Translating between locally nameless and de Bruijn indices

We reason about a translation between syntax with de Bruijn indices and locally nameless variables. This consists of a function which, given a locally closed term t, outputs a term of the same shape whose leaves are de Bruijn indices and a "key": some arbitrary permutation of the names of free variables in t. Another function accepts a key and a de Bruijn term and computes a locally nameless term of the same shape. The two functions are shown to be inverses.

Table of Contents :depth: 2

Imports and setup
Translation operations
Simplification support
Properties of LNtokey
Global operations
Properties of <<toDB>>

Imports and setup

Since we are using the Kleisli typeclass hierarchy, we import modules under the namespaces Classes.Kleisli and Theory.Kleisli.

From Tealeaves Require Export
  Backends.LN
  Backends.DB.DB
  Backends.Adapters.Key
  Functors.Option.

Import LN.Notations.

Import DecoratedTraversableMonad.UsefulInstances.

From Coq Require Import Logic.Decidable.

#[local] Generalizable Variables W T U.
#[local] Open Scope nat_scope.

Translation operations

Definition toDB_loc (k: key) '(depth, l) : option nat :=
  match l with
  | Bd n => if Nat.ltb n depth then Some n else None
  | Fr x => map (fun ix => ix + depth) (getIndex k x)
  end.

Fixpoint LNtokey_list (l: list LN): key :=
  match l with
  | [] => nil
  | (Bd n :: rest) => LNtokey_list rest
  | (Fr x :: rest) => key_insert_atom (LNtokey_list rest) x
  end.

Lemma toDB_loc_None_iff:
  forall k d l, toDB_loc k (d, l) = None <->
             (exists x, l = Fr x /\ ~ x ∈ k) \/ (exists n, l = Bd n /\ n >= d).forall (k : key) (d : nat) (l : LN),
toDB_loc k (d, l) = None <->
(exists x : atom, l = Fr x /\ ~ x ∈ k) \/
(exists n : nat, l = Bd n /\ n >= d)
Proof.forall (k : key) (d : nat) (l : LN),
toDB_loc k (d, l) = None <->
(exists x : atom, l = Fr x /\ ~ x ∈ k) \/
(exists n : nat, l = Bd n /\ n >= d)
  intros.k: key
d: nat
l: LN
toDB_loc k (d, l) = None <->
(exists x : atom, l = Fr x /\ ~ x ∈ k) \/
(exists n : nat, l = Bd n /\ n >= d)
  unfold toDB_loc.k: key
d: nat
l: LN
match l with
| Fr x => map (fun ix : nat => ix + d) (getIndex k x)
| Bd n => if Nat.ltb n d then Some n else None
end = None <->
(exists x : atom, l = Fr x /\ ~ x ∈ k) \/
(exists n : nat, l = Bd n /\ n >= d)
  destruct l as [x | n].k: key
d: nat
x: atom
map (fun ix : nat => ix + d) (getIndex k x) = None <->
(exists x0 : atom, Fr x = Fr x0 /\ ~ x0 ∈ k) \/
(exists n : nat, Fr x = Bd n /\ n >= d)
k: key
d, n: nat
(if Nat.ltb n d then Some n else None) = None <->
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
  -k: key
d: nat
x: atom
map (fun ix : nat => ix + d) (getIndex k x) = None <->
(exists x0 : atom, Fr x = Fr x0 /\ ~ x0 ∈ k) \/
(exists n : nat, Fr x = Bd n /\ n >= d) rewrite map_None_eq_iff.k: key
d: nat
x: atom
getIndex k x = None <->
(exists x0 : atom, Fr x = Fr x0 /\ ~ x0 ∈ k) \/
(exists n : nat, Fr x = Bd n /\ n >= d)
    setoid_rewrite getIndex_not_in_iff.k: key
d: nat
x: atom
getIndex k x = None <->
(exists x0 : atom,
   Fr x = Fr x0 /\ getIndex k x0 = None) \/
(exists n : nat, Fr x = Bd n /\ n >= d)
    firstorder.k: key
d: nat
x, x0: atom
H: Fr x = Fr x0
H0: getIndex k x0 = None
getIndex k x = None
k: key
d: nat
x: atom
x0: nat
H: Fr x = Bd x0
H0: x0 >= d
getIndex k x = None
    now inversion H.k: key
d: nat
x: atom
x0: nat
H: Fr x = Bd x0
H0: x0 >= d
getIndex k x = None
    now inversion H.
  -k: key
d, n: nat
(if Nat.ltb n d then Some n else None) = None <->
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d) split; intro contra.k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
k: key
d, n: nat
contra: (exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
(if Nat.ltb n d then Some n else None) = None
    +k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d) assert (Nat.ltb n d = false).k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
Nat.ltb n d = false
k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
H: Nat.ltb n d = false
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
      {k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
Nat.ltb n d = false now destruct (Nat.ltb n d). }k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
H: Nat.ltb n d = false
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
      apply PeanoNat.Nat.ltb_ge in H.k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
H: d <= n
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
      right.k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
H: d <= n
exists n0 : nat, Bd n = Bd n0 /\ n0 >= d exists n.k: key
d, n: nat
contra: (if Nat.ltb n d then Some n else None) = None
H: d <= n
Bd n = Bd n /\ n >= d auto.
    +k: key
d, n: nat
contra: (exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= d)
(if Nat.ltb n d then Some n else None) = None destruct contra as [[x [contra rest]] | [n' [Heq contra]]].k: key
d, n: nat
x: atom
contra: Bd n = Fr x
rest: ~ x ∈ k
(if Nat.ltb n d then Some n else None) = None
k: key
d, n, n': nat
Heq: Bd n = Bd n'
contra: n' >= d
(if Nat.ltb n d then Some n else None) = None
      *k: key
d, n: nat
x: atom
contra: Bd n = Fr x
rest: ~ x ∈ k
(if Nat.ltb n d then Some n else None) = None false.
      *k: key
d, n, n': nat
Heq: Bd n = Bd n'
contra: n' >= d
(if Nat.ltb n d then Some n else None) = None inversion Heq; subst.k: key
d, n': nat
Heq: Bd n' = Bd n'
contra: n' >= d
(if Nat.ltb n' d then Some n' else None) = None
        apply PeanoNat.Nat.ltb_ge in contra.k: key
d, n': nat
Heq: Bd n' = Bd n'
contra: PeanoNat.Nat.ltb n' d = false
(if Nat.ltb n' d then Some n' else None) = None
        rewrite contra.k: key
d, n': nat
Heq: Bd n' = Bd n'
contra: PeanoNat.Nat.ltb n' d = false
None = None
        reflexivity.
Qed.

Simplification support

Lemma toDB_loc_rw1 (k: key) (depth: nat) (n: nat):
  n < depth -> toDB_loc k (depth, Bd n) = Some n.k: key
depth, n: nat
n < depth -> toDB_loc k (depth, Bd n) = Some n
Proof.k: key
depth, n: nat
n < depth -> toDB_loc k (depth, Bd n) = Some n
  intros.k: key
depth, n: nat
H: n < depth
toDB_loc k (depth, Bd n) = Some n cbn.k: key
depth, n: nat
H: n < depth
(if
  match depth with
  | 0 => false
  | S m' => Nat.leb n m'
  end
 then Some n
 else None) = Some n
  destruct depth.k: key
n: nat
H: n < 0
None = Some n
k: key
depth, n: nat
H: n < S depth
(if Nat.leb n depth then Some n else None) = Some n
  -k: key
n: nat
H: n < 0
None = Some n false.k: key
n: nat
H: n < 0
False lia.
  -k: key
depth, n: nat
H: n < S depth
(if Nat.leb n depth then Some n else None) = Some n apply PeanoNat.Nat.leb_le in H.k: key
depth, n: nat
H: PeanoNat.Nat.leb (S n) (S depth) = true
(if Nat.leb n depth then Some n else None) = Some n
    cbn in H.k: key
depth, n: nat
H: PeanoNat.Nat.leb n depth = true
(if Nat.leb n depth then Some n else None) = Some n
    rewrite H.k: key
depth, n: nat
H: PeanoNat.Nat.leb n depth = true
Some n = Some n
    reflexivity.
Qed.

Lemma toDB_loc_rw2 (k: key) (depth: nat) (x: atom):
  toDB_loc k (depth, Fr x) =
    map (fun ix => ix + depth) (getIndex k x).k: key
depth: nat
x: atom
toDB_loc k (depth, Fr x) =
map (fun ix : nat => ix + depth) (getIndex k x)
Proof.k: key
depth: nat
x: atom
toDB_loc k (depth, Fr x) =
map (fun ix : nat => ix + depth) (getIndex k x)
  reflexivity.
Qed.

Properties of LNtokey

Lemma LNtokey_unique: forall l,
    unique (LNtokey_list l).forall l : list LN, unique (LNtokey_list l)
Proof.forall l : list LN, unique (LNtokey_list l)
  intros l.l: list LN
unique (LNtokey_list l) induction l as [|[x|n] rest].unique (LNtokey_list [])
x: atom
rest: list LN
IHrest: unique (LNtokey_list rest)
unique (LNtokey_list (Fr x :: rest))
n: nat
rest: list LN
IHrest: unique (LNtokey_list rest)
unique (LNtokey_list (Bd n :: rest))
  -unique (LNtokey_list []) exact I.
  -x: atom
rest: list LN
IHrest: unique (LNtokey_list rest)
unique (LNtokey_list (Fr x :: rest)) now apply key_insert_unique.
  -n: nat
rest: list LN
IHrest: unique (LNtokey_list rest)
unique (LNtokey_list (Bd n :: rest)) cbn.n: nat
rest: list LN
IHrest: unique (LNtokey_list rest)
unique (LNtokey_list rest) assumption.
Qed.

Lemma LNtokey_bijection: forall l ix a,
    getName (LNtokey_list l) ix = Some a <->
      getIndex (LNtokey_list l) a = Some ix.forall (l : list LN) (ix : nat) (a : atom),
getName (LNtokey_list l) ix = Some a <->
getIndex (LNtokey_list l) a = Some ix
Proof.forall (l : list LN) (ix : nat) (a : atom),
getName (LNtokey_list l) ix = Some a <->
getIndex (LNtokey_list l) a = Some ix
  intros.l: list LN
ix: nat
a: atom
getName (LNtokey_list l) ix = Some a <->
getIndex (LNtokey_list l) a = Some ix
  apply key_bijection.l: list LN
ix: nat
a: atom
unique (LNtokey_list l)
  apply LNtokey_unique.
Qed.

Global operations

Definition toDB
  `{Mapdt_inst: Mapdt nat T} (k: key): T LN -> option (T nat) :=
  mapdt (G := option) (toDB_loc k).

Definition LNtokey
  `{Traverse_inst: Traverse T} (t: T LN): key :=
  LNtokey_list (tolist t).

Definition toDB_default_key
  `{Traverse_inst: Traverse T}
  `{Mapdt_inst: Mapdt nat T} (t: T LN): option (T nat) :=
  toDB (LNtokey t) t.

Properties of <<toDB>>

Section theory.

  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)}.

  Definition scoped_key (k: key) (t: U LN) :=
    forall x: atom, x ∈ free t -> x ∈ k.

  Definition scoped_key_loc (k: key): LN -> Prop :=
    fun v => match v with
          | Bd _ => True
          | Fr x => x ∈ k
          end.

  Lemma decidable_scoped_key_loc (k: key) (v: LN):
    decidable (scoped_key_loc k v).T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
v: LN
decidable (scoped_key_loc k v)
  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
k: key
v: LN
decidable (scoped_key_loc k v)
    unfold decidable.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
v: LN
scoped_key_loc k v \/ ~ scoped_key_loc k v
    unfold scoped_key_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
k: key
v: LN
match v with
| Fr x => x ∈ k
| Bd _ => True
end \/
~ match v with
  | Fr x => x ∈ k
  | Bd _ => True
  end
    destruct v as [x | 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
k: key
x: atom
x ∈ k \/ ~ x ∈ k
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
n: nat
True \/ ~ 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
k: key
x: atom
x ∈ k \/ ~ x ∈ k destruct (elt_decidable x k); 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
k: key
n: nat
True \/ ~ True now left.
  Qed.

  Lemma scoped_key_equiv (k: key):
    scoped_key k = Forall (scoped_key_loc k).T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
scoped_key k = Forall (scoped_key_loc k)
  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
k: key
scoped_key k = Forall (scoped_key_loc k)
    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
k: key
t: U LN
scoped_key k t = Forall (scoped_key_loc k) t
    rewrite (forall_iff_eq (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
k: key
t: U LN
scoped_key k t =
(forall a : LN, a ∈ t -> scoped_key_loc k a)
    unfold scoped_key_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
k: key
t: U LN
scoped_key k t =
(forall a : LN,
 a ∈ t ->
 match a with
 | Fr x => x ∈ k
 | Bd _ => True
 end)
    unfold scoped_key.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(forall x : atom, x ∈ free t -> x ∈ k) =
(forall a : LN,
 a ∈ t ->
 match a with
 | Fr x => x ∈ k
 | Bd _ => True
 end) 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
k: key
t: U LN
(forall x : atom, x ∈ free t -> x ∈ k) ->
forall a : LN,
a ∈ t ->
match a with
| Fr x => x ∈ k
| Bd _ => True
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
k: key
t: U LN
(forall a : LN,
 a ∈ t ->
 match a with
 | Fr x => x ∈ k
 | Bd _ => True
 end) -> forall x : atom, x ∈ free t -> x ∈ k
    -T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(forall x : atom, x ∈ free t -> x ∈ k) ->
forall a : LN,
a ∈ t ->
match a with
| Fr x => x ∈ k
| Bd _ => True
end intros Hyp v Hvin.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
Hyp: forall x : atom, x ∈ free t -> x ∈ k
v: LN
Hvin: v ∈ t
match v with
| Fr x => x ∈ k
| Bd _ => True
end
      destruct v as [x | 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
k: key
t: U LN
Hyp: forall x : atom, x ∈ free t -> x ∈ k
x: atom
Hvin: Fr x ∈ t
x ∈ k
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
Hyp: forall x : atom, x ∈ free t -> x ∈ k
n: nat
Hvin: Bd n ∈ 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
k: key
t: U LN
Hyp: forall x : atom, x ∈ free t -> x ∈ k
x: atom
Hvin: Fr x ∈ t
x ∈ k setoid_rewrite in_free_iff 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
k: key
t: U LN
Hyp: forall x : atom, Fr x ∈ t -> x ∈ k
x: atom
Hvin: Fr x ∈ t
x ∈ k
        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
k: key
t: U LN
Hyp: forall x : atom, x ∈ free t -> x ∈ k
n: nat
Hvin: Bd n ∈ 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
k: key
t: U LN
(forall a : LN,
 a ∈ t ->
 match a with
 | Fr x => x ∈ k
 | Bd _ => True
 end) -> forall x : atom, x ∈ free t -> x ∈ k intros Hyp x 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
k: key
t: U LN
Hyp: forall a : LN,
a ∈ t ->
match a with
| Fr x => x ∈ k
| Bd _ => True
end
x: atom
Hin: x ∈ free t
x ∈ k
      rewrite in_free_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
k: key
t: U LN
Hyp: forall a : LN,
a ∈ t ->
match a with
| Fr x => x ∈ k
| Bd _ => True
end
x: atom
Hin: Fr x ∈ t
x ∈ k
      apply (Hyp (Fr x) Hin).
  Qed.

  Lemma scoped_key_decidable: forall (k: key),
      decidable_pred (scoped_key k).T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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 k : key, decidable_pred (scoped_key k)
  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 k : key, decidable_pred (scoped_key k)
    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
k: key
decidable_pred (scoped_key k)
    rewrite scoped_key_equiv.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
decidable_pred (Forall (scoped_key_loc k))
    apply decidable_Forall.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
decidable_pred (scoped_key_loc k)
    intro 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
k: key
t: LN
decidable (scoped_key_loc k t)
    apply decidable_scoped_key_loc.
  Qed.

  Lemma LCloc_decidable (gap: nat):
      decidable_pred (lc_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
decidable_pred (lc_loc 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
gap: nat
decidable_pred (lc_loc gap)
    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
gap: nat
decidable_pred
  (fun p : nat * LN =>
   let (w, l) := p in
   match l with
   | Fr _ => True
   | Bd n => n < w + gap
   end)
    unfold decidable_pred.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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
forall a : nat * LN,
decidable
  (let (w, l) := a in
   match l with
   | Fr _ => True
   | Bd n => n < w + gap
   end)
    intros [d v].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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, d: nat
v: LN
decidable
  match v with
  | Fr _ => True
  | Bd n => n < d + gap
  end
    destruct v.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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, d: nat
n: atom
decidable 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
gap, d, n: nat
decidable (n < d + 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, d: nat
n: atom
decidable True 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
gap, d, n: nat
decidable (n < d + gap) unfold decidable.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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, d, n: nat
n < d + gap \/ ~ n < d + gap
      compare naturals n and d.
  Qed.

  Lemma LC_decidable:
      decidable_pred 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
decidable_pred LC
  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
decidable_pred LC
    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
decidable_pred (LCn 0)
    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
decidable_pred (Forall_ctx (lc_loc 0))
    apply decidable_Forall_ctx.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
decidable_pred (lc_loc 0)
    apply LCloc_decidable.
  Qed.

  Lemma toDB_None_iff: forall k,
    forall (t: U LN), toDB k t = None <->
                   (exists (a: atom), a ∈ free t /\ ~ a ∈ k) \/
                     (exists (depth n: nat), (depth, Bd n) ∈d t /\ n >= depth).T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_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 (k : key) (t : U LN),
toDB k t = None <->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
  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 (k : key) (t : U LN),
toDB k t = None <->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    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
k: key
t: U LN
toDB k t = None <->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    unfold toDB.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
mapdt (toDB_loc k) t = None <->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    rewrite mapdt_option_None_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
k: key
t: U LN
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ toDB_loc k (e, a) = None) <->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    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
k: key
t: U LN
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ toDB_loc k (e, a) = None) <->
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    setoid_rewrite toDB_loc_None_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
k: key
t: U LN
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\
   ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
    (exists n : nat, a = Bd n /\ n >= e))) <->
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    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
k: key
t: U LN
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\
   ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
    (exists n : nat, a = Bd n /\ n >= e))) ->
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) ->
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ 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
k: key
t: U LN
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\
   ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
    (exists n : nat, a = Bd n /\ n >= e))) ->
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) intros [e [a [Hint rest]]].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
rest: (exists x : atom, a = Fr x /\ ~ x ∈ k) \/
(exists n : nat, a = Bd n /\ n >= e)
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
      destruct rest as [ [x [HeqX xNotIn]] | [n [Heq Hgeq]]].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
x: atom
HeqX: a = Fr x
xNotIn: ~ x ∈ k
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
n: nat
Heq: a = Bd n
Hgeq: n >= e
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
      +T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
x: atom
HeqX: a = Fr x
xNotIn: ~ x ∈ k
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
x: atom
HeqX: a = Fr x
xNotIn: ~ x ∈ k
exists a : atom, Fr a ∈ t /\ ~ a ∈ k exists 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
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
x: atom
HeqX: a = Fr x
xNotIn: ~ x ∈ k
Fr x ∈ t /\ ~ x ∈ k 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
k: key
t: U LN
e: nat
x: atom
Hint: (e, Fr x) ∈d t
xNotIn: ~ x ∈ k
Fr x ∈ t /\ ~ x ∈ k 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
k: key
t: U LN
e: nat
x: atom
Hint: (e, Fr x) ∈d t
xNotIn: ~ x ∈ k
Fr x ∈ t
        apply ind_implies_in in Hint.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: nat
x: atom
Hint: Fr x ∈ t
xNotIn: ~ x ∈ k
Fr x ∈ 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
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
n: nat
Heq: a = Bd n
Hgeq: n >= e
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
n: nat
Heq: a = Bd n
Hgeq: n >= e
exists depth n : nat, (depth, Bd n) ∈d t /\ n >= depth exists e 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
k: key
t: U LN
e: nat
a: LN
Hint: (e, a) ∈d t
n: nat
Heq: a = Bd n
Hgeq: n >= e
(e, Bd n) ∈d t /\ n >= e 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
k: key
t: U LN
(exists a : atom, Fr a ∈ t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) ->
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= e)) intros [ [e [Hin Hnotin]] | [depth [n [Hin 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
k: key
t: U LN
e: atom
Hin: Fr e ∈ t
Hnotin: ~ e ∈ k
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ 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
k: key
t: U LN
e: atom
Hin: Fr e ∈ t
Hnotin: ~ e ∈ k
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= e)) apply ind_iff_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
k: key
t: U LN
e: atom
Hin: exists e0 : nat, (e0, Fr e) ∈d t
Hnotin: ~ e ∈ k
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= e))
        destruct Hin as [d Hind].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: atom
d: nat
Hind: (d, Fr e) ∈d t
Hnotin: ~ e ∈ k
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= e))
        exists d.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
e: atom
d: nat
Hind: (d, Fr e) ∈d t
Hnotin: ~ e ∈ k
exists a : LN,
  (d, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= d)) exists (Fr 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
k: key
t: U LN
e: atom
d: nat
Hind: (d, Fr e) ∈d t
Hnotin: ~ e ∈ k
(d, Fr e) ∈d t /\
((exists x : atom, Fr e = Fr x /\ ~ x ∈ k) \/
 (exists n : nat, Fr e = Bd n /\ n >= d)) split; 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
exists (e : nat) (a : LN),
  (e, a) ∈d t /\
  ((exists x : atom, a = Fr x /\ ~ x ∈ k) \/
   (exists n : nat, a = Bd n /\ n >= e)) exists 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
(depth, Bd n) ∈d t /\
((exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
 (exists n0 : nat, Bd n = Bd n0 /\ n0 >= depth)) 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
(exists x : atom, Bd n = Fr x /\ ~ x ∈ k) \/
(exists n0 : nat, Bd n = Bd n0 /\ n0 >= depth)
        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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
exists n0 : nat, Bd n = Bd n0 /\ n0 >= depth exists 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Heq: n >= depth
Bd n = Bd n /\ n >= depth auto.
  Qed.

  Lemma toDB_None_iff2: forall k,
    forall (t: U LN),
      toDB k t = None <->
        ~ scoped_key k t \/ ~ LC 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 (k : key) (t : U LN),
toDB k t = None <-> ~ scoped_key k t \/ ~ LC 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 (k : key) (t : U LN),
toDB k t = None <-> ~ scoped_key k t \/ ~ LC 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
k: key
t: U LN
toDB k t = None <-> ~ scoped_key k t \/ ~ LC t
    rewrite toDB_None_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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
~ scoped_key k t \/ ~ LC t
    rewrite scoped_key_equiv.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
~ Forall (scoped_key_loc k) t \/ ~ LC t
    rewrite not_Forall_Forany.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
Forany (not ∘ scoped_key_loc k) t \/ ~ LC 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
k: key
t: U LN
decidable_pred (scoped_key_loc k)
    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
k: key
t: U LN
decidable_pred (scoped_key_loc k) intro v.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
v: LN
decidable (scoped_key_loc k v)
        apply decidable_scoped_key_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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
Forany (not ∘ scoped_key_loc k) t \/ ~ LC t
    unfold LC, 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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
Forany (not ∘ scoped_key_loc k) t \/
~ Forall_ctx (lc_loc 0) t
    rewrite not_Forall_ctx_Forany_ctx.T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
Forany (not ∘ scoped_key_loc k) t \/
Forany_ctx (not ∘ lc_loc 0) 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
k: key
t: U LN
decidable_pred (lc_loc 0)
    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
k: key
t: U LN
decidable_pred (lc_loc 0) apply LCloc_decidable. }T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
Forany (not ∘ scoped_key_loc k) t \/
Forany_ctx (not ∘ lc_loc 0) t
    rewrite forany_iff_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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
Forany_ctx (not ∘ lc_loc 0) t
    rewrite forany_ctx_iff_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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) <->
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) 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
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) ->
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, 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
k: key
t: U LN
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) ->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    {T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) ->
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) intros [[x [Hin Hnin]] | [depth [n [Hin Hgt]]]].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, 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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) 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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a exists (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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
Fr x ∈ t /\ (not ∘ scoped_key_loc k) (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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
(not ∘ scoped_key_loc k) (Fr x)
        rewrite in_free_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
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
(not ∘ scoped_key_loc k) (Fr x) 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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
(not ∘ scoped_key_loc k) (Fr x)
        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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
~ scoped_key_loc k (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
k: key
t: U LN
x: atom
Hin: x ∈ free t
Hnin: ~ x ∈ k
~ x ∈ k
        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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
exists (e : nat) (a : LN),
  (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)
        exists 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(depth, Bd n) ∈d t /\ (not ∘ lc_loc 0) (depth, Bd n) 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(depth, Bd n) ∈d 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(not ∘ lc_loc 0) (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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(depth, Bd n) ∈d 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
(not ∘ lc_loc 0) (depth, 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
~ lc_loc 0 (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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: n >= depth
~ n < depth + 0
          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
k: key
t: U LN
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) ->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
    {T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
(exists a : LN, a ∈ t /\ (not ∘ scoped_key_loc k) a) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ (not ∘ lc_loc 0) (e, a)) ->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) unfold compose, scoped_key_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
k: key
t: U LN
(exists a : LN,
   a ∈ t /\
   ~ match a with
     | Fr x => x ∈ k
     | Bd _ => True
     end) \/
(exists (e : nat) (a : LN),
   (e, a) ∈d t /\ ~ lc_loc 0 (e, a)) ->
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
      intros [[x [Hin Hnin]] | [depth [n [Hin Hgt]]]].T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
x: LN
Hin: x ∈ t
Hnin: ~
match x with
| Fr x => x ∈ k
| Bd _ => True
end
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
depth: nat
n: LN
Hin: (depth, n) ∈d t
Hgt: ~ lc_loc 0 (depth, n)
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
      -T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
x: LN
Hin: x ∈ t
Hnin: ~
match x with
| Fr x => x ∈ k
| Bd _ => True
end
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) destruct x as [x | 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
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
n: nat
Hin: Bd n ∈ t
Hnin: ~ True
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
        +T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
exists a : atom, a ∈ free t /\ ~ a ∈ k exists 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
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
x ∈ free t /\ ~ x ∈ k 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
k: key
t: U LN
x: atom
Hin: Fr x ∈ t
Hnin: ~ x ∈ k
Fr x ∈ t /\ ~ x ∈ k 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
k: key
t: U LN
n: nat
Hin: Bd n ∈ t
Hnin: ~ True
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
depth: nat
n: LN
Hin: (depth, n) ∈d t
Hgt: ~ lc_loc 0 (depth, n)
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
depth: nat
n: LN
Hin: (depth, n) ∈d t
Hgt: ~
match n with
| Fr _ => True
| Bd n => n < depth + 0
end
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
        destruct n as [x | 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
k: key
t: U LN
depth: nat
x: atom
Hin: (depth, Fr x) ∈d t
Hgt: ~ True
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: ~ n < depth + 0
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth)
        +T: Type -> Type
Return_T: Return T
Map_T: Map T
Bind_TT: Bind T T
Traverse_T: Traverse T
Mapd_T: Mapd nat T
Bindt_TT: Bindt T T
U: Type -> Type
Bindd_T: Bindd nat T U
Mapdt_T: Mapdt nat T
Binddt_TT: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Map_U: Map U
Bind_TU: Bind T U
Traverse_U: Traverse U
Mapd_U: Mapd nat U
Bindt_TU: Bindt T U
Bindd_TU: Bindd nat T U
Mapdt_U: Mapdt nat U
Binddt_TU: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Monad_inst: DecoratedTraversableMonad nat T
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: key
t: U LN
depth: nat
x: atom
Hin: (depth, Fr x) ∈d t
Hgt: ~ True
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: ~ n < depth + 0
(exists a : atom, a ∈ free t /\ ~ a ∈ k) \/
(exists depth n : nat,
   (depth, Bd n) ∈d t /\ n >= depth) 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: ~ n < depth + 0
exists depth n : nat, (depth, Bd n) ∈d t /\ n >= depth exists depth 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: ~ n < depth + 0
(depth, Bd n) ∈d t /\ n >= depth 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
k: key
t: U LN
depth, n: nat
Hin: (depth, Bd n) ∈d t
Hgt: ~ n < depth + 0
n >= depth
          lia.
    }
  Qed.

End theory.