESTR_EQs: a = a
u2 = u2
        reflexivity.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
ret (Fr a) '{ x2 ~> u2} =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}} rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
subst_loc x2 u2 (Fr a) =
subst_loc x2 u2 (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}}
        rewrite subst_loc_neq...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
ret (Fr a) = ret (Fr a) '{ x1 ~> u1 '{ x2 ~> u2}}
        rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
ret (Fr a) = subst_loc x1 (u1 '{ x2 ~> u2}) (Fr a)
        rewrite subst_loc_neq...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
a: atom
H1: a <> x1
DESTR_NEQ: x2 <> a
DESTR_NEQs: a <> x2
ret (Fr a) = ret (Fr a)
        reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
subst_loc x1 u1 (Bd n) '{ x2 ~> u2} =
subst_loc x2 u2 (Bd n) '{ x1 ~> u1 '{ x2 ~> u2}} rewrite subst_loc_b.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
ret (Bd n) '{ x2 ~> u2} =
subst_loc x2 u2 (Bd n) '{ x1 ~> u1 '{ x2 ~> u2}}
      rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
subst_loc x2 u2 (Bd n) =
subst_loc x2 u2 (Bd n) '{ x1 ~> u1 '{ x2 ~> u2}}
      rewrite subst_loc_b.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
ret (Bd n) = ret (Bd n) '{ x1 ~> u1 '{ x2 ~> u2}}
      rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
ret (Bd n) = subst_loc x1 (u1 '{ x2 ~> u2}) (Bd n)
      rewrite subst_loc_b.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
n: nat
ret (Bd n) = ret (Bd n)
      reflexivity.
  Qed.

  (* LNgen 8: subst-subst *)
  Theorem subst_subst: forall u1 u2 t (x1 x2: atom),
      ~ x1 ∈ free u2 ->
      x1 <> x2 ->
      t '{x1 ~> u1} '{x2 ~> u2} =
      t '{x2 ~> u2} '{x1 ~> u1 '{x2 ~> u2} }.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x1 x2,
~ x1 ∈ free u2 ->
x1 <> x2 ->
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1 '{ x2 ~> u2}}
  Proof with try assumption.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x1 x2,
~ x1 ∈ free u2 ->
x1 <> x2 ->
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1 '{ x2 ~> u2}}
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1 '{ x2 ~> u2}}
    unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
bind (subst_loc x2 u2) (bind (subst_loc x1 u1) t) =
bind (subst_loc x1 (bind (subst_loc x2 u2) u1))
  (bind (subst_loc x2 u2) t)
    compose near t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
(bind (subst_loc x2 u2) ∘ bind (subst_loc x1 u1)) t =
(bind (subst_loc x1 (bind (subst_loc x2 u2) u1))
 ∘ bind (subst_loc x2 u2)) t
    rewrite kmod_bind2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
bind (subst_loc x2 u2 ⋆ subst_loc x1 u1) t =
(bind (subst_loc x1 (bind (subst_loc x2 u2) u1))
 ∘ bind (subst_loc x2 u2)) t
    rewrite subst_subst_local...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
bind
  (subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2) t =
(bind (subst_loc x1 (bind (subst_loc x2 u2) u1))
 ∘ bind (subst_loc x2 u2)) t
    rewrite kmod_bind2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: ~ x1 ∈ free u2
H0: x1 <> x2
bind
  (subst_loc x1 (u1 '{ x2 ~> u2}) ⋆ subst_loc x2 u2) t =
bind
  (subst_loc x1 (bind (subst_loc x2 u2) u1)
   ⋆ subst_loc x2 u2) t
    reflexivity.
  Qed.

  (** ** Commuting two substitutions *)
  (******************************************************************************)
  Corollary subst_subst_comm: forall u1 u2 t (x1 x2: atom),
      x1 <> x2 ->
      ~ x1 ∈ free u2 ->
      ~ x2 ∈ free u1 ->
      t '{x1 ~> u1} '{x2 ~> u2} =
      t '{x2 ~> u2} '{x1 ~> u1}.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x1 x2,
x1 <> x2 ->
~ x1 ∈ free u2 ->
~ x2 ∈ free u1 ->
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1}
  Proof with try assumption.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x1 x2,
x1 <> x2 ->
~ x1 ∈ free u2 ->
~ x2 ∈ free u1 ->
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1}
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: x1 <> x2
H0: ~ x1 ∈ free u2
H1: ~ x2 ∈ free u1
(t '{ x1 ~> u1}) '{ x2 ~> u2} =
(t '{ x2 ~> u2}) '{ x1 ~> u1}
    rewrite subst_subst...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: x1 <> x2
H0: ~ x1 ∈ free u2
H1: ~ x2 ∈ free u1
(t '{ x2 ~> u2}) '{ x1 ~> u1 '{ x2 ~> u2}} =
(t '{ x2 ~> u2}) '{ x1 ~> u1}
    rewrite (subst_fresh_T u1 x2 u2)...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x1, x2: atom
H: x1 <> x2
H0: ~ x1 ∈ free u2
H1: ~ x2 ∈ free u1
(t '{ x2 ~> u2}) '{ x1 ~> u1} =
(t '{ x2 ~> u2}) '{ x1 ~> u1}
    reflexivity.
  Qed.

  (** ** Local closure is preserved by substitution *)
  (******************************************************************************)
  Theorem subst_lc: forall u t x,
      LC t ->
      LC u ->
      LC (subst x u t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t x, LC t -> LC u -> LC (t '{ x ~> u})
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t x, LC t -> LC u -> LC (t '{ x ~> u})
    rewrite LC_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t x,
(forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)) ->
LC u ->
forall w l, (w, l) ∈d t '{ x ~> u} -> lc_loc 0 (w, l)
    introv lct lcu hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
hin: (w, l) ∈d t '{ x ~> u}
lc_loc 0 (w, l)
    rewrite ind_subst_iff2 in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
hin: (w, l) ∈d t /\ l <> Fr x \/
(exists w1 w2,
   (w1, Fr x) ∈d t /\
   (w2, l) ∈d u /\ w = w1 ● w2)
lc_loc 0 (w, l)
    destruct hin as [[? ?] | [n1 [n2 [h1 [h2 h3]]]]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
H: (w, l) ∈d t
H0: l <> Fr x
lc_loc 0 (w, l)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
n1, n2: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
h3: w = n1 ● n2
lc_loc 0 (w, l)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
H: (w, l) ∈d t
H0: l <> Fr x
lc_loc 0 (w, l) auto.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
w: nat
l: LN
n1, n2: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
h3: w = n1 ● n2
lc_loc 0 (w, l) subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: LC u
l: LN
n1, n2: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
lc_loc 0 (n1 ● n2, l)
      rewrite LC_spec in lcu.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
l: LN
n1, n2: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
lc_loc 0 (n1 ● n2, l)
      specialize (lcu n2 l h2).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
l: LN
n2: nat
lcu: lc_loc 0 (n2, l)
n1: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
lc_loc 0 (n1 ● n2, l)
      unfold lc_loc in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l,
(w, l) ∈d t ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
l: LN
n2: nat
lcu: match l with
| Fr _ => True
| Bd n => n < n2 + 0
end
n1: nat
h1: (n1, Fr x) ∈d t
h2: (n2, l) ∈d u
match l with
| Fr _ => True
| Bd n => n < (n1 ● n2) + 0
end
      destruct l; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l,
(w, l) ∈d t ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
n, n2: nat
lcu: n < n2 + 0
n1: nat
h1: (n1, Fr x) ∈d t
h2: (n2, Bd n) ∈d u
n < (n1 ● n2) + 0 unfold_monoid.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
lct: forall w l,
(w, l) ∈d t ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
n, n2: nat
lcu: n < n2 + 0
n1: nat
h1: (n1, Fr x) ∈d t
h2: (n2, Bd n) ∈d u
n < n1 + n2 + 0 lia.
  Qed.

  (** ** Decompose substitution into closing/opening *)
  (******************************************************************************)
  Lemma subst_spec_local: forall (u: T LN) w l x,
      subst_loc x u l =
      open_loc u (cobind (close_loc x) (w, l)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u w l x,
subst_loc x u l =
open_loc u (cobind (close_loc x) (w, l))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u w l x,
subst_loc x u l =
open_loc u (cobind (close_loc x) (w, l))
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
l: LN
x: atom
subst_loc x u l =
open_loc u (cobind (close_loc x) (w, l)) compare l to atom x; autorewrite* with tea_local.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
u = open_loc u (cobind (close_loc x) (w, Fr x))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x, a: atom
H: a <> x
ret (Fr a) =
open_loc u (cobind (close_loc x) (w, Fr a))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ret (Bd n) =
open_loc u (cobind (close_loc x) (w, Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
u = open_loc u (cobind (close_loc x) (w, Fr x)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
u =
match (if x == x then Bd w else Fr x) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
DESTR_EQ, DESTR_EQs: x = x
u =
match w ?= w with
| Eq => u
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end unfold id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
DESTR_EQ, DESTR_EQs: x = x
u =
match w ?= w with
| Eq => u
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end
      compare naturals w and w.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x, a: atom
H: a <> x
ret (Fr a) =
open_loc u (cobind (close_loc x) (w, Fr a)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x, a: atom
H: a <> x
ret (Fr a) =
match (if x == a then Bd w else Fr a) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end compare values x and a. (* todo fix fragile names *)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ret (Bd n) =
open_loc u (cobind (close_loc x) (w, Bd n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ret (Bd n) =
match
  match n ?= w with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end unfold id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ret (Bd n) =
match
  match n ?= w with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
ineqrw: (w ?= w) = Eq
ret (Bd w) =
match S w ?= w with
| Eq => u
| Lt => ret (Bd (S w))
| Gt => ret (Bd (S w - 1))
end
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd n) =
match S n ?= w with
| Eq => u
| Lt => ret (Bd (S n))
| Gt => ret (Bd (S n - 1))
end
      now compare naturals (S w) and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
x: atom
n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd n) =
match S n ?= w with
| Eq => u
| Lt => ret (Bd (S n))
| Gt => ret (Bd (S n - 1))
end
      now compare naturals (S n) and w.
  Qed.

  (* LNgen 9: subst-spec *)
  Theorem subst_spec: forall x u t,
      t '{x ~> u} = ('[x] t) '(u).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x u t, t '{ x ~> u} = ('[x] t) '(u)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x u t, t '{ x ~> u} = ('[x] t) '(u)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
t '{ x ~> u} = ('[x] t) '(u) compose near t on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
t '{ x ~> u} = (open u ∘ close x) t
    unfold open, close, subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
bind (subst_loc x u) t =
(bindd (open_loc u) ∘ mapd (close_loc x)) t
    rewrite (bindd_mapd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
bind (subst_loc x u) t =
bindd (open_loc u ⋆1 close_loc x) t
    rewrite (bind_to_bindd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
bindd (subst_loc x u ∘ extract) t =
bindd (open_loc u ⋆1 close_loc x) t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
subst_loc x u ∘ extract = open_loc u ⋆1 close_loc x ext [w l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
w: nat
l: LN
(subst_loc x u ∘ extract) (w, l) =
(open_loc u ⋆1 close_loc x) (w, l)
    unfold compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
u: T LN
t: U LN
w: nat
l: LN
subst_loc x u l =
match
  match l with
  | Fr y => if x == y then Bd w else Fr y
  | Bd n =>
      match n ?= w with
      | Lt => Bd n
      | _ => Bd (S n)
      end
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end
    now erewrite subst_spec_local.
  Qed.

  (** ** Substitution when <<u>> is a LN **)
  (******************************************************************************)
  Definition subst_loc_LN 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 x l,
      subst x (ret l) = map (subst_loc_LN x l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l, subst x (ret l) = map (subst_loc_LN x l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x l, subst x (ret l) = map (subst_loc_LN x l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
subst x (ret l) = map (subst_loc_LN x l) unfold subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
bind (subst_loc x (ret l)) = map (subst_loc_LN x l) ext t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
bind (subst_loc x (ret l)) t =
map (subst_loc_LN x l) t
    apply mod_bind_respectful_map.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
forall a : LN,
a ∈ t ->
subst_loc x (ret l) a = ret (subst_loc_LN x l a)
    intros l' l'in.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
l': LN
l'in: l' ∈ t
subst_loc x (ret l) l' = ret (subst_loc_LN x l l')
    destruct l'.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
n: atom
l'in: Fr n ∈ t
subst_loc x (ret l) (Fr n) =
ret (subst_loc_LN x l (Fr n))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
n: nat
l'in: Bd n ∈ t
subst_loc x (ret l) (Bd n) =
ret (subst_loc_LN x l (Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
n: atom
l'in: Fr n ∈ t
subst_loc x (ret l) (Fr n) =
ret (subst_loc_LN x l (Fr n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
n: atom
l'in: Fr n ∈ t
(if x == n then ret l else ret (Fr n)) =
ret (if x == n then l else Fr n) compare values x and n.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
l: LN
t: U LN
n: nat
l'in: Bd n ∈ t
subst_loc x (ret l) (Bd n) =
ret (subst_loc_LN x l (Bd n)) reflexivity.
  Qed.

  (** ** Substitution by the same variable is the identity *)
  (******************************************************************************)
  Theorem subst_same: forall t x,
      t '{x ~> ret (Fr x)} = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, t '{ x ~> ret (Fr x)} = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, t '{ x ~> ret (Fr x)} = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
t '{ x ~> ret (Fr x)} = t apply subst_id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
forall l, l ∈ t -> subst_loc x (ret (Fr x)) l = ret l
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
l: LN
H: l ∈ t
subst_loc x (ret (Fr x)) l = ret l compare l to atom x; now simpl_local.
  Qed.

  (** ** Free variables after variable closing *)
  (******************************************************************************)
  Lemma in_free_close_iff_loc1: forall w l (t: U LN) x y,
      (w, l) ∈d t ->
      close_loc x (w, l) = Fr y ->
      Fr y ∈ t /\ x <> y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w l t (x y : atom),
(w, l) ∈d t ->
close_loc x (w, l) = Fr y -> Fr y ∈ t /\ x <> y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall w l t (x y : atom),
(w, l) ∈d t ->
close_loc x (w, l) = Fr y -> Fr y ∈ t /\ x <> y
    introv lin heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
l: LN
t: U LN
x, y: atom
lin: (w, l) ∈d t
heq: close_loc x (w, l) = Fr y
Fr y ∈ t /\ x <> y destruct l as [la | ln].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
la: atom
t: U LN
x, y: atom
lin: (w, Fr la) ∈d t
heq: close_loc x (w, Fr la) = Fr y
Fr y ∈ t /\ x <> y
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, ln: nat
t: U LN
x, y: atom
lin: (w, Bd ln) ∈d t
heq: close_loc x (w, Bd ln) = Fr y
Fr y ∈ t /\ x <> y
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
la: atom
t: U LN
x, y: atom
lin: (w, Fr la) ∈d t
heq: close_loc x (w, Fr la) = Fr y
Fr y ∈ t /\ x <> y cbn in heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
la: atom
t: U LN
x, y: atom
lin: (w, Fr la) ∈d t
heq: (if x == la then Bd w else Fr la) = Fr y
Fr y ∈ t /\ x <> y destruct_eq_args x la.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
la: atom
t: U LN
x, y: atom
lin: (w, Fr la) ∈d t
DESTR_NEQ: x <> la
heq: Fr la = Fr y
DESTR_NEQs: la <> x
Fr y ∈ t /\ x <> y
      inverts heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
x, y: atom
DESTR_NEQs: y <> x
DESTR_NEQ: x <> y
lin: (w, Fr y) ∈d t
Fr y ∈ t /\ x <> y
      apply ind_implies_in in lin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w: nat
t: U LN
x, y: atom
DESTR_NEQs: y <> x
DESTR_NEQ: x <> y
lin: Fr y ∈ t
Fr y ∈ t /\ x <> y
      tauto.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, ln: nat
t: U LN
x, y: atom
lin: (w, Bd ln) ∈d t
heq: close_loc x (w, Bd ln) = Fr y
Fr y ∈ t /\ x <> y cbn in heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
w, ln: nat
t: U LN
x, y: atom
lin: (w, Bd ln) ∈d t
heq: match ln ?= w with
| Lt => Bd ln
| _ => Bd (S ln)
end = Fr y
Fr y ∈ t /\ x <> y compare_nats_args ln w; discriminate.
  Qed.

  Lemma in_free_close_iff_loc2: forall t x y,
      x <> y ->
      Fr y ∈ t ->
      exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
x <> y ->
Fr y ∈ t ->
exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
x <> y ->
Fr y ∈ t ->
exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y
    introv neq yin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
yin: Fr y ∈ t
exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y
    rewrite (ind_iff_in) in yin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
yin: exists e : nat, (e, Fr y) ∈d t
exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y destruct yin as [w yin].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
w: nat
yin: (w, Fr y) ∈d t
exists w l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y
    exists w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
w: nat
yin: (w, Fr y) ∈d t
exists l, (w, l) ∈d t /\ close_loc x (w, l) = Fr y exists (Fr y).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
w: nat
yin: (w, Fr y) ∈d t
(w, Fr y) ∈d t /\ close_loc x (w, Fr y) = Fr y cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
neq: x <> y
w: nat
yin: (w, Fr y) ∈d t
(w, Fr y) ∈d t /\
(if x == y then Bd w else Fr y) = Fr y compare values x and y.
  Qed.

  Theorem in_free_close_iff: forall t x y,
      y ∈ free ('[x] t) <-> y ∈ free t /\ x <> y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y ∈ free ('[x] t) <-> y ∈ free t /\ x <> y
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y ∈ free ('[x] t) <-> y ∈ free t /\ x <> y
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
y ∈ free ('[x] t) <-> y ∈ free t /\ x <> y rewrite (free_close_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
(exists w l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y) <->
y ∈ free t /\ x <> y
    rewrite (in_free_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
(exists w l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y) <->
Fr y ∈ t /\ x <> y split.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
(exists w l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y) ->
Fr y ∈ t /\ x <> y
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
Fr y ∈ t /\ x <> y ->
exists w l1,
  (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
(exists w l1,
   (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y) ->
Fr y ∈ t /\ x <> y introv [? [? [? ?] ] ].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
x0: nat
x1: LN
H: (x0, x1) ∈d t
H0: close_loc x (x0, x1) = Fr y
Fr y ∈ t /\ x <> y eauto using in_free_close_iff_loc1.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
Fr y ∈ t /\ x <> y ->
exists w l1,
  (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y intros [? ?].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
H: Fr y ∈ t
H0: x <> y
exists w l1,
  (w, l1) ∈d t /\ close_loc x (w, l1) = Fr y eauto using in_free_close_iff_loc2.
  Qed.

  Corollary in_free_close_iff1: forall t x y,
      y <> x ->
      y ∈ free ('[x] t) <-> y ∈ free t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y <> x -> y ∈ free ('[x] t) <-> y ∈ free t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t (x y : atom),
y <> x -> y ∈ free ('[x] t) <-> y ∈ free t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
H: y <> x
y ∈ free ('[x] t) <-> y ∈ free t rewrite in_free_close_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x, y: atom
H: y <> x
y ∈ free t /\ x <> y <-> y ∈ free t intuition.
  Qed.

  (* LNgen 3: fv-close *)
  Corollary FV_close: forall t x,
      FV ('[x] t) [=] FV t \\ {{ x }}.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, FV ('[x] t) [=] FV t \\ {{x}}
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, FV ('[x] t) [=] FV t \\ {{x}}
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
FV ('[x] t) [=] FV t \\ {{x}} intro a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
a: AtomSet.elt
a `in` FV ('[x] t) <-> a `in` (FV t \\ {{x}}) rewrite AtomSet.diff_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
a: AtomSet.elt
a `in` FV ('[x] t) <-> a `in` FV t /\ a `notin` {{x}}
    rewrite <- 2(free_iff_FV).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
a: AtomSet.elt
a ∈ free ('[x] t) <-> a ∈ free t /\ a `notin` {{x}} rewrite in_free_close_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
a: AtomSet.elt
a ∈ free t /\ x <> a <-> a ∈ free t /\ a `notin` {{x}}
    rewrite AtomSet.singleton_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
a: AtomSet.elt
a ∈ free t /\ x <> a <-> a ∈ free t /\ a <> x intuition.
  Qed.

  Corollary nin_free_close: forall t x,
      ~ (x ∈ free ('[x] t)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, ~ x ∈ free ('[x] t)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, ~ x ∈ free ('[x] t)
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
~ x ∈ free ('[x] t) rewrite in_free_close_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
~ (x ∈ free t /\ x <> x) intuition.
  Qed.

  Corollary nin_FV_close: forall t x,
      ~ (x `in` FV ('[x] t)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x `notin` FV ('[x] t)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, x `notin` FV ('[x] t)
    introv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
x `notin` FV ('[x] t) rewrite <- free_iff_FV.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
~ x ∈ free ('[x] t) apply nin_free_close.
  Qed.

  (** ** Variable closing and local closure *)
  (******************************************************************************)
  Theorem close_lc: forall t x,
      LC t ->
      LCn 1 (close x t).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, LC t -> LCn 1 ('[x] t)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, LC t -> LCn 1 ('[x] t)
    rewrite LC_spec, LCn_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x,
(forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)) ->
forall w l, (w, l) ∈d '[x] t -> lc_loc 1 (w, l)
    introv lct hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
l: LN
hin: (w, l) ∈d '[x] t
lc_loc 1 (w, l)
    rewrite ind_close_iff in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
l: LN
hin: exists l1,
  (w, l1) ∈d t /\ close_loc x (w, l1) = l
lc_loc 1 (w, l)
    destruct hin as [l1 [? ?]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
l, l1: LN
H: (w, l1) ∈d t
H0: close_loc x (w, l1) = l
lc_loc 1 (w, l) compare l1 to atom x; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
H: (w, Fr x) ∈d t
lc_loc 1 (w, close_loc x (w, Fr x))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
a: atom
H: (w, Fr a) ∈d t
H1: a <> x
lc_loc 1 (w, close_loc x (w, Fr a))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w, n: nat
H: (w, Bd n) ∈d t
lc_loc 1 (w, close_loc x (w, Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
H: (w, Fr x) ∈d t
lc_loc 1 (w, close_loc x (w, Fr x)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
H: (w, Fr x) ∈d t
match (if x == x then Bd w else Fr x) with
| Fr _ => True
| Bd n => n < w + 1
end compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
H: (w, Fr x) ∈d t
DESTR_EQ, DESTR_EQs: x = x
w < w + 1 unfold_lia.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
a: atom
H: (w, Fr a) ∈d t
H1: a <> x
lc_loc 1 (w, close_loc x (w, Fr a)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
a: atom
H: (w, Fr a) ∈d t
H1: a <> x
match (if x == a then Bd w else Fr a) with
| Fr _ => True
| Bd n => n < w + 1
end compare values x and a.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w, n: nat
H: (w, Bd n) ∈d t
lc_loc 1 (w, close_loc x (w, Bd n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w, n: nat
H: (w, Bd n) ∈d t
match
  match n ?= w with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr _ => True
| Bd n => n < w + 1
end compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
ineqrw: (w ?= w) = Eq
H: (w, Bd w) ∈d t
S w < w + 1
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w, n: nat
H: (w, Bd n) ∈d t
ineqrw: (n ?= w) = Gt
ineqp: n > w
S n < w + 1
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
ineqrw: (w ?= w) = Eq
H: (w, Bd w) ∈d t
S w < w + 1 false.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
ineqrw: (w ?= w) = Eq
H: (w, Bd w) ∈d t
False
        specialize (lct w (Bd w) ltac:(assumption)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
w: nat
lct: lc_loc 0 (w, Bd w)
ineqrw: (w ?= w) = Eq
H: (w, Bd w) ∈d t
False
        cbn in lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
w: nat
lct: w < w + 0
ineqrw: (w ?= w) = Eq
H: (w, Bd w) ∈d t
False lia.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
lct: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w, n: nat
H: (w, Bd n) ∈d t
ineqrw: (n ?= w) = Gt
ineqp: n > w
S n < w + 1 specialize (lct w (Bd n) ltac:(assumption)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
w, n: nat
lct: lc_loc 0 (w, Bd n)
H: (w, Bd n) ∈d t
ineqrw: (n ?= w) = Gt
ineqp: n > w
S n < w + 1
        cbn in lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
w, n: nat
lct: n < w + 0
H: (w, Bd n) ∈d t
ineqrw: (n ?= w) = Gt
ineqp: n > w
S n < w + 1 lia.
  Qed.

  (** ** Upper and lower bounds on free variables after opening *)
  (******************************************************************************)
  Lemma free_open_upper_local: forall t (u: T LN) w l x,
      l ∈ t ->
      x ∈ free (open_loc u (w, l)) ->
      (l = Fr x /\ x ∈ free t) \/
      x ∈ free u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u w l x,
l ∈ t ->
x ∈ free (open_loc u (w, l)) ->
l = Fr x /\ x ∈ free t \/ x ∈ free u
  Proof with auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u w l x,
l ∈ t ->
x ∈ free (open_loc u (w, l)) ->
l = Fr x /\ x ∈ free t \/ x ∈ free u
    introv lin xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
l: LN
x: atom
lin: l ∈ t
xin: x ∈ free (open_loc u (w, l))
l = Fr x /\ x ∈ free t \/ x ∈ free u rewrite in_free_iff in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
l: LN
x: atom
lin: l ∈ t
xin: Fr x ∈ open_loc u (w, l)
l = Fr x /\ x ∈ free t \/ x ∈ free u
    rewrite 2(in_free_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
l: LN
x: atom
lin: l ∈ t
xin: Fr x ∈ open_loc u (w, l)
l = Fr x /\ Fr x ∈ t \/ Fr x ∈ u destruct l as [y | n].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
y, x: atom
lin: Fr y ∈ t
xin: Fr x ∈ open_loc u (w, Fr y)
Fr y = Fr x /\ Fr x ∈ t \/ Fr x ∈ u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ open_loc u (w, Bd n)
Bd n = Fr x /\ Fr x ∈ t \/ Fr x ∈ u
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
y, x: atom
lin: Fr y ∈ t
xin: Fr x ∈ open_loc u (w, Fr y)
Fr y = Fr x /\ Fr x ∈ t \/ Fr x ∈ u left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
y, x: atom
lin: Fr y ∈ t
xin: Fr x ∈ open_loc u (w, Fr y)
Fr y = Fr x /\ Fr x ∈ t autorewrite with tea_local in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
y, x: atom
lin: Fr y ∈ t
xin: x = y
Fr y = Fr x /\ Fr x ∈ t inverts xin...
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ open_loc u (w, Bd n)
Bd n = Fr x /\ Fr x ∈ t \/ Fr x ∈ u right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ open_loc u (w, Bd n)
Fr x ∈ u cbn in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x
∈ match n ?= w with
  | Eq => u
  | Lt => ret (Bd n)
  | Gt => ret (Bd (n - 1))
  end
Fr x ∈ u compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd n)
ineqrw: (n ?= w) = Lt
ineqp: n < w
Fr x ∈ u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
x: atom
lin: Bd w ∈ t
xin: Fr x ∈ u
ineqrw: (w ?= w) = Eq
Fr x ∈ u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd (n - 1))
ineqrw: (n ?= w) = Gt
ineqp: n > w
Fr x ∈ u
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd n)
ineqrw: (n ?= w) = Lt
ineqp: n < w
Fr x ∈ u contradict xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
ineqrw: (n ?= w) = Lt
ineqp: n < w
~ Fr x ∈ ret (Bd n) simpl_local.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
ineqrw: (n ?= w) = Lt
ineqp: n < w
Fr x <> Bd n easy. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
x: atom
lin: Bd w ∈ t
xin: Fr x ∈ u
ineqrw: (w ?= w) = Eq
Fr x ∈ u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd (n - 1))
ineqrw: (n ?= w) = Gt
ineqp: n > w
Fr x ∈ u
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
x: atom
lin: Bd w ∈ t
xin: Fr x ∈ u
ineqrw: (w ?= w) = Eq
Fr x ∈ u assumption. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd (n - 1))
ineqrw: (n ?= w) = Gt
ineqp: n > w
Fr x ∈ u
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
xin: Fr x ∈ ret (Bd (n - 1))
ineqrw: (n ?= w) = Gt
ineqp: n > w
Fr x ∈ u contradict xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
ineqrw: (n ?= w) = Gt
ineqp: n > w
~ Fr x ∈ ret (Bd (n - 1)) simpl_local.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w, n: nat
x: atom
lin: Bd n ∈ t
ineqrw: (n ?= w) = Gt
ineqp: n > w
Fr x <> Bd (n - 1) easy. }
  Qed.

  Theorem free_open_upper: forall t (u: T LN) x,
      x ∈ free (t '(u)) ->
      x ∈ free t \/ x ∈ free u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x,
x ∈ free (t '(u)) -> x ∈ free t \/ x ∈ free u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x,
x ∈ free (t '(u)) -> x ∈ free t \/ x ∈ free u
    introv xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
xin: x ∈ free (t '(u))
x ∈ free t \/ x ∈ free u
    rewrite free_open_iff in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
xin: exists w l1,
  (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1))
x ∈ free t \/ x ∈ free u
    destruct xin as [w [l [hin ?]]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
l: LN
hin: (w, l) ∈d t
H: x ∈ free (open_loc u (w, l))
x ∈ free t \/ x ∈ free u
    apply (ind_implies_in) in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
l: LN
hin: l ∈ t
H: x ∈ free (open_loc u (w, l))
x ∈ free t \/ x ∈ free u
    enough ((l = Fr x /\ x ∈ free t) \/ x ∈ free u) by intuition.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
l: LN
hin: l ∈ t
H: x ∈ free (open_loc u (w, l))
l = Fr x /\ x ∈ free t \/ x ∈ free u
    eauto using free_open_upper_local.
  Qed.

  (* LNgen 1: fv-open-upper *)
  Corollary FV_open_upper: forall t (u: T LN),
      FV (t '(u)) ⊆ FV t ∪ FV u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, FV (t '(u)) ⊆ FV t ∪ FV u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, FV (t '(u)) ⊆ FV t ∪ FV u
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
FV (t '(u)) ⊆ FV t ∪ FV u intro a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
a: AtomSet.elt
a `in` FV (t '(u)) -> a `in` (FV t ∪ FV u) rewrite AtomSet.union_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
a: AtomSet.elt
a `in` FV (t '(u)) -> a `in` FV t \/ a `in` FV u
    repeat rewrite <- (free_iff_FV).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
a: AtomSet.elt
a ∈ free (t '(u)) -> a ∈ free t \/ a ∈ free u
    auto using free_open_upper.
  Qed.

  Theorem free_open_lower: forall t u x,
      x ∈ free t ->
      x ∈ free (t '(u)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x, x ∈ free t -> x ∈ free (t '(u))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u x, x ∈ free t -> x ∈ free (t '(u))
    introv xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
xin: x ∈ free t
x ∈ free (t '(u))
    rewrite (in_free_iff) in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
xin: Fr x ∈ t
x ∈ free (t '(u))
    rewrite (ind_iff_in) in xin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
xin: exists e : nat, (e, Fr x) ∈d t
x ∈ free (t '(u))
    destruct xin as [w xin].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
xin: (w, Fr x) ∈d t
x ∈ free (t '(u))
    rewrite (free_open_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
xin: (w, Fr x) ∈d t
exists w l1,
  (w, l1) ∈d t /\ x ∈ free (open_loc u (w, l1))
    setoid_rewrite (in_free_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
xin: (w, Fr x) ∈d t
exists w l1, (w, l1) ∈d t /\ Fr x ∈ open_loc u (w, l1)
    exists w (Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
x: atom
w: nat
xin: (w, Fr x) ∈d t
(w, Fr x) ∈d t /\ Fr x ∈ open_loc u (w, Fr x) now autorewrite with tea_local.
  Qed.

  (* LNgen 2: fv-open-lower *)
  Corollary FV_open_lower: forall t u,
      FV t ⊆ FV (t '(u)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, FV t ⊆ FV (t '(u))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, FV t ⊆ FV (t '(u))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
FV t ⊆ FV (t '(u)) intro a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
a: AtomSet.elt
a `in` FV t -> a `in` FV (t '(u)) rewrite <- 2(free_iff_FV).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
a: AtomSet.elt
a ∈ free t -> a ∈ free (t '(u))
    apply free_open_lower.
  Qed.

  (** ** Opening a locally closed term is the identity *)
  (**************************************************************************)
  Lemma open_lc_local: forall (u: T LN) w l,
      lc_loc 0 (w, l) ->
      open_loc u (w, l) = ret l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u w l,
lc_loc 0 (w, l) -> open_loc u (w, l) = ret l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u w l,
lc_loc 0 (w, l) -> open_loc u (w, l) = ret l
    introv hyp.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
l: LN
hyp: lc_loc 0 (w, l)
open_loc u (w, l) = ret l destruct l as [a | n].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
a: atom
hyp: lc_loc 0 (w, Fr a)
open_loc u (w, Fr a) = ret (Fr a)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w, n: nat
hyp: lc_loc 0 (w, Bd n)
open_loc u (w, Bd n) = ret (Bd n)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w: nat
a: atom
hyp: lc_loc 0 (w, Fr a)
open_loc u (w, Fr a) = ret (Fr a) reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w, n: nat
hyp: lc_loc 0 (w, Bd n)
open_loc u (w, Bd n) = ret (Bd n) cbn in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
w, n: nat
hyp: n < w + 0
match n ?= w with
| Eq => u
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end = ret (Bd n) compare naturals n and w; unfold_lia.
  Qed.

  Theorem open_lc: forall t u,
      LC t ->
      t '(u) = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, LC t -> t '(u) = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, LC t -> t '(u) = t
    introv lc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
lc: LC t
t '(u) = t
    rewrite LC_spec in lc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
lc: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
t '(u) = t
    apply open_id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
lc: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
forall w l, (w, l) ∈d t -> open_loc u (w, l) = ret l introv lin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
lc: forall w l, (w, l) ∈d t -> lc_loc 0 (w, l)
w: nat
l: LN
lin: (w, l) ∈d t
open_loc u (w, l) = ret l
    specialize (lc _ _ lin).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
w: nat
l: LN
lc: lc_loc 0 (w, l)
lin: (w, l) ∈d t
open_loc u (w, l) = ret l
    destruct l; auto using open_lc_local.
  Qed.

  (** ** Opening followed by substitution *)
  (**************************************************************************)
  Lemma subst_open_local: forall u1 u2 x,
      LC u2 ->
      subst x u2 ∘ open_loc u1 =
      open_loc (u1 '{x ~> u2}) ⋆5 (subst_loc x u2 ∘ extract).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 x,
LC u2 ->
subst x u2 ∘ open_loc u1 =
open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 x,
LC u2 ->
subst x u2 ∘ open_loc u1 =
open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract
    introv lcu2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
subst x u2 ∘ open_loc u1 =
open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract ext [w l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
l: LN
(subst x u2 ∘ open_loc u1) (w, l) =
(open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract)
  (w, l) unfold kc5.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
l: LN
(subst x u2 ∘ open_loc u1) (w, l) =
bindd (open_loc (u1 '{ x ~> u2}) ⦿ w)
  ((subst_loc x u2 ∘ extract) (w, l))
    unfold preincr, compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
l: LN
match l with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => u1
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 l) compare l to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ret (Fr x) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr x))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
ret (Fr a) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr a))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
match n ?= w with
| Eq => u1
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ret (Fr x) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr x)) rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
subst_loc x u2 (Fr x) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr x)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
(if x == x then u2 else ret (Fr x)) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (if x == x then u2 else ret (Fr x))
      compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
u2 = bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) u2 symmetry.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) u2 = u2
      apply (bindd_respectful_id).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
forall w0 (a : LN),
(w0, a) ∈d u2 ->
open_loc (u1 '{ x ~> u2}) (incr w (w0, a)) = ret a
      intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0: nat
a: LN
H: (w0, a) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, a)) = ret a destruct a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0: nat
n: atom
H: (w0, Fr n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Fr n)) =
ret (Fr n)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Bd n)) =
ret (Bd n)
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0: nat
n: atom
H: (w0, Fr n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Fr n)) =
ret (Fr n) reflexivity.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Bd n)) =
ret (Bd n) rewrite LC_spec in lcu2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l, (w, l) ∈d u2 -> lc_loc 0 (w, l)
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Bd n)) =
ret (Bd n)
        unfold lc_loc in lcu2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
open_loc (u1 '{ x ~> u2}) (incr w (w0, Bd n)) =
ret (Bd n)
        cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
match n ?= w ● w0 with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end = ret (Bd n)
        compare naturals n and (w ● w0).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0: nat
ineqrw: (w ● w0 ?= w ● w0) = Eq
H: (w0, Bd (w ● w0)) ∈d u2
u1 '{ x ~> u2} = ret (Bd (w ● w0))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
ineqrw: (n ?= w ● w0) = Gt
ineqp: n > w ● w0
n - 1 = n
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0: nat
ineqrw: (w ● w0 ?= w ● w0) = Eq
H: (w0, Bd (w ● w0)) ∈d u2
u1 '{ x ~> u2} = ret (Bd (w ● w0)) specialize (lcu2 _ _ ltac:(eauto)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
w, w0: nat
lcu2: match Bd (w ● w0) with
| Fr _ => True
| Bd n => n < w0 + 0
end
DESTR_EQ, DESTR_EQs: x = x
ineqrw: (w ● w0 ?= w ● w0) = Eq
H: (w0, Bd (w ● w0)) ∈d u2
u1 '{ x ~> u2} = ret (Bd (w ● w0)) cbn in lcu2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
w, w0: nat
lcu2: w ● w0 < w0 + 0
DESTR_EQ, DESTR_EQs: x = x
ineqrw: (w ● w0 ?= w ● w0) = Eq
H: (w0, Bd (w ● w0)) ∈d u2
u1 '{ x ~> u2} = ret (Bd (w ● w0)) unfold_lia. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
ineqrw: (n ?= w ● w0) = Gt
ineqp: n > w ● w0
n - 1 = n
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: forall w l,
(w, l) ∈d u2 ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
w0, n: nat
H: (w0, Bd n) ∈d u2
ineqrw: (n ?= w ● w0) = Gt
ineqp: n > w ● w0
n - 1 = n specialize (lcu2 _ _ ltac:(eauto)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
w0, n: nat
lcu2: match Bd n with
| Fr _ => True
| Bd n => n < w0 + 0
end
w: nat
DESTR_EQ, DESTR_EQs: x = x
H: (w0, Bd n) ∈d u2
ineqrw: (n ?= w ● w0) = Gt
ineqp: n > w ● w0
n - 1 = n cbn in lcu2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
w0, n: nat
lcu2: n < w0 + 0
w: nat
DESTR_EQ, DESTR_EQs: x = x
H: (w0, Bd n) ∈d u2
ineqrw: (n ?= w ● w0) = Gt
ineqp: n > w ● w0
n - 1 = n unfold_lia. }
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
ret (Fr a) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr a)) rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
subst_loc x u2 (Fr a) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Fr a)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
(if x == a then u2 else ret (Fr a)) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (if x == a then u2 else ret (Fr a)) compare values x and a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Fr a))
      compose near (Fr a).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
a: atom
H: a <> x
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
(eq ∘ ret) (Fr a)
  ((bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) ∘ ret)
     (Fr a))
      now rewrite (kdm_bindd0).
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
match n ?= w with
| Eq => u1
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (subst_loc x u2 (Bd n)) compare naturals n and w; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
ret (Bd n) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd w))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n))
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
ret (Bd n) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n)) rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
subst_loc x u2 (Bd n) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n)) compose near (Bd n) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
subst_loc x u2 (Bd n) =
(bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) ∘ ret)
  (Bd n)
        rewrite (kdm_bindd0); unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
subst_loc x u2 (Bd n) =
open_loc (u1 '{ x ~> u2}) (incr w (ret (Bd n))) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
ret (Bd n) =
match n ?= w ● Ƶ with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end
        rewrite monoid_id_r.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Lt
ineqp: n < w
ret (Bd n) =
match n ?= w with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end compare naturals n and w.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd w)) compose near (Bd w) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
(bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) ∘ ret)
  (Bd w)
        rewrite (kdm_bindd0).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
(open_loc (u1 '{ x ~> u2}) ○ incr w ∘ ret) (Bd w) unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
open_loc (u1 '{ x ~> u2}) (incr w (ret (Bd w))) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
match w ?= w ● Ƶ with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end
        rewrite monoid_id_r.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w: nat
ineqrw: (w ?= w) = Eq
u1 '{ x ~> u2} =
match w ?= w with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end compare naturals w and w.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) '{ x ~> u2} =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n)) rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
subst_loc x u2 (Bd (n - 1)) =
bindd (open_loc (u1 '{ x ~> u2}) ○ incr w)
  (ret (Bd n))
        compose near (Bd n) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
subst_loc x u2 (Bd (n - 1)) =
(bindd (open_loc (u1 '{ x ~> u2}) ○ incr w) ∘ ret)
  (Bd n)
        rewrite (kdm_bindd0).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
subst_loc x u2 (Bd (n - 1)) =
(open_loc (u1 '{ x ~> u2}) ○ incr w ∘ ret) (Bd n) unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
subst_loc x u2 (Bd (n - 1)) =
open_loc (u1 '{ x ~> u2}) (incr w (ret (Bd n))) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) =
match n ?= w ● Ƶ with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end
        rewrite monoid_id_r.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
x: atom
lcu2: LC u2
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) =
match n ?= w with
| Eq => u1 '{ x ~> u2}
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end compare naturals n and w.
  Qed.

  (* LNgen 10: subst-open *)
  Theorem subst_open:  forall u1 u2 t x,
      LC u2 ->
      t '(u1) '{x ~> u2} =
      t '{x ~> u2} '(u1 '{x ~> u2}).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x,
LC u2 ->
(t '(u1)) '{ x ~> u2} =
(t '{ x ~> u2}) '(u1 '{ x ~> u2})
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u1 u2 t x,
LC u2 ->
(t '(u1)) '{ x ~> u2} =
(t '{ x ~> u2}) '(u1 '{ x ~> u2})
    introv lc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
(t '(u1)) '{ x ~> u2} =
(t '{ x ~> u2}) '(u1 '{ x ~> u2})
    compose near t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
(subst x u2 ∘ open u1) t =
(open (u1 '{ x ~> u2}) ∘ subst x u2) t
    unfold open, subst at 1 3.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
(bind (subst_loc x u2) ∘ bindd (open_loc u1)) t =
(bindd (open_loc (u1 '{ x ~> u2}))
 ∘ bind (subst_loc x u2)) t
    rewrite (bind_bindd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
bindd (subst_loc x u2 ⋆ open_loc u1) t =
(bindd (open_loc (u1 '{ x ~> u2}))
 ∘ bind (subst_loc x u2)) t
    rewrite (bindd_bind).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
bindd (subst_loc x u2 ⋆ open_loc u1) t =
bindd
  ((fun '(w, t) =>
    bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
   ∘ map (subst_loc x u2)) t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
subst_loc x u2 ⋆ open_loc u1 =
(fun '(w, t) =>
 bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
∘ map (subst_loc x u2)
    change_left (subst x u2 ∘ open_loc u1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
subst x u2 ∘ open_loc u1 =
(fun '(w, t) =>
 bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
∘ map (subst_loc x u2)
    #[local] Set Keyed Unification.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
subst x u2 ∘ open_loc u1 =
(fun '(w, t) =>
 bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
∘ map (subst_loc x u2)
    rewrite subst_open_local; [|auto].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract =
(fun '(w, t) =>
 bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
∘ map (subst_loc x u2)
    #[local] Unset Keyed Unification.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u1, u2: T LN
t: U LN
x: atom
lc: LC u2
open_loc (u1 '{ x ~> u2}) ⋆5 subst_loc x u2 ∘ extract =
(fun '(w, t) =>
 bindd (open_loc (u1 '{ x ~> u2}) ⦿ w) t)
∘ map (subst_loc x u2)
    now ext [w t'].
  Qed.

  (** ** Closing followed by substitution *)
  (**************************************************************************)
  Lemma close_notin_FV: forall y (u: T LN) depth,
      LC u ->
      ~ (y `in` FV u) ->
      mapd (close_loc y ⦿ depth) u = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (y : AtomSet.elt) u (depth : nat),
LC u ->
y `notin` FV u -> mapd (close_loc y ⦿ depth) u = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (y : AtomSet.elt) u (depth : nat),
LC u ->
y `notin` FV u -> mapd (close_loc y ⦿ depth) u = u
    introv HLC Hnin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
mapd (close_loc y ⦿ depth) u = u
    change u with (id u) at 2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
mapd (close_loc y ⦿ depth) u = id u
    rewrite <- kdf_mapd1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
mapd (close_loc y ⦿ depth) u = mapd extract u
    apply mapd_respectful.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
forall (e : nat) (a : LN),
(e, a) ∈d u ->
(close_loc y ⦿ depth) (e, a) = extract (e, a)
    introv Hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: LN
Hin: (e, a) ∈d u
(close_loc y ⦿ depth) (e, a) = extract (e, a)
    compare a to atom y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
Hin: (e, Fr y) ∈d u
(close_loc y ⦿ depth) (e, Fr y) = extract (e, Fr y)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
(close_loc y ⦿ depth) (e, Fr a) = extract (e, Fr a)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e, n: nat
Hin: (e, Bd n) ∈d u
(close_loc y ⦿ depth) (e, Bd n) = extract (e, Bd n)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
Hin: (e, Fr y) ∈d u
(close_loc y ⦿ depth) (e, Fr y) = extract (e, Fr y) false.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
Hin: (e, Fr y) ∈d u
False
      apply ind_implies_in in Hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
Hin: Fr y ∈ u
False
      rewrite <- free_iff_FV in Hnin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: ~ y ∈ free u
e: nat
Hin: Fr y ∈ u
False
      rewrite in_free_iff in Hnin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: ~ Fr y ∈ u
e: nat
Hin: Fr y ∈ u
False
      easy.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
(close_loc y ⦿ depth) (e, Fr a) = extract (e, Fr a) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
(if y == a then Bd (depth ● e) else Fr a) = Fr a
      compare values y and a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
DESTR_NEQ: y <> a
DESTR_NEQs: a <> y
(if y == a then Bd (depth ● e) else Fr a) = Fr a
      destruct (@eq_dec Names.atom _ y a).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
DESTR_NEQ: y <> a
DESTR_NEQs: a <> y
e0: y = a
Bd (depth ● e) = Fr a
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
DESTR_NEQ: y <> a
DESTR_NEQs: a <> y
n: y <> a
Fr a = Fr a false.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e: nat
a: atom
Hin: (e, Fr a) ∈d u
H: a <> y
DESTR_NEQ: y <> a
DESTR_NEQs: a <> y
n: y <> a
Fr a = Fr a
      easy.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e, n: nat
Hin: (e, Bd n) ∈d u
(close_loc y ⦿ depth) (e, Bd n) = extract (e, Bd n) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e, n: nat
Hin: (e, Bd n) ∈d u
match n ?= depth ● e with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n unfold transparent tcs.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: LC u
Hnin: y `notin` FV u
e, n: nat
Hin: (e, Bd n) ∈d u
match n ?= depth + e with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n
      rewrite LC_spec in HLC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth: nat
HLC: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
Hnin: y `notin` FV u
e, n: nat
Hin: (e, Bd n) ∈d u
match n ?= depth + e with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n
      specialize (HLC _ _ Hin).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth, e, n: nat
HLC: lc_loc 0 (e, Bd n)
Hnin: y `notin` FV u
Hin: (e, Bd n) ∈d u
match n ?= depth + e with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n
      cbn in HLC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: AtomSet.elt
u: T LN
depth, e, n: nat
HLC: n < e + 0
Hnin: y `notin` FV u
Hin: (e, Bd n) ∈d u
match n ?= depth + e with
| Lt => Bd n
| _ => Bd (S n)
end = Bd n
      compare naturals n and (depth + e)%nat.
  Qed.

  Lemma subst_close_local: forall x y u depth l,
      x <> y ->
      ~ y `in` FV u ->
      LC u ->
      subst_loc x u (close_loc y (depth, l)) =
        mapd (close_loc y ⦿ depth) (subst_loc x u l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x y : atom) u (depth : nat) l,
x <> y ->
y `notin` FV u ->
LC u ->
subst_loc x u (close_loc y (depth, l)) =
mapd (close_loc y ⦿ depth) (subst_loc x u l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (x y : atom) u (depth : nat) l,
x <> y ->
y `notin` FV u ->
LC u ->
subst_loc x u (close_loc y (depth, l)) =
mapd (close_loc y ⦿ depth) (subst_loc x u l)
    introv Hneq lc Hnotin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
l: LN
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u (close_loc y (depth, l)) =
mapd (close_loc y ⦿ depth) (subst_loc x u l)
    compare l to atom y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u (close_loc y (depth, Fr y)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr y))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
subst_loc x u (close_loc y (depth, Fr a)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr a))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u (close_loc y (depth, Bd n)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u (close_loc y (depth, Fr y)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr y)) rewrite close_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u (Bd depth) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr y))
      compare values x and y.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
subst_loc x u (Bd depth) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr y))
      rewrite subst_loc_neq; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
subst_loc x u (Bd depth) =
mapd (close_loc y ⦿ depth) (ret (Fr y))
      rewrite subst_loc_b.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
mapd (close_loc y ⦿ depth) (ret (Fr y))
      compose near (Fr y) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
(mapd (close_loc y ⦿ depth) ∘ ret) (Fr y)
      rewrite mapd_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
(ret ∘ close_loc y ⦿ depth ∘ ret) (Fr y)
      reassociate -> on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
(ret ∘ (close_loc y ⦿ depth ∘ ret)) (Fr y)
      rewrite preincr_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
(ret ∘ (close_loc y ∘ pair depth)) (Fr y)
      unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) = ret (close_loc y (depth, Fr y)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
DESTR_NEQ: x <> y
DESTR_NEQs: y <> x
ret (Bd depth) =
ret (if y == y then Bd depth else Fr y)
      now compare values y and y.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
subst_loc x u (close_loc y (depth, Fr a)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr a)) rewrite close_loc_neq; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
subst_loc x u (Fr a) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr a))
      compare values x and a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: atom
u: T LN
depth: nat
a: atom
Hneq: a <> y
lc: y `notin` FV u
Hnotin: LC u
H: a <> y
DESTR_EQs: a = a
subst_loc a u (Fr a) =
mapd (close_loc y ⦿ depth) (subst_loc a u (Fr a))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
subst_loc x u (Fr a) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr a))
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: atom
u: T LN
depth: nat
a: atom
Hneq: a <> y
lc: y `notin` FV u
Hnotin: LC u
H: a <> y
DESTR_EQs: a = a
subst_loc a u (Fr a) =
mapd (close_loc y ⦿ depth) (subst_loc a u (Fr a)) rewrite subst_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
y: atom
u: T LN
depth: nat
a: atom
Hneq: a <> y
lc: y `notin` FV u
Hnotin: LC u
H: a <> y
DESTR_EQs: a = a
u = mapd (close_loc y ⦿ depth) u
        rewrite close_notin_FV; auto.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
subst_loc x u (Fr a) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Fr a)) rewrite subst_loc_neq; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) = mapd (close_loc y ⦿ depth) (ret (Fr a))
        compose near (Fr a) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) = (mapd (close_loc y ⦿ depth) ∘ ret) (Fr a)
        rewrite mapd_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) = (ret ∘ close_loc y ⦿ depth ∘ ret) (Fr a)
        reassociate -> on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) =
(ret ∘ (close_loc y ⦿ depth ∘ ret)) (Fr a)
        rewrite preincr_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) = (ret ∘ (close_loc y ∘ pair depth)) (Fr a)
        unfold compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
a: atom
H: a <> y
DESTR_NEQ: x <> a
DESTR_NEQs: a <> x
ret (Fr a) = ret (if y == a then Bd depth else Fr a)
        destruct_eq_args y a.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u (close_loc y (depth, Bd n)) =
mapd (close_loc y ⦿ depth) (subst_loc x u (Bd n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end = mapd (close_loc y ⦿ depth) (ret (Bd n))
      compose near (Bd n) on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end = (mapd (close_loc y ⦿ depth) ∘ ret) (Bd n)
      rewrite mapd_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end = (ret ∘ close_loc y ⦿ depth ∘ ret) (Bd n)
      reassociate -> on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end = (ret ∘ (close_loc y ⦿ depth ∘ ret)) (Bd n)
      rewrite preincr_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end = (ret ∘ (close_loc y ∘ pair depth)) (Bd n)
      unfold compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x, y: atom
u: T LN
depth: nat
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
n: nat
subst_loc x u
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end =
ret
  match n ?= depth with
  | Lt => Bd n
  | _ => Bd (S n)
  end
      compare naturals n and depth.
  Qed.

  (* LNgen 13: subst-close *)
  Theorem subst_close:  forall (u: T LN) (t: U LN) x y,
      x <> y ->
      ~ y `in` FV u ->
      LC u ->
      ('[y] t) '{x ~> u} =
        '[y] (t '{x ~> u}).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t (x y : atom),
x <> y ->
y `notin` FV u ->
LC u -> ('[y] t) '{ x ~> u} = '[y] (t '{ x ~> u})
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t (x y : atom),
x <> y ->
y `notin` FV u ->
LC u -> ('[y] t) '{ x ~> u} = '[y] (t '{ x ~> u})
    introv Hneq lc Hnotin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
('[y] t) '{ x ~> u} = '[y] (t '{ x ~> u})
    compose near t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
(subst x u ∘ close y) t = (close y ∘ subst x u) t
    unfold close, subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
(bind (subst_loc x u) ∘ mapd (close_loc y)) t =
(mapd (close_loc y) ∘ bind (subst_loc x u)) t
    rewrite (bind_mapd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
bindd (subst_loc x u ∘ close_loc y) t =
(mapd (close_loc y) ∘ bind (subst_loc x u)) t
    rewrite (mapd_bind).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
bindd (subst_loc x u ∘ close_loc y) t =
bindd
  (fun '(w, a) =>
   mapd (close_loc y ⦿ w) (subst_loc x u a)) t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u ∘ close_loc y =
(fun '(w, a) =>
 mapd (close_loc y ⦿ w) (subst_loc x u a)) unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
subst_loc x u ○ close_loc y =
(fun '(w, a) =>
 mapd (close_loc y ⦿ w) (subst_loc x u a))
    ext [depth l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
Hneq: x <> y
lc: y `notin` FV u
Hnotin: LC u
depth: nat
l: LN
subst_loc x u (close_loc y (depth, l)) =
mapd (close_loc y ⦿ depth) (subst_loc x u l)
    apply subst_close_local; auto.
  Qed.

  (** ** Decompose opening into variable opening followed by substitution *)
  (**************************************************************************)
  Lemma open_spec_local: forall u x w l,
      l <> Fr x ->
      (subst x u ∘ open_loc (ret (Fr x))) (w, l) =
      open_loc u (w, l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u x w l,
l <> Fr x ->
(subst x u ∘ open_loc (ret (Fr x))) (w, l) =
open_loc u (w, l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u x w l,
l <> Fr x ->
(subst x u ∘ open_loc (ret (Fr x))) (w, l) =
open_loc u (w, l)
    introv neq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
l: LN
neq: l <> Fr x
(subst x u ∘ open_loc (ret (Fr x))) (w, l) =
open_loc u (w, l) unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
l: LN
neq: l <> Fr x
open_loc (ret (Fr x)) (w, l) '{ x ~> u} =
open_loc u (w, l) compare l to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
neq: Fr x <> Fr x
open_loc (ret (Fr x)) (w, Fr x) '{ x ~> u} =
open_loc u (w, Fr x)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
open_loc (ret (Fr x)) (w, Fr a) '{ x ~> u} =
open_loc u (w, Fr a)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
open_loc (ret (Fr x)) (w, Bd n) '{ x ~> u} =
open_loc u (w, Bd n)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
neq: Fr x <> Fr x
open_loc (ret (Fr x)) (w, Fr x) '{ x ~> u} =
open_loc u (w, Fr x) contradiction.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
open_loc (ret (Fr x)) (w, Fr a) '{ x ~> u} =
open_loc u (w, Fr a) autorewrite with tea_local.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
ret (Fr a) '{ x ~> u} = ret (Fr a) rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
subst_loc x u (Fr a) = ret (Fr a)
      now rewrite subst_loc_neq.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
open_loc (ret (Fr x)) (w, Bd n) '{ x ~> u} =
open_loc u (w, Bd n) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
match n ?= w with
| Eq => ret (Fr x)
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end '{ x ~> u} =
match n ?= w with
| Eq => u
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Lt
ineqp: n < w
ret (Bd n) '{ x ~> u} = ret (Bd n)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
ineqrw: (w ?= w) = Eq
neq: Bd w <> Fr x
ret (Fr x) '{ x ~> u} = u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) '{ x ~> u} = ret (Bd (n - 1))
      now rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w: nat
ineqrw: (w ?= w) = Eq
neq: Bd w <> Fr x
ret (Fr x) '{ x ~> u} = u
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) '{ x ~> u} = ret (Bd (n - 1))
      now rewrite subst_ret, subst_loc_eq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Gt
ineqp: n > w
ret (Bd (n - 1)) '{ x ~> u} = ret (Bd (n - 1))
      now rewrite subst_ret, 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. *)
  (* LNgen 14: subst-intro *)
  Theorem open_spec: forall u t x,
      ~ x `in` FV t ->
      t '(u) = t '(ret (Fr x)) '{x ~> u}.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t x,
x `notin` FV t ->
t '(u) = (t '(ret (Fr x))) '{ x ~> u}
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t x,
x `notin` FV t ->
t '(u) = (t '(ret (Fr x))) '{ x ~> u}
    introv fresh.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
t '(u) = (t '(ret (Fr x))) '{ x ~> u} compose near t on right.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
t '(u) = (subst x u ∘ open (ret (Fr x))) t
    unfold subst, open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
bindd (open_loc u) t =
(bind (subst_loc x u) ∘ bindd (open_loc (ret (Fr x))))
  t
    rewrite (bind_bindd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
bindd (open_loc u) t =
bindd (subst_loc x u ⋆ open_loc (ret (Fr x))) t
    apply (bindd_respectful).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
forall w (a : LN),
(w, a) ∈d t ->
open_loc u (w, a) =
(subst_loc x u ⋆ open_loc (ret (Fr x))) (w, a) introv hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: (w, a) ∈d t
open_loc u (w, a) =
(subst_loc x u ⋆ open_loc (ret (Fr x))) (w, a)
    assert (a <> Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: (w, a) ∈d t
a <> Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: (w, a) ∈d t
H: a <> Fr x
open_loc u (w, a) =
(subst_loc x u ⋆ open_loc (ret (Fr x))) (w, a)
    {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: (w, a) ∈d t
a <> Fr x apply (ind_implies_in) in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: a ∈ t
a <> Fr x
      rewrite <- free_iff_FV in fresh.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: ~ x ∈ free t
w: nat
a: LN
hin: a ∈ t
a <> Fr x
      eapply ninf_in_neq in fresh; eauto. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x: atom
fresh: x `notin` FV t
w: nat
a: LN
hin: (w, a) ∈d t
H: a <> Fr x
open_loc u (w, a) =
(subst_loc x u ⋆ open_loc (ret (Fr x))) (w, a)
    now rewrite <- (open_spec_local u x).
  Qed.

  (** ** Opening by a variable, followed by non-equal substitution *)
  (**************************************************************************)
  Lemma subst_open_var_loc: forall u x y,
      x <> y ->
      LC u ->
      subst x u ∘ open_loc (ret (Fr y)) =
      open_loc (ret (Fr y)) ⋆5 (subst_loc x u ∘ extract).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u (x y : atom),
x <> y ->
LC u ->
subst x u ∘ open_loc (ret (Fr y)) =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract
  Proof with auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u (x y : atom),
x <> y ->
LC u ->
subst x u ∘ open_loc (ret (Fr y)) =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract
    introv neq lc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
neq: x <> y
lc: LC u
subst x u ∘ open_loc (ret (Fr y)) =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract rewrite subst_open_local...
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
neq: x <> y
lc: LC u
open_loc (ret (Fr y) '{ x ~> u})
⋆5 subst_loc x u ∘ extract =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract
    rewrite subst_ret.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
neq: x <> y
lc: LC u
open_loc (subst_loc x u (Fr y))
⋆5 subst_loc x u ∘ extract =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
x, y: atom
neq: x <> y
lc: LC u
open_loc (if x == y then u else ret (Fr y))
⋆5 subst_loc x u ∘ extract =
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract compare values x and y.
  Qed.

  (* LNgen 11: subst-open-var *)
  Theorem subst_open_var: forall u t x y,
      x <> y ->
      LC u ->
      t '(ret (Fr y)) '{x ~> u} =
      t '{x ~> u} '(ret (Fr y)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t (x y : atom),
x <> y ->
LC u ->
(t '(ret (Fr y))) '{ x ~> u} =
(t '{ x ~> u}) '(ret (Fr y))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall u t (x y : atom),
x <> y ->
LC u ->
(t '(ret (Fr y))) '{ x ~> u} =
(t '{ x ~> u}) '(ret (Fr y))
    introv neq lc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
(t '(ret (Fr y))) '{ x ~> u} =
(t '{ x ~> u}) '(ret (Fr y)) compose near t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
(subst x u ∘ open (ret (Fr y))) t =
(open (ret (Fr y)) ∘ subst x u) t
    unfold open, subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
(bind (subst_loc x u) ∘ bindd (open_loc (ret (Fr y))))
  t =
(bindd (open_loc (ret (Fr y))) ∘ bind (subst_loc x u))
  t
    rewrite (bind_bindd), (bindd_bind).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
bindd (subst_loc x u ⋆ open_loc (ret (Fr y))) t =
bindd
  ((fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
   ∘ map (subst_loc x u)) t
    fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
subst_loc x u ⋆ open_loc (ret (Fr y)) =
(fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
∘ map (subst_loc x u)
    change_left (subst x u ∘ open_loc (ret (Fr y))).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
subst x u ∘ open_loc (ret (Fr y)) =
(fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
∘ map (subst_loc x u)
    #[local] Set Keyed Unification.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
subst x u ∘ open_loc (ret (Fr y)) =
(fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
∘ map (subst_loc x u)
    rewrite subst_open_var_loc; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract =
(fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
∘ map (subst_loc x u)
    #[local] Unset Keyed Unification.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
u: T LN
t: U LN
x, y: atom
neq: x <> y
lc: LC u
open_loc (ret (Fr y)) ⋆5 subst_loc x u ∘ extract =
(fun '(w, t) => bindd (open_loc (ret (Fr y)) ⦿ w) t)
∘ map (subst_loc x u)
    now ext [w t'].
  Qed.

  (* LNgen 12: doesn't make sense here because it mentions concrete syntax *)

  (** ** Closing, followed by opening *)
  (**************************************************************************)
  Lemma open_close_local: forall x w l,
      (open_loc (ret (Fr x)) ⋆1 close_loc x)
      (w, l) = ret l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x w l,
(open_loc (ret (Fr x)) ⋆1 close_loc x) (w, l) = ret l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x w l,
(open_loc (ret (Fr x)) ⋆1 close_loc x) (w, l) = ret l
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
(open_loc (ret (Fr x)) ⋆1 close_loc x) (w, l) = ret l cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
match
  match l with
  | Fr y => if x == y then Bd w else Fr y
  | Bd n =>
      match n ?= w with
      | Lt => Bd n
      | _ => Bd (S n)
      end
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret l unfold id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
match
  match l with
  | Fr y => if x == y then Bd w else Fr y
  | Bd n =>
      match n ?= w with
      | Lt => Bd n
      | _ => Bd (S n)
      end
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret l compare l to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
match (if x == x then Bd w else Fr x) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Fr x)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
a: atom
H: a <> x
match (if x == a then Bd w else Fr a) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Fr a)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
match
  match n ?= w with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Bd n)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
match (if x == x then Bd w else Fr x) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Fr x) compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
DESTR_EQ, DESTR_EQs: x = x
match w ?= w with
| Eq => ret (Fr x)
| Lt => ret (Bd w)
| Gt => ret (Bd (w - 1))
end = ret (Fr x) compare naturals w and w.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
a: atom
H: a <> x
match (if x == a then Bd w else Fr a) with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Fr a) compare values x and a.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
match
  match n ?= w with
  | Lt => Bd n
  | _ => Bd (S n)
  end
with
| Fr x => ret (Fr x)
| Bd n =>
    match n ?= w with
    | Eq => ret (Fr x)
    | Lt => ret (Bd n)
    | Gt => ret (Bd (n - 1))
    end
end = ret (Bd n) compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
ineqrw: (w ?= w) = Eq
match S w ?= w with
| Eq => ret (Fr x)
| Lt => ret (Bd (S w))
| Gt => ret (Bd (S w - 1))
end = ret (Bd w)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
match S n ?= w with
| Eq => ret (Fr x)
| Lt => ret (Bd (S n))
| Gt => ret (Bd (S n - 1))
end = ret (Bd n)
      compare naturals (S w) and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
ineqrw: (n ?= w) = Gt
ineqp: n > w
match S n ?= w with
| Eq => ret (Fr x)
| Lt => ret (Bd (S n))
| Gt => ret (Bd (S n - 1))
end = ret (Bd n)
      compare naturals (S n) and w.
  Qed.

  (* LNgen 15: open_close *)
  Theorem open_close: forall x t,
      ('[x] t) '(ret (Fr x)) = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x t, ('[x] t) '(ret (Fr x)) = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x t, ('[x] t) '(ret (Fr x)) = t
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
('[x] t) '(ret (Fr x)) = t compose near t on left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
(open (ret (Fr x)) ∘ close x) t = t
    unfold open, close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
(bindd (open_loc (ret (Fr x))) ∘ mapd (close_loc x)) t =
t
    rewrite (bindd_mapd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
bindd (open_loc (ret (Fr x)) ⋆1 close_loc x) t = t
    apply (bindd_respectful_id); intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
w: nat
a: LN
H: (w, a) ∈d t
(open_loc (ret (Fr x)) ⋆1 close_loc x) (w, a) = ret a
    auto using open_close_local.
  Qed.

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

  Lemma open_LN_loc_spec: forall (x: LN),
      open_loc (ret x) = ret (T := T) ∘ open_LN_loc x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x : LN, open_loc (ret x) = ret ∘ open_LN_loc x
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x : LN, open_loc (ret x) = ret ∘ open_LN_loc x
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
open_loc (ret x) = ret ∘ open_LN_loc x ext [depth l].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth: nat
l: LN
open_loc (ret x) (depth, l) =
(ret ∘ open_LN_loc x) (depth, l) unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth: nat
l: LN
open_loc (ret x) (depth, l) =
ret (open_LN_loc x (depth, l))
    destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth: nat
n: atom
open_loc (ret x) (depth, Fr n) =
ret (open_LN_loc x (depth, Fr n))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth, n: nat
open_loc (ret x) (depth, Bd n) =
ret (open_LN_loc x (depth, Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth: nat
n: atom
open_loc (ret x) (depth, Fr n) =
ret (open_LN_loc x (depth, Fr n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth: nat
n: atom
ret (Fr n) = ret (Fr n) reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth, n: nat
open_loc (ret x) (depth, Bd n) =
ret (open_LN_loc x (depth, Bd n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: LN
depth, n: nat
match n ?= depth with
| Eq => ret x
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end =
ret
  match n ?= depth with
  | Eq => x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end compare naturals n and depth.
  Qed.

  Lemma open_by_LN_spec: forall l,
      open (ret l) = mapd (open_LN_loc l).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l, open (ret l) = mapd (open_LN_loc l)
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall l, open (ret l) = mapd (open_LN_loc l)
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
open (ret l) = mapd (open_LN_loc l) unfold open.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
bindd (open_loc (ret l)) = mapd (open_LN_loc l) ext t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
bindd (open_loc (ret l)) t = mapd (open_LN_loc l) t
    apply (bindd_respectful_mapd).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
forall w (a : LN),
(w, a) ∈d t ->
open_loc (ret l) (w, a) = ret (open_LN_loc l (w, a))
    intros w l' l'in.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w: nat
l': LN
l'in: (w, l') ∈d t
open_loc (ret l) (w, l') = ret (open_LN_loc l (w, l')) destruct l'.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w: nat
n: atom
l'in: (w, Fr n) ∈d t
open_loc (ret l) (w, Fr n) =
ret (open_LN_loc l (w, Fr n))
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w, n: nat
l'in: (w, Bd n) ∈d t
open_loc (ret l) (w, Bd n) =
ret (open_LN_loc l (w, Bd n))
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w: nat
n: atom
l'in: (w, Fr n) ∈d t
open_loc (ret l) (w, Fr n) =
ret (open_LN_loc l (w, Fr n)) reflexivity.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w, n: nat
l'in: (w, Bd n) ∈d t
open_loc (ret l) (w, Bd n) =
ret (open_LN_loc l (w, Bd n)) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l: LN
t: U LN
w, n: nat
l'in: (w, Bd n) ∈d t
match n ?= w with
| Eq => ret l
| Lt => ret (Bd n)
| Gt => ret (Bd (n - 1))
end =
ret
  match n ?= w with
  | Eq => l
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end compare naturals n and w.
  Qed.

  (** ** Opening, followed by closing *)
  (**************************************************************************)
  Lemma close_open_local: forall x w l,
      l <> Fr x ->
      (close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x w l,
l <> Fr x ->
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x w l,
l <> Fr x ->
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l
    introv neq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
neq: l <> Fr x
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
neq: l <> Fr x
match
  match l with
  | Fr x => Fr x
  | Bd n =>
      match n ?= w with
      | Eq => Fr x
      | Lt => Bd n
      | Gt => Bd (n - 1)
      end
  end
with
| Fr y => if x == y then Bd w else Fr y
| Bd n =>
    match n ?= w with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = l unfold id.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
l: LN
neq: l <> Fr x
match
  match l with
  | Fr x => Fr x
  | Bd n =>
      match n ?= w with
      | Eq => Fr x
      | Lt => Bd n
      | Gt => Bd (n - 1)
      end
  end
with
| Fr y => if x == y then Bd w else Fr y
| Bd n =>
    match n ?= w with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = l compare l to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
neq: Fr x <> Fr x
(if x == x then Bd w else Fr x) = Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
(if x == a then Bd w else Fr a) = Fr a
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
neq: Bd n <> Fr x
match
  match n ?= w with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd w else Fr y
| Bd n =>
    match n ?= w with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
neq: Fr x <> Fr x
(if x == x then Bd w else Fr x) = Fr x contradiction.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
(if x == a then Bd w else Fr a) = Fr a unfold compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
a: atom
neq: Fr a <> Fr x
H: a <> x
(if x == a then Bd w else Fr a) = Fr a now compare values x and a.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
neq: Bd n <> Fr x
match
  match n ?= w with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd w else Fr y
| Bd n =>
    match n ?= w with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n unfold compose; cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
neq: Bd n <> Fr x
match
  match n ?= w with
  | Eq => Fr x
  | Lt => Bd n
  | Gt => Bd (n - 1)
  end
with
| Fr y => if x == y then Bd w else Fr y
| Bd n =>
    match n ?= w with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end = Bd n
      compare naturals n and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w: nat
ineqrw: (w ?= w) = Eq
neq: Bd w <> Fr x
(if x == x then Bd w else Fr x) = Bd w
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Gt
ineqp: n > w
match n - 1 ?= w with
| Lt => Bd (n - 1)
| _ => Bd (S (n - 1))
end = Bd n compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
w, n: nat
neq: Bd n <> Fr x
ineqrw: (n ?= w) = Gt
ineqp: n > w
match n - 1 ?= w with
| Lt => Bd (n - 1)
| _ => Bd (S (n - 1))
end = Bd n
      compare naturals (n - 1) and w.
  Qed.

  (* LNgen 16: close_open *)
  Theorem close_open: forall x t,
      ~ x ∈ free t ->
      '[x] (t '(ret (Fr x))) = t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x t, ~ x ∈ free t -> '[x] (t '(ret (Fr x))) = t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x t, ~ x ∈ free t -> '[x] (t '(ret (Fr x))) = t
    introv fresh.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
'[x] (t '(ret (Fr x))) = t compose near t on left.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
(close x ∘ open (ret (Fr x))) t = t
    rewrite open_by_LN_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
(close x ∘ mapd (open_LN_loc (Fr x))) t = t unfold close.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
(mapd (close_loc x) ∘ mapd (open_LN_loc (Fr x))) t = t
    rewrite (kdf_mapd2).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
mapd (close_loc x ⋆1 open_LN_loc (Fr x)) t = t
    apply (mapd_respectful_id).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
forall (e : nat) (a : LN),
(e, a) ∈d t ->
(close_loc x ⋆1 open_LN_loc (Fr x)) (e, a) = a
    intros w l lin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l
    assert (l <> Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
l <> Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
H: l <> Fr x
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l
    {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
l <> Fr x rewrite neq_symmetry.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
Fr x <> l apply (ind_implies_in) in lin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: l ∈ t
Fr x <> l
      eauto using (ninf_in_neq (T := T)). }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t: U LN
fresh: ~ x ∈ free t
w: nat
l: LN
lin: (w, l) ∈d t
H: l <> Fr x
(close_loc x ⋆1 open_LN_loc (Fr x)) (w, l) = l
    now apply close_open_local.
  Qed.

  (** ** Opening and local closure *)
  (**************************************************************************)
  Lemma open_lcn_1: forall n t u,
      LC u ->
      LCn n t ->
      LCn (n - 1) (t '(u)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u, LC u -> LCn n t -> LCn (n - 1) (t '(u))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u, LC u -> LCn n t -> LCn (n - 1) (t '(u))
    introv lcu lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: LC u
lct: LCn n t
LCn (n - 1) (t '(u))
    rewrite LC_spec, LCn_spec in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
forall w l, (w, l) ∈d t '(u) -> lc_loc (n - 1) (w, l)
    introv Hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
Hin: (w, l) ∈d t '(u)
lc_loc (n - 1) (w, l) rewrite ind_open_iff in Hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
Hin: exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2
lc_loc (n - 1) (w, l)
    destruct Hin as [n1 [n2 [l1 [h1 [h2 h3]]]]].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2: nat
l1: LN
h1: (n1, l1) ∈d t
h2: (n2, l) ∈d open_loc u (n1, l1)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l)
    destruct l1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2: nat
n0: atom
h1: (n1, Fr n0) ∈d t
h2: (n2, l) ∈d open_loc u (n1, Fr n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2, n0: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d open_loc u (n1, Bd n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2: nat
n0: atom
h1: (n1, Fr n0) ∈d t
h2: (n2, l) ∈d open_loc u (n1, Fr n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l) cbn in h2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2: nat
n0: atom
h1: (n1, Fr n0) ∈d t
h2: (n2, l) ∈d ret (Fr n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l) rewrite (ind_ret_iff) in h2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2: nat
n0: atom
h1: (n1, Fr n0) ∈d t
h2: n2 = Ƶ /\ l = Fr n0
h3: w = n1 ● n2
lc_loc (n - 1) (w, l)
      destruct h2; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
n1: nat
n0: atom
h1: (n1, Fr n0) ∈d t
lc_loc (n - 1) (n1 ● Ƶ, Fr n0) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
n1: nat
n0: atom
h1: (n1, Fr n0) ∈d t
True trivial.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l, (w, l) ∈d t -> lc_loc n (w, l)
w: nat
l: LN
n1, n2, n0: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d open_loc u (n1, Bd n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l) specialize (lct _ _ h1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d open_loc u (n1, Bd n0)
h3: w = n1 ● n2
lc_loc (n - 1) (w, l) cbn in h2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l)
∈d match n0 ?= n1 with
   | Eq => u
   | Lt => ret (Bd n0)
   | Gt => ret (Bd (n0 - 1))
   end
h3: w = n1 ● n2
lc_loc (n - 1) (w, l) compare naturals n0 and n1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d ret (Bd n0)
h3: w = n1 ● n2
ineqrw: (n0 ?= n1) = Lt
ineqp: n0 < n1
lc_loc (n - 1) (w, l)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1: nat
lct: lc_loc n (n1, Bd n1)
l: LN
n2: nat
h1: (n1, Bd n1) ∈d t
h2: (n2, l) ∈d u
ineqrw: (n1 ?= n1) = Eq
lc_loc (n - 1) (n1 ● n2, l)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d ret (Bd (n0 - 1))
h3: w = n1 ● n2
ineqrw: (n0 ?= n1) = Gt
ineqp: n0 > n1
lc_loc (n - 1) (w, l)
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d ret (Bd n0)
h3: w = n1 ● n2
ineqrw: (n0 ?= n1) = Lt
ineqp: n0 < n1
lc_loc (n - 1) (w, l) rewrite (ind_ret_iff) in h2; destruct h2; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
h1: (n1, Bd n0) ∈d t
ineqrw: (n0 ?= n1) = Lt
ineqp: n0 < n1
lc_loc (n - 1) (n1 ● Ƶ, Bd n0)
        cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
h1: (n1, Bd n0) ∈d t
ineqrw: (n0 ?= n1) = Lt
ineqp: n0 < n1
n0 < (n1 ● Ƶ) + (n - 1) unfold_monoid.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
h1: (n1, Bd n0) ∈d t
ineqrw: (n0 ?= n1) = Lt
ineqp: n0 < n1
n0 < n1 + 0 + (n - 1) lia.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1: nat
lct: lc_loc n (n1, Bd n1)
l: LN
n2: nat
h1: (n1, Bd n1) ∈d t
h2: (n2, l) ∈d u
ineqrw: (n1 ?= n1) = Eq
lc_loc (n - 1) (n1 ● n2, l) specialize (lcu n2 l h2).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
l: LN
n2: nat
lcu: lc_loc 0 (n2, l)
n1: nat
lct: lc_loc n (n1, Bd n1)
h1: (n1, Bd n1) ∈d t
h2: (n2, l) ∈d u
ineqrw: (n1 ?= n1) = Eq
lc_loc (n - 1) (n1 ● n2, l) unfold lc_loc in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
l: LN
n2: nat
lcu: match l with
| Fr _ => True
| Bd n => n < n2 + 0
end
n1: nat
lct: n1 < n1 + n
h1: (n1, Bd n1) ∈d t
h2: (n2, l) ∈d u
ineqrw: (n1 ?= n1) = Eq
match l with
| Fr _ => True
| Bd n0 => n0 < (n1 ● n2) + (n - 1)
end
        destruct l; [trivial|].
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
n0, n2: nat
lcu: n0 < n2 + 0
n1: nat
lct: n1 < n1 + n
h1: (n1, Bd n1) ∈d t
h2: (n2, Bd n0) ∈d u
ineqrw: (n1 ?= n1) = Eq
n0 < (n1 ● n2) + (n - 1) unfold_monoid.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
n0, n2: nat
lcu: n0 < n2 + 0
n1: nat
lct: n1 < n1 + n
h1: (n1, Bd n1) ∈d t
h2: (n2, Bd n0) ∈d u
ineqrw: (n1 ?= n1) = Eq
n0 < n1 + n2 + (n - 1) lia.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: (n2, l) ∈d ret (Bd (n0 - 1))
h3: w = n1 ● n2
ineqrw: (n0 ?= n1) = Gt
ineqp: n0 > n1
lc_loc (n - 1) (w, l) rewrite (ind_ret_iff) in h2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
w: nat
l: LN
n2: nat
h1: (n1, Bd n0) ∈d t
h2: n2 = Ƶ /\ l = Bd (n0 - 1)
h3: w = n1 ● n2
ineqrw: (n0 ?= n1) = Gt
ineqp: n0 > n1
lc_loc (n - 1) (w, l) destruct h2; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
n1, n0: nat
lct: lc_loc n (n1, Bd n0)
h1: (n1, Bd n0) ∈d t
ineqrw: (n0 ?= n1) = Gt
ineqp: n0 > n1
lc_loc (n - 1) (n1 ● Ƶ, Bd (n0 - 1))
        unfold lc_loc in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
lcu: forall w l,
(w, l) ∈d u ->
match l with
| Fr _ => True
| Bd n => n < w + 0
end
n1, n0: nat
lct: n0 < n1 + n
h1: (n1, Bd n0) ∈d t
ineqrw: (n0 ?= n1) = Gt
ineqp: n0 > n1
n0 - 1 < (n1 ● Ƶ) + (n - 1) unfold_lia.
  Qed.

  Lemma open_lcn_2: forall n t u,
      n > 0 ->
      LC u ->
      LCn (n - 1) (t '(u)) ->
      LCn n t.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u,
n > 0 -> LC u -> LCn (n - 1) (t '(u)) -> LCn n t
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u,
n > 0 -> LC u -> LCn (n - 1) (t '(u)) -> LCn n t
    introv ngt lcu lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: LC u
lct: LCn (n - 1) (t '(u))
LCn n t
    rewrite LC_spec, LCn_spec in *.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(w, l) ∈d t '(u) -> lc_loc (n - 1) (w, l)
forall w l, (w, l) ∈d t -> lc_loc n (w, l)
    introv Hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(w, l) ∈d t '(u) -> lc_loc (n - 1) (w, l)
w: nat
l: LN
Hin: (w, l) ∈d t
lc_loc n (w, l) setoid_rewrite (ind_open_iff) in lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
l: LN
Hin: (w, l) ∈d t
lc_loc n (w, l)
    destruct l.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
n0: atom
Hin: (w, Fr n0) ∈d t
lc_loc n (w, Fr n0)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
lc_loc n (w, Bd n0)
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
n0: atom
Hin: (w, Fr n0) ∈d t
lc_loc n (w, Fr n0) cbv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
n0: atom
Hin: (w, Fr n0) ∈d t
True trivial.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
lc_loc n (w, Bd n0) compare naturals n0 and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
lc_loc n (w, Bd n0)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
ineqrw: (w ?= w) = Eq
Hin: (w, Bd w) ∈d t
lc_loc n (w, Bd w)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
lc_loc n (w, Bd n0)
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
lc_loc n (w, Bd n0) specialize (lct w (Bd n0)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
lc_loc n (w, Bd n0)
        lapply lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
lc_loc (n - 1) (w, Bd n0) -> lc_loc n (w, Bd n0)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd n0) ∈d open_loc u (n1, l1) /\ w = n1 ● n2
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
lc_loc (n - 1) (w, Bd n0) -> lc_loc n (w, Bd n0) cbn; unfold_lia. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd n0) ∈d open_loc u (n1, l1) /\ w = n1 ● n2
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd n0) ∈d open_loc u (n1, l1) /\ w = n1 ● n2 exists w (Ƶ: nat) (Bd n0).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
(w, Bd n0) ∈d t /\
(Ƶ, Bd n0) ∈d open_loc u (w, Bd n0) /\ w = w ● Ƶ
          split; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
(Ƶ, Bd n0) ∈d open_loc u (w, Bd n0) /\ w = w ● Ƶ cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Lt
ineqp: n0 < w
(Ƶ, Bd n0)
∈d match n0 ?= w with
   | Eq => u
   | Lt => ret (Bd n0)
   | Gt => ret (Bd (n0 - 1))
   end /\ w = w ● Ƶ compare naturals n0 and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: Lt = Lt
ineqp: n0 < w
ineqrw0: (n0 ?= w) = Lt
ineqp0: n0 < w
(Ƶ, Bd n0) ∈d ret (Bd n0) /\ w = w ● Ƶ
          rewrite (ind_ret_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd n0) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) -> lc_loc (n - 1) (w, Bd n0)
Hin: (w, Bd n0) ∈d t
ineqrw: Lt = Lt
ineqp: n0 < w
ineqrw0: (n0 ?= w) = Lt
ineqp0: n0 < w
(Ƶ = Ƶ /\ Bd n0 = Bd n0) /\ w = w ● Ƶ now simpl_monoid. }
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
ineqrw: (w ?= w) = Eq
Hin: (w, Bd w) ∈d t
lc_loc n (w, Bd w) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w: nat
ineqrw: (w ?= w) = Eq
Hin: (w, Bd w) ∈d t
w < w + n unfold_lia.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
lc_loc n (w, Bd n0) cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
lct: forall w l,
(exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, l) ∈d open_loc u (n1, l1) /\ w = n1 ● n2) ->
lc_loc (n - 1) (w, l)
w, n0: nat
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
n0 < w + n specialize (lct w (Bd (n0 - 1))).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
n0 < w + n
        lapply lct.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
lc_loc (n - 1) (w, Bd (n0 - 1)) -> n0 < w + n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
  w = n1 ● n2
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
lc_loc (n - 1) (w, Bd (n0 - 1)) -> n0 < w + n cbn; unfold_lia. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
  w = n1 ● n2
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
exists n1 n2 l1,
  (n1, l1) ∈d t /\
  (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
  w = n1 ● n2 exists w (Ƶ: nat) (Bd n0).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
(w, Bd n0) ∈d t /\
(Ƶ, Bd (n0 - 1)) ∈d open_loc u (w, Bd n0) /\ w = w ● Ƶ
          split; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
(Ƶ, Bd (n0 - 1)) ∈d open_loc u (w, Bd n0) /\ w = w ● Ƶ cbn.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: (n0 ?= w) = Gt
ineqp: n0 > w
(Ƶ, Bd (n0 - 1))
∈d match n0 ?= w with
   | Eq => u
   | Lt => ret (Bd n0)
   | Gt => ret (Bd (n0 - 1))
   end /\ w = w ● Ƶ compare naturals n0 and w.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: Gt = Gt
ineqp: n0 > w
ineqrw0: (n0 ?= w) = Gt
ineqp0: n0 > w
(Ƶ, Bd (n0 - 1)) ∈d ret (Bd (n0 - 1)) /\ w = w ● Ƶ
          rewrite (ind_ret_iff).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
u: T LN
ngt: n > 0
lcu: forall w l, (w, l) ∈d u -> lc_loc 0 (w, l)
w, n0: nat
lct: (exists n1 n2 l1,
   (n1, l1) ∈d t /\
   (n2, Bd (n0 - 1)) ∈d open_loc u (n1, l1) /\
   w = n1 ● n2) ->
lc_loc (n - 1) (w, Bd (n0 - 1))
Hin: (w, Bd n0) ∈d t
ineqrw: Gt = Gt
ineqp: n0 > w
ineqrw0: (n0 ?= w) = Gt
ineqp0: n0 > w
(Ƶ = Ƶ /\ Bd (n0 - 1) = Bd (n0 - 1)) /\ w = w ● Ƶ now simpl_monoid. }
  Qed.

  Theorem open_lcn_iff: forall n t u,
      n > 0 ->
      LC u ->
      LCn n t <->
      LCn (n - 1) (t '(u)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u,
n > 0 -> LC u -> LCn n t <-> LCn (n - 1) (t '(u))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t u,
n > 0 -> LC u -> LCn n t <-> LCn (n - 1) (t '(u))
    intros; intuition (eauto using open_lcn_1, open_lcn_2).
  Qed.

  Corollary open_var_lcn: forall n t x,
      n > 0 ->
      LCn n t <->
      LCn (n - 1) (t '(ret (Fr x))).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t x,
n > 0 -> LCn n t <-> LCn (n - 1) (t '(ret (Fr x)))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n t x,
n > 0 -> LCn n t <-> LCn (n - 1) (t '(ret (Fr x)))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
LCn n t <-> LCn (n - 1) (t '(ret (Fr x))) apply open_lcn_iff.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
n > 0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
LC (ret (Fr x)) auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
LC (ret (Fr x))
    rewrite LC_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
forall w l, (w, l) ∈d ret (Fr x) -> lc_loc 0 (w, l)
    intros w l hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
w: nat
l: LN
hin: (w, l) ∈d ret (Fr x)
lc_loc 0 (w, l) rewrite (ind_ret_iff) in hin.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
w: nat
l: LN
hin: w = Ƶ /\ l = Fr x
lc_loc 0 (w, l)
    destruct hin; subst.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
lc_loc 0 (Ƶ, Fr x) cbv.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
t: U LN
x: atom
H: n > 0
True trivial.
  Qed.

  Corollary open_lcn_iff_1: forall t u,
      LC u ->
      LCn 1 t <->
      LC (t '(u)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, LC u -> LCn 1 t <-> LC (t '(u))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t u, LC u -> LCn 1 t <-> LC (t '(u))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: LC u
LCn 1 t <-> LC (t '(u)) unfold LC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: LC u
LCn 1 t <-> LCn 0 (t '(u))
    change 0 with (1 - 1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: LC u
LCn (S (1 - 1)) t <-> LCn (1 - 1) (t '(u))
    rewrite <- open_lcn_iff; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
u: T LN
H: LC u
LCn (S (1 - 1)) t <-> LCn 1 t
    reflexivity.
  Qed.

  Corollary lc_open_var: forall t x,
      LCn 1 t <->
      LC (t '(ret (Fr x))).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, LCn 1 t <-> LC (t '(ret (Fr x)))
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall t x, LCn 1 t <-> LC (t '(ret (Fr x)))
    intros.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
LCn 1 t <-> LC (t '(ret (Fr x))) unfold LC.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
LCn 1 t <-> LCn 0 (t '(ret (Fr x)))
    change 0 with (1 - 1).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
LCn (S (1 - 1)) t <-> LCn (1 - 1) (t '(ret (Fr x)))
    rewrite <- open_var_lcn; auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U LN
x: atom
LCn (S (1 - 1)) t <-> LCn 1 t
    reflexivity.
  Qed.

  (** ** Injectivity of opening *)
  (******************************************************************************)

  (* LNgen 17: open-inj *)
  Lemma open_inj: forall (x: atom) (t u: U LN),
      ~ x `in` (FV t ∪ FV u) ->
      t '(ret (Fr x)) = u '(ret (Fr x)) ->
      t = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x (t u : U LN),
x `notin` (FV t ∪ FV u) ->
t '(ret (Fr x)) = u '(ret (Fr x)) -> t = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x (t u : U LN),
x `notin` (FV t ∪ FV u) ->
t '(ret (Fr x)) = u '(ret (Fr x)) -> t = u
    introv Hnin Heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: t '(ret (Fr x)) = u '(ret (Fr x))
t = u
    rewrite open_by_LN_spec in Heq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
t = u
    apply (mapd_injective2 (open_LN_loc (Fr x))); auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
forall (e : nat) (a1 a2 : LN),
(e, a1) ∈d t ->
(e, a2) ∈d u ->
open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2) -> a1 = a2
    intros e a1 a2 Hint1 Hint2 Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
a1 = a2
    assert (Heq_loc2:
             open_loc (ret (T := T) (Fr x)) (e, a1) =
               open_loc (ret (Fr x)) (e, a2)).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 = a2
    {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2) rewrite open_LN_loc_spec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
(ret ∘ open_LN_loc (Fr x)) (e, a1) =
(ret ∘ open_LN_loc (Fr x)) (e, a2)
      unfold compose.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
ret (open_LN_loc (Fr x) (e, a1)) =
ret (open_LN_loc (Fr x) (e, a2)) fequal.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2) rewrite Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
open_LN_loc (Fr x) (e, a2) =
open_LN_loc (Fr x) (e, a2) reflexivity. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 = a2
    enough (Hneq: a1 <> Fr x /\ a2 <> Fr x).
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
Hneq: a1 <> Fr x /\ a2 <> Fr x
a1 = a2
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 <> Fr x /\ a2 <> Fr x
    {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
Hneq: a1 <> Fr x /\ a2 <> Fr x
a1 = a2 destruct Hneq.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
H: a1 <> Fr x
H0: a2 <> Fr x
a1 = a2
      cbn in Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match a1 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
H: a1 <> Fr x
H0: a2 <> Fr x
a1 = a2
      destruct a1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
a2: LN
Hint1: (e, Fr n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: Fr n =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, a2)
H: Fr n <> Fr x
H0: a2 <> Fr x
Fr n = a2
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
a2: LN
Hint1: (e, Bd n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, a2)
H: Bd n <> Fr x
H0: a2 <> Fr x
Bd n = a2
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
a2: LN
Hint1: (e, Fr n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: Fr n =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, a2)
H: Fr n <> Fr x
H0: a2 <> Fr x
Fr n = a2 destruct a2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n, n0: atom
Hint1: (e, Fr n) ∈d t
Hint2: (e, Fr n0) ∈d u
Heq_loc: Fr n = Fr n0
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Fr n0)
H: Fr n <> Fr x
H0: Fr n0 <> Fr x
Fr n = Fr n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
n0: nat
Hint1: (e, Fr n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Fr n <> Fr x
H0: Bd n0 <> Fr x
Fr n = Bd n0
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n, n0: atom
Hint1: (e, Fr n) ∈d t
Hint2: (e, Fr n0) ∈d u
Heq_loc: Fr n = Fr n0
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Fr n0)
H: Fr n <> Fr x
H0: Fr n0 <> Fr x
Fr n = Fr n0 assumption. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
n0: nat
Hint1: (e, Fr n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Fr n <> Fr x
H0: Bd n0 <> Fr x
Fr n = Bd n0
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
n0: nat
Hint1: (e, Fr n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Fr n <> Fr x
H0: Bd n0 <> Fr x
Fr n = Bd n0 false.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
n: atom
n0: nat
Hint1: (e, Fr n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Fr n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Fr n <> Fr x
H0: Bd n0 <> Fr x
False
          destruct (compare n0 e); easy.
        }
      }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
a2: LN
Hint1: (e, Bd n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, a2)
H: Bd n <> Fr x
H0: a2 <> Fr x
Bd n = a2
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
a2: LN
Hint1: (e, Bd n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match a2 with
| Fr x => Fr x
| Bd n =>
    match n ?= e with
    | Eq => Fr x
    | Lt => Bd n
    | Gt => Bd (n - 1)
    end
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, a2)
H: Bd n <> Fr x
H0: a2 <> Fr x
Bd n = a2 destruct a2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
n0: atom
Hint1: (e, Bd n) ∈d t
Hint2: (e, Fr n0) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end = Fr n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Fr n0)
H: Bd n <> Fr x
H0: Fr n0 <> Fr x
Bd n = Fr n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
Bd n = Bd n0
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
n0: atom
Hint1: (e, Bd n) ∈d t
Hint2: (e, Fr n0) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end = Fr n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Fr n0)
H: Bd n <> Fr x
H0: Fr n0 <> Fr x
Bd n = Fr n0 compare naturals n and e; false. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
Bd n = Bd n0
        {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Eq => Fr x
| Lt => Bd n
| Gt => Bd (n - 1)
end =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
Bd n = Bd n0 compare naturals n and e.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
n = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
e = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
n = n0
          {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
n = n0 compare naturals n0 and e.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n = Bd n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (n0 ?= e) = Lt
ineqp0: n0 < e
n = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd e) ∈d u
Heq_loc: Bd n = Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd e)
H: Bd n <> Fr x
H0: Bd e <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (e ?= e) = Eq
n = e
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
n = n0
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n = Bd n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (n0 ?= e) = Lt
ineqp0: n0 < e
n = n0 now inversion Heq_loc.
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd e) ∈d u
Heq_loc: Bd n = Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd e)
H: Bd n <> Fr x
H0: Bd e <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (e ?= e) = Eq
n = e false.
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
n = n0 inversion Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd n = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Lt
ineqp: n < e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
H2: n = n0 - 1
n0 - 1 = n0 lia.
          }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
e = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
n = n0
          {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
e = n0 compare naturals n0 and e.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x = Bd n0
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
ineqrw0: (n0 ?= e) = Lt
ineqp: n0 < e
e = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x = Bd (n0 - 1)
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
ineqrw0: (n0 ?= e) = Gt
ineqp: n0 > e
e = n0
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x = Bd n0
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
ineqrw0: (n0 ?= e) = Lt
ineqp: n0 < e
e = n0 now inversion Heq_loc.
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n0: nat
Hint1: (e, Bd e) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Fr x = Bd (n0 - 1)
H: Bd e <> Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd e) =
open_loc (ret (Fr x)) (e, Bd n0)
H0: Bd n0 <> Fr x
ineqrw: (e ?= e) = Eq
ineqrw0: (n0 ?= e) = Gt
ineqp: n0 > e
e = n0 false.
          }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
n = n0
          {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) =
match n0 ?= e with
| Eq => Fr x
| Lt => Bd n0
| Gt => Bd (n0 - 1)
end
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
n = n0 compare naturals n0 and e.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Lt
ineqp0: n0 < e
n = n0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd e) ∈d u
Heq_loc: Bd (n - 1) = Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd e)
H: Bd n <> Fr x
H0: Bd e <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (e ?= e) = Eq
n = e
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
n = n0
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Lt
ineqp0: n0 < e
n = n0 inversion Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd n0
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Lt
ineqp0: n0 < e
H2: n - 1 = n0
n = n - 1 lia.
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd e) ∈d u
Heq_loc: Bd (n - 1) = Fr x
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd e)
H: Bd n <> Fr x
H0: Bd e <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (e ?= e) = Eq
n = e false.
            -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
n = n0 inversion Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e, n, n0: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Bd n0) ∈d u
Heq_loc: Bd (n - 1) = Bd (n0 - 1)
Heq_loc2: open_loc (ret (Fr x)) (e, Bd n) =
open_loc (ret (Fr x)) (e, Bd n0)
H: Bd n <> Fr x
H0: Bd n0 <> Fr x
ineqrw: (n ?= e) = Gt
ineqp: n > e
ineqrw0: (n0 ?= e) = Gt
ineqp0: n0 > e
H2: n - 1 = n0 - 1
n = n0 lia. }
        }
      }
    }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 <> Fr x /\ a2 <> Fr x
    split.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 <> Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a2 <> Fr x
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a1 <> Fr x assert (cut: ~ (x `in` FV t)) by fsetdec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: x `notin` FV t
a1 <> Fr x
      rewrite <- free_iff_FV in cut.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: ~ x ∈ free t
a1 <> Fr x
      apply ind_implies_in in Hint1.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: a1 ∈ t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: ~ x ∈ free t
a1 <> Fr x
      eapply ninf_in_neq in Hint1; eauto.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
a2 <> Fr x assert (cut: ~ (x `in` FV u)) by fsetdec.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: x `notin` FV u
a2 <> Fr x
      rewrite <- free_iff_FV in cut.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: ~ x ∈ free u
a2 <> Fr x
      apply ind_implies_in in Hint2.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
Hnin: x `notin` (FV t ∪ FV u)
Heq: mapd (open_LN_loc (Fr x)) t =
mapd (open_LN_loc (Fr x)) u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: a2 ∈ u
Heq_loc: open_LN_loc (Fr x) (e, a1) =
open_LN_loc (Fr x) (e, a2)
Heq_loc2: open_loc (ret (Fr x)) (e, a1) =
open_loc (ret (Fr x)) (e, a2)
cut: ~ x ∈ free u
a2 <> Fr x
      eapply ninf_in_neq in Hint2; eauto.
  Qed.

  (* LNgen 18: close-inj *)
  Lemma close_inj: forall (x: atom) (t u: U LN),
      '[x] t = '[x]  u ->
      t = u.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x (t u : U LN), '[x] t = '[x] u -> t = u
  Proof.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall x (t u : U LN), '[x] t = '[x] u -> t = u
    introv premise.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
t = u
    apply (mapd_injective2 (close_loc x)); auto.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
forall (e : nat) (a1 a2 : LN),
(e, a1) ∈d t ->
(e, a2) ∈d u ->
close_loc x (e, a1) = close_loc x (e, a2) -> a1 = a2
    intros e a1 a2 Hint1 Hint2 Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: close_loc x (e, a1) = close_loc x (e, a2)
a1 = a2
    cbn in Heq_loc.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a1, a2: LN
Hint1: (e, a1) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match a1 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
a1 = a2
    compare a1 to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
Hint1: (e, Fr x) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: (if x == x then Bd e else Fr x) =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
Fr x = a2
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: (if x == a then Bd e else Fr a) =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
H: a <> x
Fr a = a2
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
Bd n = a2
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
Hint1: (e, Fr x) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: (if x == x then Bd e else Fr x) =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
Fr x = a2 compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
Hint1: (e, Fr x) ∈d t
Hint2: (e, a2) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
DESTR_EQs: x = x
Fr x = a2
      compare a2 to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
Hint2: (e, Fr x) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e = (if x == x then Bd e else Fr x)
DESTR_EQs: x = x
Fr x = Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
a: atom
Hint2: (e, Fr a) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e = (if x == a then Bd e else Fr a)
DESTR_EQs: x = x
H: a <> x
Fr x = Fr a
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e =
match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end
DESTR_EQs: x = x
Fr x = Bd n
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
Hint2: (e, Fr x) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e = (if x == x then Bd e else Fr x)
DESTR_EQs: x = x
Fr x = Fr x reflexivity.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
a: atom
Hint2: (e, Fr a) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e = (if x == a then Bd e else Fr a)
DESTR_EQs: x = x
H: a <> x
Fr x = Fr a compare values x and a.
      +
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
Hint1: (e, Fr x) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_EQ: x = x
Heq_loc: Bd e =
match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end
DESTR_EQs: x = x
Fr x = Bd n compare naturals n and e;
          inversion Heq_loc; lia.
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: (if x == a then Bd e else Fr a) =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
H: a <> x
Fr a = a2 compare values x and a.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, a2) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
H, DESTR_NEQs: a <> x
Fr a = a2
      compare a2 to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, Fr x) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = (if x == x then Bd e else Fr x)
H, DESTR_NEQs: a <> x
Fr a = Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
a0: atom
Hint2: (e, Fr a0) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = (if x == a0 then Bd e else Fr a0)
H, DESTR_NEQs: a <> x
H0: a0 <> x
Fr a = Fr a0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a =
match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end
H, DESTR_NEQs: a <> x
Fr a = Bd n
      compare values x and x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
a0: atom
Hint2: (e, Fr a0) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = (if x == a0 then Bd e else Fr a0)
H, DESTR_NEQs: a <> x
H0: a0 <> x
Fr a = Fr a0
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a =
match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end
H, DESTR_NEQs: a <> x
Fr a = Bd n
      compare values x and a0.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a =
match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end
H, DESTR_NEQs: a <> x
Fr a = Bd n
      compare naturals n and e.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd n
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Lt
ineqp: n < e
a = n
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, Bd e) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S e)
H, DESTR_NEQs: a <> x
ineqrw: (e ?= e) = Eq
a = e
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S n)
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Gt
ineqp: n > e
a = n
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd n
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Lt
ineqp: n < e
a = n false. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, Bd e) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S e)
H, DESTR_NEQs: a <> x
ineqrw: (e ?= e) = Eq
a = e
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S n)
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Gt
ineqp: n > e
a = n
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
Hint2: (e, Bd e) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S e)
H, DESTR_NEQs: a <> x
ineqrw: (e ?= e) = Eq
a = e false. }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S n)
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Gt
ineqp: n > e
a = n
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a: atom
Hint1: (e, Fr a) ∈d t
n: nat
Hint2: (e, Bd n) ∈d u
DESTR_NEQ: x <> a
Heq_loc: Fr a = Bd (S n)
H, DESTR_NEQs: a <> x
ineqrw: (n ?= e) = Gt
ineqp: n > e
a = n false. }
    -
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e: nat
a2: LN
n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, a2) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match a2 with
| Fr y => if x == y then Bd e else Fr y
| Bd n =>
    match n ?= e with
    | Lt => Bd n
    | _ => Bd (S n)
    end
end
Bd n = a2 compare a2 to atom x.
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Fr x) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end = (if x == x then Bd e else Fr x)
Bd n = Fr x
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
a: atom
Hint2: (e, Fr a) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end = (if x == a then Bd e else Fr a)
H: a <> x
Bd n = Fr a
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
n0: nat
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match n0 ?= e with
| Lt => Bd n0
| _ => Bd (S n0)
end
Bd n = Bd n0
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
Hint2: (e, Fr x) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end = (if x == x then Bd e else Fr x)
Bd n = Fr x compare naturals n and e;
          compare values x and x;
          inversion Heq_loc; try lia.
      }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
a: atom
Hint2: (e, Fr a) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end = (if x == a then Bd e else Fr a)
H: a <> x
Bd n = Fr a
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
n0: nat
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match n0 ?= e with
| Lt => Bd n0
| _ => Bd (S n0)
end
Bd n = Bd n0
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
a: atom
Hint2: (e, Fr a) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end = (if x == a then Bd e else Fr a)
H: a <> x
Bd n = Fr a compare naturals n and e;
          compare values x and a.
      }
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
n0: nat
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match n0 ?= e with
| Lt => Bd n0
| _ => Bd (S n0)
end
Bd n = Bd n0
      {
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
x: atom
t, u: U LN
premise: '[x] t = '[x] u
e, n: nat
Hint1: (e, Bd n) ∈d t
n0: nat
Hint2: (e, Bd n0) ∈d u
Heq_loc: match n ?= e with
| Lt => Bd n
| _ => Bd (S n)
end =
match n0 ?= e with
| Lt => Bd n0
| _ => Bd (S n0)
end
Bd n = Bd n0 compare naturals n and e;
          compare naturals n0 and e;
          inversion Heq_loc; try lia; easy.
      }
  Qed.

End locally_nameless_metatheory.