From Tealeaves.Misc Require Export
  NaturalNumbers
  Iterate.
From Tealeaves.Theory Require Export
  DecoratedTraversableMonad.

Import DecoratedContainerMonad.
Import Kleisli.DecoratedTraversableMonad.
Import Kleisli.Theory.DecoratedTraversableFunctor.
Import DecoratedTraversableMonad.UsefulInstances.

Import PeanoNat.
#[local] Open Scope nat_scope.

Import DecoratedContainerFunctor.Notations.
Import DecoratedMonad.Notations.
Import Kleisli.Comonad.Notations.
Import Monoid.Notations.

Implicit Types (ρ: nat -> nat).

#[local] Generalizable Variables W T U.

(** * Closure *)
(**********************************************************************)
Section section.

  Context
    `{ret_inst: Return T}
    `{Mapd_T_inst: Mapd nat T}
    `{Mapd_U_inst: Mapd nat U}
    `{Mapdt_U_inst: Mapdt nat U}
    `{Bindd_U_inst: Bindd nat T U}
    `{ToCtxset_U_inst: ToCtxset nat U}
    `{! Compat_ToCtxset_Mapdt nat U}.

  (** ** Level *)
  (********************************************************************)
  Definition level_loc: nat * nat -> nat :=
    fun p => match p with
          | (d, n) => n + 1 - d
          end.

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

  (** ** Boolean versions *)
  (********************************************************************)
  Definition bound_within_b (gap: nat): nat -> nat -> bool :=
    fun ix depth => ix <? depth + gap.

  Definition bound_b: nat -> nat -> bool :=
    bound_within_b 0.

  Definition cl_at_loc_b (gap: nat) (p: nat * nat): bool :=
    match p with
    | (depth, ix) => bound_within_b gap ix depth
    end.

  Definition cl_at_b (gap: nat) (t: U nat): bool :=
    Forall_ctx_b (cl_at_loc_b gap) t.

  Definition closed_b (t: U nat): bool :=
    cl_at_b 0 t.

  (** ** Propositional versions *)
  (********************************************************************)
  Definition bound_within (gap: nat): nat -> nat -> Prop :=
    fun ix depth => ix < depth + gap.

  Definition bound: nat -> nat -> Prop :=
    bound_within 0.

  Definition cl_at_loc (gap: nat) (p: nat * nat): Prop :=
    match p with
    | (depth, ix) => bound_within gap ix depth
    end.

  Definition cl_at (gap: nat) (t: U nat): Prop :=
    Forall_ctx (cl_at_loc gap) t.

  Definition closed (t: U nat): Prop :=
    cl_at 0 t.

  (** ** Relating the two *)
  (********************************************************************)
  Lemma bound_within_spec (gap: nat): forall (ix depth: nat),
      bound_within_b gap ix depth = true <-> ix < gap + depth.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall ix depth : nat,
bound_within_b gap ix depth = true <->
ix < gap + depth
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall ix depth : nat,
bound_within_b gap ix depth = true <->
ix < gap + depth
    intros.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
bound_within_b gap ix depth = true <->
ix < gap + depth
    unfold bound_within_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
(ix <? depth + gap) = true <-> ix < gap + depth
    rewrite Nat.ltb_lt.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
ix < depth + gap <-> ix < gap + depth
    lia.
  Qed.
  Lemma bound_within_spec_not (gap: nat): forall (ix depth: nat),
      bound_within_b gap ix depth = false <-> ~ (ix < gap + depth).
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall ix depth : nat,
bound_within_b gap ix depth = false <->
~ ix < gap + depth
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall ix depth : nat,
bound_within_b gap ix depth = false <->
~ ix < gap + depth
    intros.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
bound_within_b gap ix depth = false <->
~ ix < gap + depth
    unfold bound_within_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
(ix <? depth + gap) = false <-> ~ ix < gap + depth
    rewrite Nat.ltb_nlt.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, ix, depth: nat
~ ix < depth + gap <-> ~ ix < gap + depth
    lia.
  Qed.

  Lemma bound_spec: forall (ix depth: nat),
      bound_b ix depth = true <-> ix < depth.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
forall ix depth : nat,
bound_b ix depth = true <-> ix < depth
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
forall ix depth : nat,
bound_b ix depth = true <-> ix < depth
    intros.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
bound_b ix depth = true <-> ix < depth
    unfold bound_b, bound_within_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
(ix <? depth + 0) = true <-> ix < depth
    rewrite Nat.ltb_lt.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
ix < depth + 0 <-> ix < depth
    lia.
  Qed.
  Lemma bound_spec_not: forall (ix depth: nat),
      bound_b ix depth = false <-> ~ (ix < depth).
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
forall ix depth : nat,
bound_b ix depth = false <-> ~ ix < depth
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
forall ix depth : nat,
bound_b ix depth = false <-> ~ ix < depth
    intros.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
bound_b ix depth = false <-> ~ ix < depth
    unfold bound_b, bound_within_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
(ix <? depth + 0) = false <-> ~ ix < depth
    rewrite Nat.ltb_nlt.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
ix, depth: nat
~ ix < depth + 0 <-> ~ ix < depth
    lia.
  Qed.

  Lemma cl_at_loc_spec {gap: nat}: forall (p: nat * nat),
      cl_at_loc_b gap p = true <-> cl_at_loc gap p.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall p : nat * nat,
cl_at_loc_b gap p = true <-> cl_at_loc gap p
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
forall p : nat * nat,
cl_at_loc_b gap p = true <-> cl_at_loc gap p
    intros.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
p: (nat * nat)%type
cl_at_loc_b gap p = true <-> cl_at_loc gap p
    unfold cl_at_loc, cl_at_loc_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
p: (nat * nat)%type
(let (depth, ix) := p in bound_within_b gap ix depth) =
true <->
(let (depth, ix) := p in bound_within gap ix depth)
    unfold bound_within_b, bound_within.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap: nat
p: (nat * nat)%type
(let (depth, ix) := p in ix <? depth + gap) = true <->
(let (depth, ix) := p in ix < depth + gap)
    destruct p as [d n].
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, d, n: nat
(n <? d + gap) = true <-> n < d + gap
    rewrite Nat.ltb_lt.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
gap, d, n: nat
n < d + gap <-> n < d + gap
    reflexivity.
  Qed.

  Corollary closed_at_spec `{! DecoratedTraversableFunctor nat U}:
    forall (gap: nat) (t: U nat),
      cl_at_b gap t = true <-> cl_at gap t.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
forall (gap : nat) (t : U nat),
cl_at_b gap t = true <-> cl_at gap t
  Proof.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
forall (gap : nat) (t : U nat),
cl_at_b gap t = true <-> cl_at gap t
    intros gap t.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
gap: nat
t: U nat
cl_at_b gap t = true <-> cl_at gap t
    unfold cl_at_b, cl_at.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
gap: nat
t: U nat
Forall_ctx_b (cl_at_loc_b gap) t = true <->
Forall_ctx (cl_at_loc gap) t
    rewrite decidable_Forall_ctx_b.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
gap: nat
t: U nat
Forall_ctx_b (cl_at_loc_b gap) t = true <->
Forall_ctx_b ?Goal0 t = true
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
gap: nat
t: U nat
forall p : nat * nat,
?Goal0 p = true <-> cl_at_loc gap p
    reflexivity.
T: Type -> Type
ret_inst: Return T
Mapd_T_inst: Mapd nat T
U: Type -> Type
Mapd_U_inst: Mapd nat U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
ToCtxset_U_inst: ToCtxset nat U
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt nat U
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  nat U
gap: nat
t: U nat
forall p : nat * nat,
cl_at_loc_b gap p = true <-> cl_at_loc gap p
    apply cl_at_loc_spec.
  Qed.

End section.

#[local] Infix "`bound in`" := (bound_b) (at level 10): tealeaves_scope.

(** ** Boundedness and closure *)
(**********************************************************************)
Lemma bound_within_spec_b: forall ix d k,
    bound_within_b k ix d =
      ix `bound in` (d + k).
forall ix d k : nat,
bound_within_b k ix d = ix `bound in` (d + k)
Proof.
forall ix d k : nat,
bound_within_b k ix d = ix `bound in` (d + k)
  intros.
ix, d, k: nat
bound_within_b k ix d = ix `bound in` (d + k) cbn.
ix, d, k: nat
match d + k with
| 0 => false
| S m' => ix <=? m'
end =
match d + k + 0 with
| 0 => false
| S m' => ix <=? m'
end replace (d + k + 0) with (d + k) by lia.
ix, d, k: nat
match d + k with
| 0 => false
| S m' => ix <=? m'
end =
match d + k with
| 0 => false
| S m' => ix <=? m'
end
  reflexivity.
Qed.

Lemma bound_zero: forall ix,
    ix `bound in` 0 = false.
forall ix : nat, ix `bound in` 0 = false
Proof.
forall ix : nat, ix `bound in` 0 = false
  reflexivity.
Qed.

Lemma bound_lt: forall ix n,
    ix < n -> ix `bound in` n = true.
forall ix n : nat, ix < n -> ix `bound in` n = true
Proof.
forall ix n : nat, ix < n -> ix `bound in` n = true
  introv Hlt.
ix, n: nat
Hlt: ix < n
ix `bound in` n = true
  rewrite <- Nat.ltb_lt in Hlt.
ix, n: nat
Hlt: (ix <? n) = true
ix `bound in` n = true
  unfold bound_b, bound_within_b.
ix, n: nat
Hlt: (ix <? n) = true
(ix <? n + 0) = true
  replace (n + 0) with n by lia.
ix, n: nat
Hlt: (ix <? n) = true
(ix <? n) = true
  destruct n; auto.
Qed.

Lemma bound_lt_iff: forall ix n,
    ix < n <-> ix `bound in` n = true.
forall ix n : nat, ix < n <-> ix `bound in` n = true
Proof.
forall ix n : nat, ix < n <-> ix `bound in` n = true
  intros.
ix, n: nat
ix < n <-> ix `bound in` n = true split.
ix, n: nat
ix < n -> ix `bound in` n = true
ix, n: nat
ix `bound in` n = true -> ix < n
  -
ix, n: nat
ix < n -> ix `bound in` n = true apply bound_lt.
  -
ix, n: nat
ix `bound in` n = true -> ix < n cbn.
ix, n: nat
match n + 0 with
| 0 => false
| S m' => ix <=? m'
end = true -> ix < n destruct n.
ix: nat
match 0 + 0 with
| 0 => false
| S m' => ix <=? m'
end = true -> ix < 0
ix, n: nat
match S n + 0 with
| 0 => false
| S m' => ix <=? m'
end = true -> ix < S n
    +
ix: nat
match 0 + 0 with
| 0 => false
| S m' => ix <=? m'
end = true -> ix < 0 inversion 1.
    +
ix, n: nat
match S n + 0 with
| 0 => false
| S m' => ix <=? m'
end = true -> ix < S n cbn.
ix, n: nat
(ix <=? n + 0) = true -> ix < S n rewrite Nat.leb_le.
ix, n: nat
ix <= n + 0 -> ix < S n lia.
Qed.

Lemma bound_ge: forall ix n,
    ix >= n -> ix `bound in` n = false.
forall ix n : nat, ix >= n -> ix `bound in` n = false
Proof.
forall ix n : nat, ix >= n -> ix `bound in` n = false
  introv Hlt.
ix, n: nat
Hlt: ix >= n
ix `bound in` n = false
  unfold bound_b, bound_within_b.
ix, n: nat
Hlt: ix >= n
(ix <? n + 0) = false
  replace (n + 0) with n by lia.
ix, n: nat
Hlt: ix >= n
(ix <? n) = false
  destruct n; cbn; auto.
ix, n: nat
Hlt: ix >= S n
(ix <=? n) = false
  now rewrite Nat.leb_gt.
Qed.

Lemma bound_ge_iff: forall ix n,
    ix >= n <-> ix `bound in` n = false.
forall ix n : nat, ix >= n <-> ix `bound in` n = false
Proof.
forall ix n : nat, ix >= n <-> ix `bound in` n = false
  intros.
ix, n: nat
ix >= n <-> ix `bound in` n = false split.
ix, n: nat
ix >= n -> ix `bound in` n = false
ix, n: nat
ix `bound in` n = false -> ix >= n
  -
ix, n: nat
ix >= n -> ix `bound in` n = false apply bound_ge.
  -
ix, n: nat
ix `bound in` n = false -> ix >= n cbn.
ix, n: nat
match n + 0 with
| 0 => false
| S m' => ix <=? m'
end = false -> ix >= n replace (n + 0) with n by lia.
ix, n: nat
match n with
| 0 => false
| S m' => ix <=? m'
end = false -> ix >= n
    destruct n.
ix: nat
false = false -> ix >= 0
ix, n: nat
(ix <=? n) = false -> ix >= S n
    +
ix: nat
false = false -> ix >= 0 lia.
    +
ix, n: nat
(ix <=? n) = false -> ix >= S n intros.
ix, n: nat
H: (ix <=? n) = false
ix >= S n
      rewrite Nat.leb_gt in H.
ix, n: nat
H: n < ix
ix >= S n
      lia.
Qed.

Lemma bound_ind: forall ix n (P:Prop),
    (ix >= n -> ix `bound in` n = false -> P) ->
    (ix < n -> ix `bound in` n = true -> P) ->
    P.
forall (ix n : nat) (P : Prop),
(ix >= n -> ix `bound in` n = false -> P) ->
(ix < n -> ix `bound in` n = true -> P) -> P
Proof.
forall (ix n : nat) (P : Prop),
(ix >= n -> ix `bound in` n = false -> P) ->
(ix < n -> ix `bound in` n = true -> P) -> P
  introv.
ix, n: nat
P: Prop
(ix >= n -> ix `bound in` n = false -> P) ->
(ix < n -> ix `bound in` n = true -> P) -> P
  rewrite <- bound_ge_iff.
ix, n: nat
P: Prop
(ix >= n -> ix >= n -> P) ->
(ix < n -> ix `bound in` n = true -> P) -> P
  rewrite <- bound_lt_iff.
ix, n: nat
P: Prop
(ix >= n -> ix >= n -> P) ->
(ix < n -> ix < n -> P) -> P
  introv Hge Hlt.
ix, n: nat
P: Prop
Hge: ix >= n -> ix >= n -> P
Hlt: ix < n -> ix < n -> P
P
  compare naturals ix and n; auto.
ix, n: nat
P: Prop
Hge: ix >= n -> ix >= n -> P
Hlt: ix < n -> ix < n -> P
ineqrw: (ix ?= n) = Gt
ineqp: ix > n
P
  apply Hge; lia.
Qed.

Lemma bound_mono: forall ix n m,
    m > n ->
    (ix `bound in` n = true) ->
    ix `bound in` m = true.
forall ix n m : nat,
m > n ->
ix `bound in` n = true -> ix `bound in` m = true
Proof.
forall ix n m : nat,
m > n ->
ix `bound in` n = true -> ix `bound in` m = true
  introv Hlt Hbound.
ix, n, m: nat
Hlt: m > n
Hbound: ix `bound in` n = true
ix `bound in` m = true
  rewrite <- bound_lt_iff in Hbound.
ix, n, m: nat
Hlt: m > n
Hbound: ix < n
ix `bound in` m = true
  rewrite <- bound_lt_iff.
ix, n, m: nat
Hlt: m > n
Hbound: ix < n
ix < m
  lia.
Qed.

Lemma bound_rev_mono: forall ix n m,
    m < n ->
    bound_b ix n = false ->
    bound_b ix m = false.
forall ix n m : nat,
m < n ->
ix `bound in` n = false -> ix `bound in` m = false
Proof.
forall ix n m : nat,
m < n ->
ix `bound in` n = false -> ix `bound in` m = false
  introv Hlt Hbound.
ix, n, m: nat
Hlt: m < n
Hbound: ix `bound in` n = false
ix `bound in` m = false
  rewrite <- bound_ge_iff in Hbound.
ix, n, m: nat
Hlt: m < n
Hbound: ix >= n
ix `bound in` m = false
  rewrite <- bound_ge_iff.
ix, n, m: nat
Hlt: m < n
Hbound: ix >= n
ix >= m
  lia.
Qed.

Lemma bound_1: forall ix,
    bound_b ix 1 = true <-> ix = 0.
forall ix : nat, ix `bound in` 1 = true <-> ix = 0
Proof.
forall ix : nat, ix `bound in` 1 = true <-> ix = 0
  destruct ix; intuition.
Qed.

Ltac bound_induction :=
  match goal with
  | |- context[bound_b ?ix ?n] =>
      apply (bound_ind ix n);
      let Hord := fresh "Hord" in
      let Hbound := fresh "Hbound" in
      introv Hord Hbound;
      try rewrite Hbound in *;
      try solve [lia | easy]
  end.

Ltac bound_induction_in H :=
  match type of H with
  | context[bound_b ?ix ?n] =>
      apply (bound_ind ix n);
      let Hord := fresh "Hord" in
      let Hbound := fresh "Hbound" in
      introv Hord Hbound;
      try rewrite Hbound in *;
      try solve [lia | easy]
  end.

(** * Incrementing/lifting indices *)
(**********************************************************************)
Definition lift (x y: nat): nat := plus x y.
#[local] Notation "( + x )" := (lift x) (format "( + x )").
#[global] Arguments lift x y/.

Lemma lift0: (+0) = id.
(+0) = id reflexivity. Qed.

Lemma lift_comp :forall m n, (+m) ∘ (+n) = (+ (m + n)).
forall m n : nat, (+m) ∘ (+n) = (+m + n)
Proof.
forall m n : nat, (+m) ∘ (+n) = (+m + n)
  intros.
m, n: nat
(+m) ∘ (+n) = (+m + n) unfold compose, lift.
m, n: nat
Init.Nat.add m ○ Init.Nat.add n =
(fun y : nat => m + n + y) ext y.
m, n, y: nat
m + (n + y) = m + n + y
  lia.
Qed.

Lemma lift_compR n m {X}(f: nat -> X): (f ∘ (+m)) ∘ (+n) = f ∘ (+ (m + n)).
n, m: nat
X: Type
f: nat -> X
f ∘ (+m) ∘ (+n) = f ∘ (+m + n)
Proof.
n, m: nat
X: Type
f: nat -> X
f ∘ (+m) ∘ (+n) = f ∘ (+m + n)
  now rewrite <- lift_comp.
Qed.

Lemma plusSn n m: S n + m = S (n + m).
n, m: nat
S n + m = S (n + m) reflexivity. Qed.
Lemma plusnS n m: n + S m = S (n + m).
n, m: nat
n + S m = S (n + m) symmetry.
n, m: nat
S (n + m) = n + S m apply plus_n_Sm. Qed.
Lemma plusOn n: O + n = n.
n: nat
0 + n = n reflexivity. Qed.
Lemma plusnO n: n + O = n.
n: nat
n + 0 = n symmetry.
n: nat
n = n + 0 apply plus_n_O. Qed.

Ltac simplify_lift :=
  progress repeat match goal with
    | [|- context[(+0)]] => change (+0) with (@id nat)
    | [|- context[?s S]] => change (s S) with (s (+1))
    | [|- context[S ?n + ?m]] => rewrite (plusSn n m)
    | [|- context[?n + S ?m]] => rewrite (plusnS n m)
    | [|- context[?n + 0]] => rewrite (plusnO n)
    | [|- context[0 + ?n]] => rewrite (plusOn n)
    | [|- context[(+ ?m) ∘ (+ ?n)]] => rewrite (lift_comp m n)
    | [|- context[(?f ∘ (+ ?m)) ∘ (+ ?n)]] => rewrite (lift_compR m n f)
    end.

(** * Cons operation *)
(**********************************************************************)
Definition scons {X: Type}: X -> (nat -> X) -> (nat -> X)  :=
  fun new sub n => match n with
                | O => new
                | S n' => sub n'
                end.

#[local] Infix "⋅" := (scons) (at level 10).

(** ** Properties of scons *)
(**********************************************************************)
Lemma scons_rw0 {A}: forall `(x: A) (σ: nat -> A),
    (x ⋅ σ) 0 = x.
A: Type
forall (x : A) (σ : nat -> A), (x ⋅ σ) 0 = x
Proof.
A: Type
forall (x : A) (σ : nat -> A), (x ⋅ σ) 0 = x
  reflexivity.
Qed.

Lemma scons_rw1 {A}: forall `(x: A) (n: nat) (σ: nat -> A),
    (x ⋅ σ) (S n) = σ n.
A: Type
forall (x : A) (n : nat) (σ : nat -> A),
(x ⋅ σ) (S n) = σ n
Proof.
A: Type
forall (x : A) (n : nat) (σ : nat -> A),
(x ⋅ σ) (S n) = σ n
  reflexivity.
Qed.

Lemma scons_sub_id {X}: forall (σ: nat -> X),
    (σ 0) ⋅ (σ ∘ S) = σ.
X: Type
forall σ : nat -> X, (σ 0) ⋅ (σ ∘ S) = σ
Proof.
X: Type
forall σ : nat -> X, (σ 0) ⋅ (σ ∘ S) = σ
  intros.
X: Type
σ: nat -> X
(σ 0) ⋅ (σ ∘ S) = σ
  ext ix.
X: Type
σ: nat -> X
ix: nat
((σ 0) ⋅ (σ ∘ S)) ix = σ ix
  destruct ix.
X: Type
σ: nat -> X
((σ 0) ⋅ (σ ∘ S)) 0 = σ 0
X: Type
σ: nat -> X
ix: nat
((σ 0) ⋅ (σ ∘ S)) (S ix) = σ (S ix)
  -
X: Type
σ: nat -> X
((σ 0) ⋅ (σ ∘ S)) 0 = σ 0 rewrite scons_rw0.
X: Type
σ: nat -> X
σ 0 = σ 0
    reflexivity.
  -
X: Type
σ: nat -> X
ix: nat
((σ 0) ⋅ (σ ∘ S)) (S ix) = σ (S ix) rewrite scons_rw1.
X: Type
σ: nat -> X
ix: nat
(σ ∘ S) ix = σ (S ix)
    reflexivity.
Qed.

Lemma cons_compose {X Y}: forall (σ: nat -> X) (f: X -> Y) (x: X),
    f ∘ (x ⋅ σ) = (f x) ⋅ (f ∘ σ).
X, Y: Type
forall (σ : nat -> X) (f : X -> Y) (x : X),
f ∘ x ⋅ σ = (f x) ⋅ (f ∘ σ)
Proof.
X, Y: Type
forall (σ : nat -> X) (f : X -> Y) (x : X),
f ∘ x ⋅ σ = (f x) ⋅ (f ∘ σ)
  intros; now ext [|n].
Qed.

(** ** Cons and lifting *)
(**********************************************************************)
Lemma cons_eta {X}: forall (σ: nat -> X) (n: nat),
    (σ n) ⋅ (σ ∘ (+ (S n))) = σ ∘ (+ n).
X: Type
forall (σ : nat -> X) (n : nat),
(σ n) ⋅ (σ ∘ (+S n)) = σ ∘ (+n)
Proof.
X: Type
forall (σ : nat -> X) (n : nat),
(σ n) ⋅ (σ ∘ (+S n)) = σ ∘ (+n)
  intros.
X: Type
σ: nat -> X
n: nat
(σ n) ⋅ (σ ∘ (+S n)) = σ ∘ (+n)
  unfold compose, lift.
X: Type
σ: nat -> X
n: nat
(σ n) ⋅ (σ ○ Init.Nat.add (S n)) = σ ○ Init.Nat.add n
  ext [|m].
X: Type
σ: nat -> X
n: nat
((σ n) ⋅ (σ ○ Init.Nat.add (S n))) 0 = σ (n + 0)
X: Type
σ: nat -> X
n, m: nat
((σ n) ⋅ (σ ○ Init.Nat.add (S n))) (S m) = σ (n + S m)
  -
X: Type
σ: nat -> X
n: nat
((σ n) ⋅ (σ ○ Init.Nat.add (S n))) 0 = σ (n + 0) cbn.
X: Type
σ: nat -> X
n: nat
σ n = σ (n + 0) simplify_lift.
X: Type
σ: nat -> X
n: nat
σ n = σ n reflexivity.
  -
X: Type
σ: nat -> X
n, m: nat
((σ n) ⋅ (σ ○ Init.Nat.add (S n))) (S m) = σ (n + S m) cbn.
X: Type
σ: nat -> X
n, m: nat
σ (S (n + m)) = σ (n + S m) simplify_lift.
X: Type
σ: nat -> X
n, m: nat
σ (S (n + m)) = σ (S (n + m)) reflexivity.
Qed.

Lemma lift_eta n: n ⋅ (+ (S n)) = (+ n).
n: nat
n ⋅ (+S n) = (+n)
Proof.
n: nat
n ⋅ (+S n) = (+n)
  apply (cons_eta id).
Qed.

(** * Renaming *)
(**********************************************************************)
(* Given a depth and renaming ρ, adjust ρ to account for the
     depth *)
Definition lift__ren: nat -> (nat -> nat) -> (nat -> nat) :=
  fun depth ρ ix =>
    if bound_b ix depth
    then ix
    else let free_ix := ix - depth
         in ρ free_ix + depth.

Definition local__ren (ρ: nat -> nat) (p: nat * nat): nat :=
  match p with
  | (depth, ix) => lift__ren depth ρ ix
  end.

Definition rename `{Mapd_T_inst: Mapd nat T} (ρ: nat -> nat): T nat -> T nat :=
  mapd (T := T) (local__ren ρ).

(* adjust a renaming to go under one binder *)
Definition up__ren (ρ: nat -> nat): nat -> nat :=
  0 ⋅ (S ∘ ρ).

(** ** Properties of level*)
(**********************************************************************)
Section level_theory.

  Context
    `{Mapdt_T_inst: Mapdt nat T}
    `{Functor_inst: ! DecoratedTraversableFunctor nat T}.

  Lemma level1: forall (gap: nat) (t: T nat),
      level t <= gap ->
      cl_at gap t.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
level t <= gap -> cl_at gap t
  Proof.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
level t <= gap -> cl_at gap t
    intros gap t.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
level t <= gap -> cl_at gap t
    unfold level.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapdReduce level_loc t <= gap -> cl_at gap t
    unfold cl_at.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapdReduce level_loc t <= gap ->
Forall_ctx (cl_at_loc gap) t
    rewrite mapdReduce_through_toctxlist.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
(mapReduce level_loc ∘ toctxlist) t <= gap ->
Forall_ctx (cl_at_loc gap) t
    unfold Forall_ctx.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
(mapReduce level_loc ∘ toctxlist) t <= gap ->
mapdReduce (cl_at_loc gap) t
    rewrite (mapdReduce_through_toctxlist _ (cl_at_loc gap)).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
(mapReduce level_loc ∘ toctxlist) t <= gap ->
(mapReduce (cl_at_loc gap) ∘ toctxlist) t
    unfold compose.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce level_loc (toctxlist t) <= gap ->
mapReduce (cl_at_loc gap) (toctxlist t)
    induction (toctxlist t).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce level_loc nil <= gap ->
mapReduce (cl_at_loc gap) nil
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
mapReduce level_loc (a :: e) <= gap ->
mapReduce (cl_at_loc gap) (a :: e)
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce level_loc nil <= gap ->
mapReduce (cl_at_loc gap) nil cbv.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
0 <= gap -> True easy.
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
mapReduce level_loc (a :: e) <= gap ->
mapReduce (cl_at_loc gap) (a :: e) cbn.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
level_loc a ● mapReduce level_loc e <= gap ->
(pure cons ● cl_at_loc gap a)
● mapReduce (cl_at_loc gap) e
      do 2 unfold transparent tcs.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap ->
(True /\ cl_at_loc gap a) /\
mapReduce (cl_at_loc gap) e
      intros.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
(True /\ cl_at_loc gap a) /\
mapReduce (cl_at_loc gap) e split.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
True /\ cl_at_loc gap a
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
mapReduce (cl_at_loc gap) e
      +
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
True /\ cl_at_loc gap a split; auto.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
cl_at_loc gap a
        unfold cl_at_loc.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
let (depth, ix) := a in bound_within gap ix depth
        destruct a as (d, n).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc (d, n))
  (mapReduce level_loc e) <= gap
bound_within gap n d
        unfold bound_within.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc (d, n))
  (mapReduce level_loc e) <= gap
n < d + gap
        unfold level_loc in H.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (n + 1 - d)
  (mapReduce
     (fun p : nat * nat =>
      let (d, n) := p in n + 1 - d) e) <= gap
n < d + gap
        lia.
      +
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
mapReduce (cl_at_loc gap) e apply IHe.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap ->
mapReduce (cl_at_loc gap) e
H: Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
mapReduce level_loc e <= gap lia.
  Qed.

  Lemma level2: forall (gap: nat) (t: T nat),
      cl_at gap t ->
      level t <= gap.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
cl_at gap t -> level t <= gap
  Proof.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
cl_at gap t -> level t <= gap
    intros gap t.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
cl_at gap t -> level t <= gap
    unfold level.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
cl_at gap t -> mapdReduce level_loc t <= gap
    unfold cl_at.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
Forall_ctx (cl_at_loc gap) t ->
mapdReduce level_loc t <= gap
    rewrite mapdReduce_through_toctxlist.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
Forall_ctx (cl_at_loc gap) t ->
(mapReduce level_loc ∘ toctxlist) t <= gap
    unfold Forall_ctx.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapdReduce (cl_at_loc gap) t ->
(mapReduce level_loc ∘ toctxlist) t <= gap
    rewrite (mapdReduce_through_toctxlist _ (cl_at_loc gap)).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
(mapReduce (cl_at_loc gap) ∘ toctxlist) t ->
(mapReduce level_loc ∘ toctxlist) t <= gap
    unfold compose.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce (cl_at_loc gap) (toctxlist t) ->
mapReduce level_loc (toctxlist t) <= gap
    induction (toctxlist t).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce (cl_at_loc gap) nil ->
mapReduce level_loc nil <= gap
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce (cl_at_loc gap) (a :: e) ->
mapReduce level_loc (a :: e) <= gap
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
mapReduce (cl_at_loc gap) nil ->
mapReduce level_loc nil <= gap cbv.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
True -> 0 <= gap lia.
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
mapReduce (cl_at_loc gap) (a :: e) ->
mapReduce level_loc (a :: e) <= gap cbn.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
(pure cons ● cl_at_loc gap a)
● mapReduce (cl_at_loc gap) e ->
level_loc a ● mapReduce level_loc e <= gap
      do 2 unfold transparent tcs.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
(True /\ cl_at_loc gap a) /\
mapReduce (cl_at_loc gap) e ->
Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
      intros [[_ H1] H2].
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
H1: cl_at_loc gap a
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
      unfold cl_at_loc in H1.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
a: (nat * nat)%type
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
H1: let (depth, ix) := a in bound_within gap ix depth
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc a) (mapReduce level_loc e) <=
gap
      destruct a as (d, n).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
H1: bound_within gap n d
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc (d, n))
  (mapReduce level_loc e) <= gap
      unfold bound_within in H1.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce (cl_at_loc gap) e ->
mapReduce level_loc e <= gap
H1: n < d + gap
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc (d, n))
  (mapReduce level_loc e) <= gap
      specialize (IHe H2).
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap
H1: n < d + gap
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (level_loc (d, n))
  (mapReduce level_loc e) <= gap
      unfold level_loc at 1.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
d, n: nat
e: list (nat * nat)
IHe: mapReduce level_loc e <= gap
H1: n < d + gap
H2: mapReduce (cl_at_loc gap) e
Init.Nat.max (n + 1 - d) (mapReduce level_loc e) <=
gap
      lia.
  Qed.

  Lemma level_iff: forall (gap: nat) (t: T nat),
      cl_at gap t <-> level t <= gap.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
cl_at gap t <-> level t <= gap
  Proof.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
forall (gap : nat) (t : T nat),
cl_at gap t <-> level t <= gap
    intros.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
cl_at gap t <-> level t <= gap split.
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
cl_at gap t -> level t <= gap
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
level t <= gap -> cl_at gap t
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
cl_at gap t -> level t <= gap apply level2.
    -
T: Type -> Type
Mapdt_T_inst: Mapdt nat T
Functor_inst: DecoratedTraversableFunctor nat T
gap: nat
t: T nat
level t <= gap -> cl_at gap t apply level1.
Qed.

End level_theory.

(** ** Properties of renaming *)
(**********************************************************************)
Section renaming_theory.

  Context
    `{Mapd_T_inst: Mapd nat T}
    `{Functor_inst: ! DecoratedFunctor nat T}.

  (** ** Renaming and <<ret>> *)
  (********************************************************************)
  Lemma lift__ren_zero:
    lift__ren 0 = id.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = id
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = id
    unfold lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
(fun ρ (ix : nat) =>
 if ix `bound in` 0 then ix else ρ (ix - 0) + 0) = id
    ext ρ ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
(if ix `bound in` 0 then ix else ρ (ix - 0) + 0) =
id ρ ix
    bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 0
Hbound: ix `bound in` 0 = false
ρ (ix - 0) + 0 = id ρ ix
    replace (ix - 0) with ix by lia.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 0
Hbound: ix `bound in` 0 = false
ρ ix + 0 = id ρ ix
    replace (ρ ix + 0) with (ρ ix) by lia.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 0
Hbound: ix `bound in` 0 = false
ρ ix = id ρ ix
    reflexivity.
  Qed.

  Corollary local__ren_zero: forall ρ ix,
    local__ren ρ (0, ix) = ρ ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (ix : nat), local__ren ρ (0, ix) = ρ ix
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (ix : nat), local__ren ρ (0, ix) = ρ ix
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
local__ren ρ (0, ix) = ρ ix
    unfold local__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
lift__ren 0 ρ ix = ρ ix
    rewrite lift__ren_zero.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
id ρ ix = ρ ix
    reflexivity.
  Qed.

  Corollary rename_ret
    `{Return T}
    `{Bindd nat T T}
    `{! Compat_Mapd_Bindd nat T T}
    `{! DecoratedMonad nat T}: forall ρ ix,
      rename ρ (ret ix) = ret (ρ ix).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
forall ρ (ix : nat), rename ρ (ret ix) = ret (ρ ix)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
forall ρ (ix : nat), rename ρ (ret ix) = ret (ρ ix)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
rename ρ (ret ix) = ret (ρ ix)
    unfold rename.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
mapd (local__ren ρ) (ret ix) = ret (ρ ix)
    compose near ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
(mapd (local__ren ρ) ∘ ret) ix = (ret ∘ ρ) ix
    rewrite mapd_ret.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
(ret ∘ local__ren ρ ∘ ret) ix = (ret ∘ ρ) ix
    unfold compose.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
ret (local__ren ρ (ret ix)) = ret (ρ ix)
    unfold_ops @Return_Writer.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
ret (local__ren ρ (Ƶ, ix)) = ret (ρ ix)
    unfold_ops @Monoid_unit_zero.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
ret (local__ren ρ (0, ix)) = ret (ρ ix)
    rewrite local__ren_zero.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
H: Return T
H0: Bindd nat T T
Compat_Mapd_Bindd0: Compat_Mapd_Bindd nat T T
DecoratedMonad0: DecoratedMonad nat T
ρ: nat -> nat
ix: nat
ret (ρ ix) = ret (ρ ix)
    reflexivity.
  Qed.

  (** ** Renaming identity *)
  (********************************************************************)
  Lemma lift__ren_id: forall depth,
      lift__ren depth id = id.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall depth : nat, lift__ren depth id = id
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall depth : nat, lift__ren depth id = id
    unfold lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall depth : nat,
(fun ix : nat =>
 if ix `bound in` depth
 then ix
 else id (ix - depth) + depth) = id
    intro depth.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
(fun ix : nat =>
 if ix `bound in` depth
 then ix
 else id (ix - depth) + depth) = id ext ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
(if ix `bound in` depth
 then ix
 else id (ix - depth) + depth) = id ix
    unfold id.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
(if ix `bound in` depth
 then ix
 else ix - depth + depth) = ix
    bound_induction.
  Qed.

  Corollary local__ren_id:
    local__ren id = extract.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
local__ren id = extract
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
local__ren id = extract
    ext [depth ix].
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
local__ren id (depth, ix) = extract (depth, ix)
    unfold local__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
lift__ren depth id ix = extract (depth, ix)
    rewrite lift__ren_id.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
id ix = extract (depth, ix)
    reflexivity.
  Qed.

  Lemma remame_id:
    rename id = (@id (T nat)).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
rename id = id
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
rename id = id
    unfold rename.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
mapd (local__ren id) = id
    rewrite local__ren_id.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
mapd extract = id
    apply kdf_mapd1.
  Qed.

  (** ** Composition of renaming *)
  (********************************************************************)
  Lemma lift__ren_loc_compose1: forall depth ix ρ1 ρ2,
      ix >= depth ->
      lift__ren depth ρ2 (ρ1 (ix - depth) + depth) =
        ρ2 (ρ1 (ix - depth)) + depth.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (depth ix : nat) ρ1 ρ2,
ix >= depth ->
lift__ren depth ρ2 (ρ1 (ix - depth) + depth) =
ρ2 (ρ1 (ix - depth)) + depth
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (depth ix : nat) ρ1 ρ2,
ix >= depth ->
lift__ren depth ρ2 (ρ1 (ix - depth) + depth) =
ρ2 (ρ1 (ix - depth)) + depth
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
lift__ren depth ρ2 (ρ1 (ix - depth) + depth) =
ρ2 (ρ1 (ix - depth)) + depth unfold lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
(if (ρ1 (ix - depth) + depth) `bound in` depth
 then ρ1 (ix - depth) + depth
 else ρ2 (ρ1 (ix - depth) + depth - depth) + depth) =
ρ2 (ρ1 (ix - depth)) + depth
    bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
Hord: ρ1 (ix - depth) + depth >= depth
Hbound: (ρ1 (ix - depth) + depth) `bound in` depth =
false
ρ2 (ρ1 (ix - depth) + depth - depth) + depth =
ρ2 (ρ1 (ix - depth)) + depth
    cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
Hord: ρ1 (ix - depth) + depth >= depth
Hbound: (ρ1 (ix - depth) + depth) `bound in` depth =
false
ρ2 (ρ1 (ix - depth) + depth - depth) + depth =
ρ2 (ρ1 (ix - depth)) + depth fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
Hord: ρ1 (ix - depth) + depth >= depth
Hbound: (ρ1 (ix - depth) + depth) `bound in` depth =
false
ρ2 (ρ1 (ix - depth) + depth - depth) =
ρ2 (ρ1 (ix - depth)) fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth, ix: nat
ρ1, ρ2: nat -> nat
H: ix >= depth
Hord: ρ1 (ix - depth) + depth >= depth
Hbound: (ρ1 (ix - depth) + depth) `bound in` depth =
false
ρ1 (ix - depth) + depth - depth = ρ1 (ix - depth) lia.
  Qed.

  Lemma lift__ren_loc_compose2: forall ρ depth ix,
      ix < depth ->
      lift__ren depth ρ ix = ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (depth ix : nat),
ix < depth -> lift__ren depth ρ ix = ix
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (depth ix : nat),
ix < depth -> lift__ren depth ρ ix = ix
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
H: ix < depth
lift__ren depth ρ ix = ix unfold lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
H: ix < depth
(if ix `bound in` depth
 then ix
 else ρ (ix - depth) + depth) = ix
    bound_induction.
  Qed.

  Lemma lift__ren_loc_compose: forall depth ρ1 ρ2,
      lift__ren depth ρ2 ∘ lift__ren depth ρ1 =
        lift__ren depth (ρ2 ∘ ρ1).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (depth : nat) ρ1 ρ2,
lift__ren depth ρ2 ∘ lift__ren depth ρ1 =
lift__ren depth (ρ2 ∘ ρ1)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (depth : nat) ρ1 ρ2,
lift__ren depth ρ2 ∘ lift__ren depth ρ1 =
lift__ren depth (ρ2 ∘ ρ1)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
ρ1, ρ2: nat -> nat
lift__ren depth ρ2 ∘ lift__ren depth ρ1 =
lift__ren depth (ρ2 ∘ ρ1) ext ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
ρ1, ρ2: nat -> nat
ix: nat
(lift__ren depth ρ2 ∘ lift__ren depth ρ1) ix =
lift__ren depth (ρ2 ∘ ρ1) ix
    unfold compose.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
ρ1, ρ2: nat -> nat
ix: nat
lift__ren depth ρ2 (lift__ren depth ρ1 ix) =
lift__ren depth (ρ2 ○ ρ1) ix
    unfold lift__ren at 2 3.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
ρ1, ρ2: nat -> nat
ix: nat
lift__ren depth ρ2
  (if ix `bound in` depth
   then ix
   else ρ1 (ix - depth) + depth) =
(if ix `bound in` depth
 then ix
 else ρ2 (ρ1 (ix - depth)) + depth)
    bound_induction;
      auto using lift__ren_loc_compose1, lift__ren_loc_compose2.
  Qed.

  Lemma rename_rename_loc: forall ρ2 ρ1,
      local__ren ρ2 ⋆1 local__ren ρ1 = local__ren (ρ2 ∘ ρ1).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ2 ρ1,
local__ren ρ2 ⋆1 local__ren ρ1 = local__ren (ρ2 ∘ ρ1)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ2 ρ1,
local__ren ρ2 ⋆1 local__ren ρ1 = local__ren (ρ2 ∘ ρ1)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
local__ren ρ2 ⋆1 local__ren ρ1 = local__ren (ρ2 ∘ ρ1)
    ext [depth ix].
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
depth, ix: nat
(local__ren ρ2 ⋆1 local__ren ρ1) (depth, ix) =
local__ren (ρ2 ∘ ρ1) (depth, ix) cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
depth, ix: nat
lift__ren depth ρ2 (lift__ren depth ρ1 ix) =
lift__ren depth (ρ2 ∘ ρ1) ix
    compose near ix on left.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
depth, ix: nat
(lift__ren depth ρ2 ∘ lift__ren depth ρ1) ix =
lift__ren depth (ρ2 ∘ ρ1) ix
    rewrite lift__ren_loc_compose.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
depth, ix: nat
lift__ren depth (ρ2 ∘ ρ1) ix =
lift__ren depth (ρ2 ∘ ρ1) ix
    reflexivity.
  Qed.

  Lemma rename_rename: forall ρ2 ρ1,
      rename ρ2 ∘ rename ρ1 = rename (T := T) (ρ2 ∘ ρ1).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ2 ρ1, rename ρ2 ∘ rename ρ1 = rename (ρ2 ∘ ρ1)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ2 ρ1, rename ρ2 ∘ rename ρ1 = rename (ρ2 ∘ ρ1)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
rename ρ2 ∘ rename ρ1 = rename (ρ2 ∘ ρ1)
    unfold rename.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
mapd (local__ren ρ2) ∘ mapd (local__ren ρ1) =
mapd (local__ren (ρ2 ∘ ρ1))
    rewrite kdf_mapd2.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
mapd (local__ren ρ2 ⋆1 local__ren ρ1) =
mapd (local__ren (ρ2 ∘ ρ1))
    rewrite rename_rename_loc.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ2, ρ1: nat -> nat
mapd (local__ren (ρ2 ∘ ρ1)) =
mapd (local__ren (ρ2 ∘ ρ1))
    reflexivity.
  Qed.

  (** ** Iterating lifting *)
  (********************************************************************)
  Lemma lift__ren_compose: forall (n m: nat),
      lift__ren m ∘ lift__ren n = lift__ren (m + n).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall n m : nat,
lift__ren m ∘ lift__ren n = lift__ren (m + n)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall n m : nat,
lift__ren m ∘ lift__ren n = lift__ren (m + n)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
lift__ren m ∘ lift__ren n = lift__ren (m + n) ext σ ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
(lift__ren m ∘ lift__ren n) σ ix =
lift__ren (m + n) σ ix
    unfold compose, lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
(if ix `bound in` m
 then ix
 else
  (if (ix - m) `bound in` n
   then ix - m
   else σ (ix - m - n) + n) + m) =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n))
    bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
(if (ix - m) `bound in` n
 then ix - m
 else σ (ix - m - n) + n) + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n))
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix < m
Hbound: ix `bound in` m = true
ix =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n))
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
(if (ix - m) `bound in` n
 then ix - m
 else σ (ix - m - n) + n) + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n)) bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
σ (ix - m - n) + n + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n))
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
ix - m + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n))
      +
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
σ (ix - m - n) + n + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n)) bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) + n + m = σ (ix - (m + n)) + (m + n)
        rewrite (Nat.add_comm m n).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) + n + m = σ (ix - (n + m)) + (n + m)
        rewrite Nat.add_assoc.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) + n + m = σ (ix - (n + m)) + n + m
        fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) + n = σ (ix - (n + m)) + n fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) = σ (ix - (n + m)) fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
ix - m - n = ix - (n + m) lia.
      +
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
ix - m + m =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n)) bound_induction.
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n, m: nat
σ: nat -> nat
ix: nat
Hord: ix < m
Hbound: ix `bound in` m = true
ix =
(if ix `bound in` (m + n)
 then ix
 else σ (ix - (m + n)) + (m + n)) bound_induction.
  Qed.

  Corollary lift__ren_S: forall (n: nat) ρ,
      lift__ren (S n) ρ = lift__ren n (lift__ren 1 ρ).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (n : nat) ρ,
lift__ren (S n) ρ = lift__ren n (lift__ren 1 ρ)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall (n : nat) ρ,
lift__ren (S n) ρ = lift__ren n (lift__ren 1 ρ)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
ρ: nat -> nat
lift__ren (S n) ρ = lift__ren n (lift__ren 1 ρ)
    replace (S n) with (n + 1) by lia.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
ρ: nat -> nat
lift__ren (n + 1) ρ = lift__ren n (lift__ren 1 ρ)
    compose near ρ on right.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
ρ: nat -> nat
lift__ren (n + 1) ρ = (lift__ren n ∘ lift__ren 1) ρ
    now rewrite lift__ren_compose.
  Qed.

  Lemma lift__ren_iter: forall n,
      lift__ren n = iterate n (lift__ren 1).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall n : nat, lift__ren n = iterate n (lift__ren 1)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall n : nat, lift__ren n = iterate n (lift__ren 1)
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
lift__ren n = iterate n (lift__ren 1)
    induction n.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = iterate 0 (lift__ren 1)
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
lift__ren (S n) = iterate (S n) (lift__ren 1)
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = iterate 0 (lift__ren 1) rewrite lift__ren_zero.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
id = iterate 0 (lift__ren 1)
      reflexivity.
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
lift__ren (S n) = iterate (S n) (lift__ren 1) ext ρ.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
ρ: nat -> nat
lift__ren (S n) ρ = iterate (S n) (lift__ren 1) ρ
      rewrite lift__ren_S.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
ρ: nat -> nat
lift__ren n (lift__ren 1 ρ) =
iterate (S n) (lift__ren 1) ρ
      rewrite IHn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
ρ: nat -> nat
iterate n (lift__ren 1) (lift__ren 1 ρ) =
iterate (S n) (lift__ren 1) ρ
      rewrite iterate_rw1.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
n: nat
IHn: lift__ren n = iterate n (lift__ren 1)
ρ: nat -> nat
iterate n (lift__ren 1) (lift__ren 1 ρ) =
(iterate n (lift__ren 1) ∘ lift__ren 1) ρ
      reflexivity.
  Qed.

  Lemma local__ren_preincr (ρ: nat -> nat) (n: nat):
    (local__ren ρ) ⦿ n =
      local__ren (lift__ren n ρ).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n: nat
local__ren ρ ⦿ n = local__ren (lift__ren n ρ)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n: nat
local__ren ρ ⦿ n = local__ren (lift__ren n ρ)
    ext (depth, ix); cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n, depth, ix: nat
lift__ren (n ● depth) ρ ix =
lift__ren depth (lift__ren n ρ) ix
    change (?n ● ?m) with (n + m).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n, depth, ix: nat
lift__ren (n + depth) ρ ix =
lift__ren depth (lift__ren n ρ) ix
    compose near ρ.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n, depth, ix: nat
lift__ren (n + depth) ρ ix =
(lift__ren depth ∘ lift__ren n) ρ ix
    rewrite (lift__ren_compose).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
n, depth, ix: nat
lift__ren (n + depth) ρ ix =
lift__ren (depth + n) ρ ix
    fequal; lia.
  Qed.

  (** ** Relating <<lift__ren>> to <<up__ren>> *)
  (********************************************************************)
  Lemma lift__ren_1:
    lift__ren 1 = up__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 1 = up__ren
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 1 = up__ren
    ext ρ ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
lift__ren 1 ρ ix = up__ren ρ ix
    unfold lift__ren, up__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
(if ix `bound in` 1 then ix else ρ (ix - 1) + 1) =
(0 ⋅ (S ∘ ρ)) ix
    bound_induction.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 1
Hbound: ix `bound in` 1 = false
ρ (ix - 1) + 1 = (0 ⋅ (S ∘ ρ)) ix
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix < 1
Hbound: ix `bound in` 1 = true
ix = (0 ⋅ (S ∘ ρ)) ix
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 1
Hbound: ix `bound in` 1 = false
ρ (ix - 1) + 1 = (0 ⋅ (S ∘ ρ)) ix cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix >= 1
Hbound: ix `bound in` 1 = false
ρ (ix - 1) + 1 = (0 ⋅ (S ∘ ρ)) ix destruct ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
Hord: 0 >= 1
Hbound: 0 `bound in` 1 = false
ρ (0 - 1) + 1 = (0 ⋅ (S ∘ ρ)) 0
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
ρ (S ix - 1) + 1 = (0 ⋅ (S ∘ ρ)) (S ix)
      +
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
Hord: 0 >= 1
Hbound: 0 `bound in` 1 = false
ρ (0 - 1) + 1 = (0 ⋅ (S ∘ ρ)) 0 false.
      +
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
ρ (S ix - 1) + 1 = (0 ⋅ (S ∘ ρ)) (S ix) cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
ρ (ix - 0) + 1 = (S ∘ ρ) ix unfold compose.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
ρ (ix - 0) + 1 = S (ρ ix)
        rewrite Nat.add_1_r.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
S (ρ (ix - 0)) = S (ρ ix)
        do 2 fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
ix - 0 = ix lia.
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix < 1
Hbound: ix `bound in` 1 = true
ix = (0 ⋅ (S ∘ ρ)) ix apply bound_1 in Hbound.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
ix: nat
Hord: ix < 1
Hbound: ix = 0
ix = (0 ⋅ (S ∘ ρ)) ix
      subst.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
Hord: 0 < 1
0 = (0 ⋅ (S ∘ ρ)) 0 reflexivity.
  Qed.

  Lemma lift__ren_repr: forall depth,
      lift__ren depth = iterate depth up__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall depth : nat,
lift__ren depth = iterate depth up__ren
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall depth : nat,
lift__ren depth = iterate depth up__ren
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
lift__ren depth = iterate depth up__ren
    induction depth.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = iterate 0 up__ren
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
lift__ren (S depth) = iterate (S depth) up__ren
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
lift__ren 0 = iterate 0 up__ren rewrite lift__ren_zero.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
id = iterate 0 up__ren
      rewrite iterate_rw0.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
id = id
      reflexivity.
    -
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
lift__ren (S depth) = iterate (S depth) up__ren ext ρ.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
ρ: nat -> nat
lift__ren (S depth) ρ = iterate (S depth) up__ren ρ
      rewrite lift__ren_S.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
ρ: nat -> nat
lift__ren depth (lift__ren 1 ρ) =
iterate (S depth) up__ren ρ
      rewrite iterate_rw1.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
ρ: nat -> nat
lift__ren depth (lift__ren 1 ρ) =
(iterate depth up__ren ∘ up__ren) ρ
      rewrite IHdepth.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
ρ: nat -> nat
iterate depth up__ren (lift__ren 1 ρ) =
(iterate depth up__ren ∘ up__ren) ρ
      rewrite lift__ren_1.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
depth: nat
IHdepth: lift__ren depth = iterate depth up__ren
ρ: nat -> nat
iterate depth up__ren (up__ren ρ) =
(iterate depth up__ren ∘ up__ren) ρ
      reflexivity.
  Qed.

  Corollary local__ren_repr: forall ρ depth ix,
      local__ren ρ (depth, ix) = iterate depth up__ren ρ ix.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (depth ix : nat),
local__ren ρ (depth, ix) = iterate depth up__ren ρ ix
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
forall ρ (depth ix : nat),
local__ren ρ (depth, ix) = iterate depth up__ren ρ ix
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
local__ren ρ (depth, ix) = iterate depth up__ren ρ ix cbn.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
lift__ren depth ρ ix = iterate depth up__ren ρ ix
    rewrite lift__ren_repr.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
iterate depth up__ren ρ ix =
iterate depth up__ren ρ ix
    reflexivity.
  Qed.

  (** ** Operations with policy *)
  (********************************************************************)
  Definition map_with_policy `{Mapd nat T}
    (policy: (nat -> nat) -> (nat -> nat)) (ρ: nat -> nat): T nat -> T nat :=
    mapd (fun '(depth, ix) => iterate depth policy ρ ix).

  Lemma rename_policy_repr (ρ: nat -> nat):
    rename (T := T) ρ = map_with_policy up__ren ρ.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
rename ρ = map_with_policy up__ren ρ
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
rename ρ = map_with_policy up__ren ρ
    unfold rename, map_with_policy.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
mapd (local__ren ρ) =
mapd (fun '(depth, ix) => iterate depth up__ren ρ ix)
    fequal.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
local__ren ρ =
(fun '(depth, ix) => iterate depth up__ren ρ ix) ext [depth ix].
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
depth, ix: nat
local__ren ρ (depth, ix) = iterate depth up__ren ρ ix
    apply local__ren_repr.
  Qed.

  Lemma local__ren_preincr_1 (ρ: nat -> nat):
    (local__ren ρ) ⦿ 1 =
      local__ren (up__ren ρ).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
local__ren ρ ⦿ 1 = local__ren (up__ren ρ)
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
local__ren ρ ⦿ 1 = local__ren (up__ren ρ)
    rewrite local__ren_preincr.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
local__ren (lift__ren 1 ρ) = local__ren (up__ren ρ)
    rewrite lift__ren_1.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
ρ: nat -> nat
local__ren (up__ren ρ) = local__ren (up__ren ρ)
    reflexivity.
  Qed.

  (** ** Characterizing occurrence *)
  (********************************************************************)

  Context
    `{Mapdt_inst: Mapdt nat T}
    `{! Compat_Mapd_Mapdt nat T}
    `{DTF_inst: ! DecoratedTraversableFunctor nat T}.

  Lemma ind_rename_iff: forall l n (t:T nat) ρ,
      (n, l) ∈d rename ρ t <-> exists l1,
        (n, l1) ∈d t /\
          l = (if l1 `bound in` n then l1
               else n + ρ (l1 - n)).
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
forall (l n : nat) (t : T nat) ρ,
(n, l) ∈d rename ρ t <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
  Proof.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
forall (l n : nat) (t : T nat) ρ,
(n, l) ∈d rename ρ t <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    intros.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
l, n: nat
t: T nat
ρ: nat -> nat
(n, l) ∈d rename ρ t <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    unfold rename in *.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
l, n: nat
t: T nat
ρ: nat -> nat
(n, l) ∈d mapd (local__ren ρ) t <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    rewrite ind_mapd_iff.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
l, n: nat
t: T nat
ρ: nat -> nat
(exists a : nat,
   (n, a) ∈d t /\ local__ren ρ (n, a) = l) <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    unfold local__ren, lift__ren.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
l, n: nat
t: T nat
ρ: nat -> nat
(exists a : nat,
   (n, a) ∈d t /\
   (if a `bound in` n then a else ρ (a - n) + n) = l) <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    setoid_rewrite Nat.add_comm at 1.
T: Type -> Type
Mapd_T_inst: Mapd nat T
Functor_inst: DecoratedFunctor nat T
Mapdt_inst: Mapdt nat T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt nat T
DTF_inst: DecoratedTraversableFunctor nat T
l, n: nat
t: T nat
ρ: nat -> nat
(exists a : nat,
   (n, a) ∈d t /\
   (if a `bound in` n then a else n + ρ (a - n)) = l) <->
(exists l1 : nat,
   (n, l1) ∈d t /\
   l =
   (if l1 `bound in` n then l1 else n + ρ (l1 - n)))
    intuition; preprocess; eauto.
  Qed.

End renaming_theory.

(** * De Bruijn operations *)
(**********************************************************************)
Section ops.

  Context
    `{ret_inst: Return T}
    `{Mapd_T_inst: Mapd nat T}
    `{Bindd_U_inst: Bindd nat T U}.

  (* Given a depth and substitution σ, adjust σ to account for the
       depth *)
  Definition lift__sub: nat -> (nat -> T nat) -> (nat -> T nat) :=
    fun depth σ ix =>
      if ix `bound in` depth
      then ret ix
      else let free_ix := ix - depth
           in rename (+ depth) (σ free_ix).

  Definition local__sub (σ: nat -> T nat) (p: nat * nat): T nat :=
    match p with
    | (depth, ix) => lift__sub depth σ ix
    end.

  Definition subst (σ: nat -> T nat): U nat -> U nat :=
    bindd (local__sub σ).

  (* adjust a substitution to go under one binder *)
  Definition up__sub (σ: nat -> T nat): nat -> T nat :=
    (ret 0) ⋅ (rename (+1) ∘ σ).

End ops.

Section theory.

  Context
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd nat T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt nat T}
    `{Bindd_T_inst: Bindd nat T T}
    `{Bindt_T_inst: Bindt T T}
    `{Binddt_T_inst: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{Monad_inst: ! DecoratedTraversableMonad nat T}.

  Context
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd nat U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt nat U}
    `{Bindd_U_inst: Bindd nat T U}
    `{Bindt_U_inst: Bindt T U}
    `{Binddt_U_inst: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
    (unit := Monoid_unit_zero)
    (op := Monoid_op_plus)}.

  (** * Properties of substitution *)
  (********************************************************************)

  (** ** Substitution and ret *)
  (********************************************************************)
  Lemma lift__sub_zero:
    lift__sub 0 = id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 0 = id
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 0 = id
    ext σ ix.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
lift__sub 0 σ ix = id σ ix unfold id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
lift__sub 0 σ ix = σ ix
    unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
(if ix `bound in` 0
 then ret ix
 else rename (+0) (σ (ix - 0))) = σ ix
    cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
rename (+0) (σ (ix - 0)) = σ ix
    replace (ix - 0) with ix by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
rename (+0) (σ ix) = σ ix
    replace (+ 0) with (@id nat).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
rename id (σ ix) = σ ix
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
id = (+0)
    2:{
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
id = (+0) ext n.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix, n: nat
id n = (+0) n unfold lift, id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix, n: nat
n = 0 + n lia. }
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
rename id (σ ix) = σ ix
    unfold rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
mapd (local__ren id) (σ ix) = σ ix
    rewrite local__ren_id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
mapd extract (σ ix) = σ ix
    rewrite mapd_id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
id (σ ix) = σ ix
    reflexivity.
  Qed.

  Lemma local__sub_zero: forall σ n,
    local__sub σ (0, n) = σ n.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n : nat),
local__sub σ (0, n) = σ n
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n : nat),
local__sub σ (0, n) = σ n
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
local__sub σ (0, n) = σ n unfold local__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
lift__sub 0 σ n = σ n
    rewrite lift__sub_zero.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
id σ n = σ n
    reflexivity.
  Qed.

  Lemma local__sub_compose_ret: forall σ x,
      (local__sub σ ∘ ret) x = σ x.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (x : nat),
(local__sub σ ∘ ret) x = σ x
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (x : nat),
(local__sub σ ∘ ret) x = σ x
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
(local__sub σ ∘ ret) x = σ x
    unfold_ops @Return_Writer @Monoid_unit_zero.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
(local__sub σ ∘ (fun a : nat => (0, a))) x = σ x
    unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
local__sub σ (0, x) = σ x
    rewrite local__sub_zero.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
σ x = σ x
    reflexivity.
  Qed.

  Lemma subst_ret: forall σ x,
      subst σ (ret x) = σ x.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (x : nat),
subst σ (ret x) = σ x
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (x : nat),
subst σ (ret x) = σ x
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
subst σ (ret x) = σ x
    unfold subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
bindd (local__sub σ) (ret x) = σ x
    compose near x on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
(bindd (local__sub σ) ∘ ret) x = σ x
    rewrite bindd_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
(local__sub σ ∘ ret) x = σ x
    apply local__sub_compose_ret.
  Qed.

  Lemma subst_compose_ret: forall σ,
      subst σ ∘ ret = σ.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ : nat -> T nat, subst σ ∘ ret = σ
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ : nat -> T nat, subst σ ∘ ret = σ
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
subst σ ∘ ret = σ ext x.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
(subst σ ∘ ret) x = σ x
    unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
x: nat
subst σ (ret x) = σ x apply subst_ret.
  Qed.

  (** ** Substitution and identity *)
  (********************************************************************)
  Lemma lift__sub_id: forall depth ix,
      lift__sub depth ret ix = (ret ∘ extract) (depth, ix).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall depth ix : nat,
lift__sub depth ret ix = (ret ∘ extract) (depth, ix)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall depth ix : nat,
lift__sub depth ret ix = (ret ∘ extract) (depth, ix)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
lift__sub depth ret ix = (ret ∘ extract) (depth, ix)
    unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
(if ix `bound in` depth
 then ret ix
 else rename (+depth) (ret (ix - depth))) =
(ret ∘ extract) (depth, ix)
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
rename (+depth) (ret (ix - depth)) =
(ret ∘ extract) (depth, ix)
    rewrite rename_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
ret ((+depth) (ix - depth)) =
(ret ∘ extract) (depth, ix)
    unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
ret ((+depth) (ix - depth)) =
ret (extract (depth, ix)) cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
ret (depth + (ix - depth)) = ret ix
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth + (ix - depth) = ix lia.
  Qed.

  Corollary local__subst_id:
    local__sub (T := T) (@ret T _ nat) = ret ∘ extract.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
local__sub ret = ret ∘ extract
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
local__sub ret = ret ∘ extract
    unfold local__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
(fun p : nat * nat =>
 let (depth, ix) := p in lift__sub depth ret ix) =
ret ∘ extract ext [depth ix].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
lift__sub depth ret ix = (ret ∘ extract) (depth, ix)
    apply lift__sub_id.
  Qed.

  Lemma subst_id:
      subst (@ret T _ nat) = id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
subst ret = id
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
subst ret = id
    unfold subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
bindd (local__sub ret) = id
    rewrite local__subst_id.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
bindd (ret ∘ extract) = id
    apply bindd_id.
  Qed.

  (** ** Iterated lifting *)
  (********************************************************************)
  Lemma lift__sub_compose: forall (n m: nat),
      lift__sub m ∘ lift__sub n =
        lift__sub (m + n).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n m : nat,
lift__sub m ∘ lift__sub n = lift__sub (m + n)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n m : nat,
lift__sub m ∘ lift__sub n = lift__sub (m + n)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
lift__sub m ∘ lift__sub n = lift__sub (m + n) ext σ ix.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
(lift__sub m ∘ lift__sub n) σ ix =
lift__sub (m + n) σ ix
    unfold compose, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
(if ix `bound in` m
 then ret ix
 else
  rename (+m)
    (if (ix - m) `bound in` n
     then ret (ix - m)
     else rename (+n) (σ (ix - m - n)))) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n))))
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
rename (+m)
  (if (ix - m) `bound in` n
   then ret (ix - m)
   else rename (+n) (σ (ix - m - n))) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n))))
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix < m
Hbound: ix `bound in` m = true
ret ix =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n))))
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
rename (+m)
  (if (ix - m) `bound in` n
   then ret (ix - m)
   else rename (+n) (σ (ix - m - n))) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n)))) bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
rename (+m) (rename (+n) (σ (ix - m - n))) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n))))
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
rename (+m) (ret (ix - m)) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n))))
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
rename (+m) (rename (+n) (σ (ix - m - n))) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n)))) bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
rename (+m) (rename (+n) (σ (ix - m - n))) =
rename (+m + n) (σ (ix - (m + n)))
        compose near (σ (ix - m - n)) on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
(rename (+m) ∘ rename (+n)) (σ (ix - m - n)) =
rename (+m + n) (σ (ix - (m + n)))
        rewrite rename_rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
rename ((+m) ∘ (+n)) (σ (ix - m - n)) =
rename (+m + n) (σ (ix - (m + n)))
        rewrite lift_comp.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
rename (+m + n) (σ (ix - m - n)) =
rename (+m + n) (σ (ix - (m + n)))
        fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
σ (ix - m - n) = σ (ix - (m + n)) fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m >= n
Hbound0: (ix - m) `bound in` n = false
Hord1: ix >= m + n
Hbound1: ix `bound in` (m + n) = false
ix - m - n = ix - (m + n)
        lia.
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
rename (+m) (ret (ix - m)) =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n)))) bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
rename (+m) (ret (ix - m)) = ret ix
        compose near (ix - m).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
(rename (+m) ∘ ret) (ix - m) = ret ix
        unfold rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
(mapd (local__ren (+m)) ∘ ret) (ix - m) = ret ix
        rewrite mapd_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
(ret ∘ local__ren (+m) ∘ ret) (ix - m) = ret ix
        unfold compose; cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
ret (m + (ix - m - Ƶ) + Ƶ) = ret ix
        fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
m + (ix - m - Ƶ) + Ƶ = ix
        unfold_ops @Monoid_unit_zero.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix >= m
Hbound: ix `bound in` m = false
Hord0: ix - m < n
Hbound0: (ix - m) `bound in` n = true
Hord1: ix < m + n
Hbound1: ix `bound in` (m + n) = true
m + (ix - m - 0) + 0 = ix
        lia.
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, m: nat
σ: nat -> T nat
ix: nat
Hord: ix < m
Hbound: ix `bound in` m = true
ret ix =
(if ix `bound in` (m + n)
 then ret ix
 else rename (+m + n) (σ (ix - (m + n)))) bound_induction.
  Qed.

  Corollary lift__sub_S: forall (n: nat) σ,
      lift__sub (S n) σ =
        lift__sub n (lift__sub 1 σ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) (σ : nat -> T nat),
lift__sub (S n) σ = lift__sub n (lift__sub 1 σ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) (σ : nat -> T nat),
lift__sub (S n) σ = lift__sub n (lift__sub 1 σ)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
σ: nat -> T nat
lift__sub (S n) σ = lift__sub n (lift__sub 1 σ)
    replace (S n) with (n + 1) by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
σ: nat -> T nat
lift__sub (n + 1) σ = lift__sub n (lift__sub 1 σ)
    compose near σ on right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
σ: nat -> T nat
lift__sub (n + 1) σ = (lift__sub n ∘ lift__sub 1) σ
    now rewrite lift__sub_compose.
  Qed.

  Lemma lift__sub_iter: forall n,
      lift__sub n = iterate n (lift__sub 1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, lift__sub n = iterate n (lift__sub 1)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, lift__sub n = iterate n (lift__sub 1)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
lift__sub n = iterate n (lift__sub 1) induction n.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 0 = iterate 0 (lift__sub 1)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
lift__sub (S n) = iterate (S n) (lift__sub 1)
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 0 = iterate 0 (lift__sub 1) rewrite lift__sub_zero.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
id = iterate 0 (lift__sub 1)
      reflexivity.
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
lift__sub (S n) = iterate (S n) (lift__sub 1) ext σ.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
σ: nat -> T nat
lift__sub (S n) σ = iterate (S n) (lift__sub 1) σ
      rewrite lift__sub_S.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
σ: nat -> T nat
lift__sub n (lift__sub 1 σ) =
iterate (S n) (lift__sub 1) σ
      rewrite iterate_rw1A.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
σ: nat -> T nat
lift__sub n (lift__sub 1 σ) =
iterate n (lift__sub 1) (lift__sub 1 σ)
      rewrite IHn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
IHn: lift__sub n = iterate n (lift__sub 1)
σ: nat -> T nat
iterate n (lift__sub 1) (lift__sub 1 σ) =
iterate n (lift__sub 1) (lift__sub 1 σ)
      reflexivity.
  Qed.

  Lemma local__sub_preincr (σ: nat -> T nat) (n: nat):
    (local__sub σ) ⦿ n =
      local__sub (lift__sub n σ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
local__sub σ ⦿ n = local__sub (lift__sub n σ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
local__sub σ ⦿ n = local__sub (lift__sub n σ)
    unfold local__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
(fun p : nat * nat =>
 let (depth, ix) := p in lift__sub depth σ ix) ⦿ n =
(fun p : nat * nat =>
 let (depth, ix) := p in
 lift__sub depth (lift__sub n σ) ix)
    ext (depth, ix); cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n, depth, ix: nat
lift__sub (n ● depth) σ ix =
lift__sub depth (lift__sub n σ) ix
    compose near σ on right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n, depth, ix: nat
lift__sub (n ● depth) σ ix =
(lift__sub depth ∘ lift__sub n) σ ix
    rewrite (lift__sub_compose).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n, depth, ix: nat
lift__sub (n ● depth) σ ix =
lift__sub (depth + n) σ ix
    change (?n ● ?m) with (n + m).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n, depth, ix: nat
lift__sub (n + depth) σ ix =
lift__sub (depth + n) σ ix
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n, depth, ix: nat
n + depth = depth + n lia.
  Qed.

  (** ** Substitution and composition *)
  (********************************************************************)
  Lemma subst_subst_loc: forall (σ2 σ1: nat -> T nat),
      local__sub (T := T) σ2 ⋆5 local__sub (T := T) σ1 =
        local__sub (T := T) (subst σ2 ∘ σ1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ2 σ1 : nat -> T nat,
local__sub σ2 ⋆5 local__sub σ1 =
local__sub (subst σ2 ∘ σ1)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ2 σ1 : nat -> T nat,
local__sub σ2 ⋆5 local__sub σ1 =
local__sub (subst σ2 ∘ σ1)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
local__sub σ2 ⋆5 local__sub σ1 =
local__sub (subst σ2 ∘ σ1)
    unfold kc5.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
(fun '(w, a) =>
 bindd (local__sub σ2 ⦿ w) (local__sub σ1 (w, a))) =
local__sub (subst σ2 ∘ σ1)
    ext [depth ix].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
bindd (local__sub σ2 ⦿ depth)
  (local__sub σ1 (depth, ix)) =
local__sub (subst σ2 ∘ σ1) (depth, ix)
    unfold local__sub at 2 3.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
bindd (local__sub σ2 ⦿ depth) (lift__sub depth σ1 ix) =
lift__sub depth (subst σ2 ∘ σ1) ix
    unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
bindd (local__sub σ2 ⦿ depth)
  (if ix `bound in` depth
   then ret ix
   else rename (+depth) (σ1 (ix - depth))) =
(if ix `bound in` depth
 then ret ix
 else rename (+depth) ((subst σ2 ∘ σ1) (ix - depth)))
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub σ2 ⦿ depth)
  (rename (+depth) (σ1 (ix - depth))) =
rename (+depth) ((subst σ2 ∘ σ1) (ix - depth))
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
bindd (local__sub σ2 ⦿ depth) (ret ix) = ret ix
    2:{
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
bindd (local__sub σ2 ⦿ depth) (ret ix) = ret ix compose near ix on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
(bindd (local__sub σ2 ⦿ depth) ∘ ret) ix = ret ix
        rewrite bindd_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
(local__sub σ2 ⦿ depth ∘ ret) ix = ret ix
        unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
(local__sub σ2 ⦿ depth) (ret ix) = ret ix
        unfold_ops @Return_Writer.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
(local__sub σ2 ⦿ depth) (Ƶ, ix) = ret ix
        unfold local__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
((fun p : nat * nat =>
  let (depth, ix) := p in lift__sub depth σ2 ix)
 ⦿ depth) (Ƶ, ix) = ret ix cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
lift__sub (depth ● Ƶ) σ2 ix = ret ix
        rewrite monoid_id_r.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
lift__sub depth σ2 ix = ret ix
        unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
(if ix `bound in` depth
 then ret ix
 else rename (+depth) (σ2 (ix - depth))) = ret ix
        rewrite Hbound.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix < depth
Hbound: ix `bound in` depth = true
ret ix = ret ix reflexivity. }
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub σ2 ⦿ depth)
  (rename (+depth) (σ1 (ix - depth))) =
rename (+depth) ((subst σ2 ∘ σ1) (ix - depth))
    {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub σ2 ⦿ depth)
  (rename (+depth) (σ1 (ix - depth))) =
rename (+depth) ((subst σ2 ∘ σ1) (ix - depth)) unfold compose at 1.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub σ2 ⦿ depth)
  (rename (+depth) (σ1 (ix - depth))) =
rename (+depth) (subst σ2 (σ1 (ix - depth)))
      unfold rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub σ2 ⦿ depth)
  (mapd (local__ren (+depth)) (σ1 (ix - depth))) =
mapd (local__ren (+depth))
  (subst σ2 (σ1 (ix - depth)))
      rewrite local__sub_preincr.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd (local__sub (lift__sub depth σ2))
  (mapd (local__ren (+depth)) (σ1 (ix - depth))) =
mapd (local__ren (+depth))
  (subst σ2 (σ1 (ix - depth)))
      compose near (σ1 (ix - depth)) on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
(bindd (local__sub (lift__sub depth σ2))
 ∘ mapd (local__ren (+depth))) (σ1 (ix - depth)) =
mapd (local__ren (+depth))
  (subst σ2 (σ1 (ix - depth)))
      compose near (σ1 (ix - depth)) on right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
(bindd (local__sub (lift__sub depth σ2))
 ∘ mapd (local__ren (+depth))) (σ1 (ix - depth)) =
(mapd (local__ren (+depth)) ∘ subst σ2)
  (σ1 (ix - depth))
      rewrite bindd_mapd.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd
  (local__sub (lift__sub depth σ2)
   ⋆1 local__ren (+depth)) (σ1 (ix - depth)) =
(mapd (local__ren (+depth)) ∘ subst σ2)
  (σ1 (ix - depth))
      unfold subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd
  (local__sub (lift__sub depth σ2)
   ⋆1 local__ren (+depth)) (σ1 (ix - depth)) =
(mapd (local__ren (+depth)) ∘ bindd (local__sub σ2))
  (σ1 (ix - depth))
      rewrite mapd_bindd.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
bindd
  (local__sub (lift__sub depth σ2)
   ⋆1 local__ren (+depth)) (σ1 (ix - depth)) =
bindd
  (fun '(w, t) =>
   mapd (local__ren (+depth) ⦿ w)
     (local__sub σ2 (w, t))) (σ1 (ix - depth))
      fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
local__sub (lift__sub depth σ2) ⋆1 local__ren (+depth) =
(fun '(w, t) =>
 mapd (local__ren (+depth) ⦿ w) (local__sub σ2 (w, t)))
      ext [depth' ix'].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
(local__sub (lift__sub depth σ2)
 ⋆1 local__ren (+depth)) (depth', ix') =
mapd (local__ren (+depth) ⦿ depth')
  (local__sub σ2 (depth', ix'))
      {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
(local__sub (lift__sub depth σ2)
 ⋆1 local__ren (+depth)) (depth', ix') =
mapd (local__ren (+depth) ⦿ depth')
  (local__sub σ2 (depth', ix')) rewrite local__ren_preincr.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
(local__sub (lift__sub depth σ2)
 ⋆1 local__ren (+depth)) (depth', ix') =
mapd (local__ren (lift__ren depth' (+depth)))
  (local__sub σ2 (depth', ix'))
        cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
lift__sub depth' (lift__sub depth σ2)
  (lift__ren depth' (+depth) ix') =
mapd (local__ren (lift__ren depth' (+depth)))
  (lift__sub depth' σ2 ix') unfold lift__ren at 1.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
lift__sub depth' (lift__sub depth σ2)
  (if ix' `bound in` depth'
   then ix'
   else (+depth) (ix' - depth') + depth') =
mapd (local__ren (lift__ren depth' (+depth)))
  (lift__sub depth' σ2 ix')
        unfold lift__sub at 3.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
lift__sub depth' (lift__sub depth σ2)
  (if ix' `bound in` depth'
   then ix'
   else (+depth) (ix' - depth') + depth') =
mapd (local__ren (lift__ren depth' (+depth)))
  (if ix' `bound in` depth'
   then ret ix'
   else rename (+depth') (σ2 (ix' - depth')))
        bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2)
  ((+depth) (ix' - depth') + depth') =
mapd (local__ren (lift__ren depth' (+depth)))
  (rename (+depth') (σ2 (ix' - depth')))
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub depth' (lift__sub depth σ2) ix' =
mapd (local__ren (lift__ren depth' (+depth)))
  (ret ix')
        -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2)
  ((+depth) (ix' - depth') + depth') =
mapd (local__ren (lift__ren depth' (+depth)))
  (rename (+depth') (σ2 (ix' - depth')))  unfold lift at 1.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2)
  (depth + (ix' - depth') + depth') =
mapd (local__ren (lift__ren depth' (+depth)))
  (rename (+depth') (σ2 (ix' - depth')))
           replace (depth + (ix' - depth') + depth') with (ix' + depth) by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2) (ix' + depth) =
mapd (local__ren (lift__ren depth' (+depth)))
  (rename (+depth') (σ2 (ix' - depth')))
           unfold rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2) (ix' + depth) =
mapd (local__ren (lift__ren depth' (+depth)))
  (mapd (local__ren (+depth')) (σ2 (ix' - depth')))
           compose near (σ2 (ix' - depth')).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2) (ix' + depth) =
(mapd (local__ren (lift__ren depth' (+depth)))
 ∘ mapd (local__ren (+depth'))) (σ2 (ix' - depth'))
           rewrite kdf_mapd2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2) (ix' + depth) =
mapd
  (local__ren (lift__ren depth' (+depth))
   ⋆1 local__ren (+depth')) (σ2 (ix' - depth'))
           rewrite rename_rename_loc.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
lift__sub depth' (lift__sub depth σ2) (ix' + depth) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
(if (ix' + depth) `bound in` depth'
 then ret (ix' + depth)
 else
  rename (+depth')
    (if (ix' + depth - depth') `bound in` depth
     then ret (ix' + depth - depth')
     else
      rename (+depth)
        (σ2 (ix' + depth - depth' - depth)))) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
rename (+depth')
  (if (ix' + depth - depth') `bound in` depth
   then ret (ix' + depth - depth')
   else
    rename (+depth)
      (σ2 (ix' + depth - depth' - depth))) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
rename (+depth')
  (rename (+depth) (σ2 (ix' + depth - depth' - depth))) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           replace (ix' + depth - depth' - depth) with (ix' - depth') by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
rename (+depth') (rename (+depth) (σ2 (ix' - depth'))) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           compose near (σ2 (ix' - depth')) on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
(rename (+depth') ∘ rename (+depth))
  (σ2 (ix' - depth')) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           rewrite rename_rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
rename ((+depth') ∘ (+depth)) (σ2 (ix' - depth')) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth'))
           unfold rename.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
mapd (local__ren ((+depth') ∘ (+depth)))
  (σ2 (ix' - depth')) =
mapd
  (local__ren (lift__ren depth' (+depth) ∘ (+depth')))
  (σ2 (ix' - depth')) fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
local__ren ((+depth') ∘ (+depth)) =
local__ren (lift__ren depth' (+depth) ∘ (+depth'))
           unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
local__ren ((+depth') ○ (+depth)) =
local__ren (lift__ren depth' (+depth) ○ (+depth'))
           fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
(+depth') ○ (+depth) =
lift__ren depth' (+depth) ○ (+depth')
           ext m.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
m: nat
(+depth') ((+depth) m) =
lift__ren depth' (+depth) ((+depth') m) unfold lift__ren.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
m: nat
(+depth') ((+depth) m) =
(if ((+depth') m) `bound in` depth'
 then (+depth') m
 else (+depth) ((+depth') m - depth') + depth')
           unfold lift.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' >= depth'
Hbound0: ix' `bound in` depth' = false
Hord1: ix' + depth >= depth'
Hbound1: (ix' + depth) `bound in` depth' = false
Hord2: ix' + depth - depth' >= depth
Hbound2: (ix' + depth - depth') `bound in` depth =
false
m: nat
depth' + (depth + m) =
(if (depth' + m) `bound in` depth'
 then depth' + m
 else depth + (depth' + m - depth') + depth')
           bound_induction.
        -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub depth' (lift__sub depth σ2) ix' =
mapd (local__ren (lift__ren depth' (+depth)))
  (ret ix') compose near ix' on right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub depth' (lift__sub depth σ2) ix' =
(mapd (local__ren (lift__ren depth' (+depth))) ∘ ret)
  ix'
          rewrite mapd_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub depth' (lift__sub depth σ2) ix' =
(ret ∘ local__ren (lift__ren depth' (+depth)) ∘ ret)
  ix'
          compose near σ2 on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
(lift__sub depth' ∘ lift__sub depth) σ2 ix' =
(ret ∘ local__ren (lift__ren depth' (+depth)) ∘ ret)
  ix'
          rewrite lift__sub_compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub (depth' + depth) σ2 ix' =
(ret ∘ local__ren (lift__ren depth' (+depth)) ∘ ret)
  ix'
          unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub (depth' + depth) σ2 ix' =
ret (local__ren (lift__ren depth' (+depth)) (ret ix')) unfold transparent tcs.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub (depth' + depth) σ2 ix' =
ret (local__ren (lift__ren depth' (+depth)) (0, ix'))
          cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
lift__sub (depth' + depth) σ2 ix' =
ret (lift__ren depth' (+depth) (ix' - 0) + 0)
          unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
(if ix' `bound in` (depth' + depth)
 then ret ix'
 else
  rename (+depth' + depth)
    (σ2 (ix' - (depth' + depth)))) =
ret (lift__ren depth' (+depth) (ix' - 0) + 0)
          bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
Hord1: ix' < depth' + depth
Hbound1: ix' `bound in` (depth' + depth) = true
ret ix' =
ret (lift__ren depth' (+depth) (ix' - 0) + 0)
          fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
Hord1: ix' < depth' + depth
Hbound1: ix' `bound in` (depth' + depth) = true
ix' = lift__ren depth' (+depth) (ix' - 0) + 0
          unfold lift__ren.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
Hord1: ix' < depth' + depth
Hbound1: ix' `bound in` (depth' + depth) = true
ix' =
(if (ix' - 0) `bound in` depth'
 then ix' - 0
 else (+depth) (ix' - 0 - depth') + depth') + 0
          replace (ix' - 0) with ix' by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
depth', ix': nat
Hord0: ix' < depth'
Hbound0: ix' `bound in` depth' = true
Hord1: ix' < depth' + depth
Hbound1: ix' `bound in` (depth' + depth) = true
ix' =
(if ix' `bound in` depth'
 then ix'
 else (+depth) (ix' - depth') + depth') + 0 bound_induction.
      }
    }
  Qed.

  Lemma subst_subst: forall (σ2 σ1: nat -> T nat),
      subst σ2 ∘ subst σ1 = subst (T := T) (subst σ2 ∘ σ1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ2 σ1 : nat -> T nat,
subst σ2 ∘ subst σ1 = subst (subst σ2 ∘ σ1)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall σ2 σ1 : nat -> T nat,
subst σ2 ∘ subst σ1 = subst (subst σ2 ∘ σ1)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
subst σ2 ∘ subst σ1 = subst (subst σ2 ∘ σ1)
    unfold subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
bindd (local__sub σ2) ∘ bindd (local__sub σ1) =
bindd (local__sub (bindd (local__sub σ2) ∘ σ1))
    rewrite bindd_bindd.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
bindd (local__sub σ2 ⋆5 local__sub σ1) =
bindd (local__sub (bindd (local__sub σ2) ∘ σ1))
    rewrite subst_subst_loc.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ2, σ1: nat -> T nat
bindd (local__sub (subst σ2 ∘ σ1)) =
bindd (local__sub (bindd (local__sub σ2) ∘ σ1))
    reflexivity.
  Qed.

  (** ** Substitution and renaming *)
  (********************************************************************)
  Lemma lift__ren_to_sub: forall n ρ,
      ret ∘ lift__ren n ρ = lift__sub n (ret ∘ ρ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) ρ,
ret ∘ lift__ren n ρ = lift__sub n (ret ∘ ρ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (n : nat) ρ,
ret ∘ lift__ren n ρ = lift__sub n (ret ∘ ρ)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
ret ∘ lift__ren n ρ = lift__sub n (ret ∘ ρ) ext m.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
(ret ∘ lift__ren n ρ) m = lift__sub n (ret ∘ ρ) m
    unfold compose; cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
ret (lift__ren n ρ m) = lift__sub n (ret ○ ρ) m
    unfold lift__sub, lift__ren.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
ret (if m `bound in` n then m else ρ (m - n) + n) =
(if m `bound in` n
 then ret m
 else rename (+n) (ret (ρ (m - n))))
    unfold lift.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
ret (if m `bound in` n then m else ρ (m - n) + n) =
(if m `bound in` n
 then ret m
 else rename (fun y : nat => n + y) (ret (ρ (m - n))))
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
Hord: m >= n
Hbound: m `bound in` n = false
ret (ρ (m - n) + n) =
rename (fun y : nat => n + y) (ret (ρ (m - n)))
    rewrite rename_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
Hord: m >= n
Hbound: m `bound in` n = false
ret (ρ (m - n) + n) = ret (n + ρ (m - n))
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
ρ: nat -> nat
m: nat
Hord: m >= n
Hbound: m `bound in` n = false
ρ (m - n) + n = n + ρ (m - n) lia.
  Qed.

  Lemma rename_to_subst_loc: forall ρ,
      ret ∘ local__ren ρ = local__sub (T := T) (ret ∘ ρ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall ρ, ret ∘ local__ren ρ = local__sub (ret ∘ ρ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall ρ, ret ∘ local__ren ρ = local__sub (ret ∘ ρ)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
ret ∘ local__ren ρ = local__sub (ret ∘ ρ) ext [depth ix].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
(ret ∘ local__ren ρ) (depth, ix) =
local__sub (ret ∘ ρ) (depth, ix)
    unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret (local__ren ρ (depth, ix)) =
local__sub (ret ○ ρ) (depth, ix) cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret (lift__ren depth ρ ix) =
lift__sub depth (ret ○ ρ) ix
    compose near ix on left.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
(ret ∘ lift__ren depth ρ) ix =
lift__sub depth (ret ○ ρ) ix
    rewrite lift__ren_to_sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
lift__sub depth (ret ∘ ρ) ix =
lift__sub depth (ret ○ ρ) ix
    reflexivity.
  Qed.

  Lemma rename_to_subst: forall ρ,
      rename ρ = subst (U := T) (ret ∘ ρ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall ρ, rename ρ = subst (ret ∘ ρ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall ρ, rename ρ = subst (ret ∘ ρ)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
rename ρ = subst (ret ∘ ρ)
    unfold rename, subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
mapd (local__ren ρ) = bindd (local__sub (ret ∘ ρ))
    rewrite mapd_to_bindd.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
bindd (ret ∘ local__ren ρ) =
bindd (local__sub (ret ∘ ρ))
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
ret ∘ local__ren ρ = local__sub (ret ∘ ρ)
    (* Fails with a universe error for some reason
    now rewrite (rename_to_subst_loc ρ).
     *)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
ret ∘ local__ren ρ = local__sub (ret ∘ ρ) ext [depth ix].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
(ret ∘ local__ren ρ) (depth, ix) =
local__sub (ret ∘ ρ) (depth, ix) unfold compose.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret (local__ren ρ (depth, ix)) =
local__sub (ret ○ ρ) (depth, ix)
    cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret (lift__ren depth ρ ix) =
lift__sub depth (ret ○ ρ) ix unfold lift__ren, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret
  (if ix `bound in` depth
   then ix
   else ρ (ix - depth) + depth) =
(if ix `bound in` depth
 then ret ix
 else rename (+depth) (ret (ρ (ix - depth))))
    unfold lift.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
ret
  (if ix `bound in` depth
   then ix
   else ρ (ix - depth) + depth) =
(if ix `bound in` depth
 then ret ix
 else
  rename (fun y : nat => depth + y)
    (ret (ρ (ix - depth))))
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
ret (ρ (ix - depth) + depth) =
rename (fun y : nat => depth + y)
  (ret (ρ (ix - depth)))
    rewrite rename_ret.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
ρ: nat -> nat
depth, ix: nat
Hord: ix >= depth
Hbound: ix `bound in` depth = false
ret (ρ (ix - depth) + depth) =
ret (depth + ρ (ix - depth))
    fequal; lia.
  Qed.

  (** ** up__sub and lift__spec *)
  (********************************************************************)
  Lemma up__sub_unfold (σ: nat -> T nat):
    up__sub σ = ret 0 ⋅ (subst (ret ∘ (+1)) ∘ σ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
up__sub σ = (ret 0) ⋅ (subst (ret ∘ (+1)) ∘ σ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
up__sub σ = (ret 0) ⋅ (subst (ret ∘ (+1)) ∘ σ)
    unfold up__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
(ret 0) ⋅ (rename (+1) ∘ σ) =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ σ)
    rewrite rename_to_subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ σ) =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ σ)
    reflexivity.
  Qed.

  Lemma up_spec:
    lift__sub 1 = up__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 1 = up__sub
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
lift__sub 1 = up__sub
    ext σ ix.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
lift__sub 1 σ ix = up__sub σ ix
    unfold up__sub, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
(if ix `bound in` 1
 then ret ix
 else rename (+1) (σ (ix - 1))) =
((ret 0) ⋅ (rename (+1) ∘ σ)) ix
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: ix >= 1
Hbound: ix `bound in` 1 = false
rename (+1) (σ (ix - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) ix
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: ix < 1
Hbound: ix `bound in` 1 = true
ret ix = ((ret 0) ⋅ (rename (+1) ∘ σ)) ix
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: ix >= 1
Hbound: ix `bound in` 1 = false
rename (+1) (σ (ix - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) ix destruct ix.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
Hord: 0 >= 1
Hbound: 0 `bound in` 1 = false
rename (+1) (σ (0 - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) 0
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
rename (+1) (σ (S ix - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) (S ix)
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
Hord: 0 >= 1
Hbound: 0 `bound in` 1 = false
rename (+1) (σ (0 - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) 0 false.
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
rename (+1) (σ (S ix - 1)) =
((ret 0) ⋅ (rename (+1) ∘ σ)) (S ix) replace (S ix - 1) with ix by lia.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: S ix >= 1
Hbound: (S ix) `bound in` 1 = false
rename (+1) (σ ix) =
((ret 0) ⋅ (rename (+1) ∘ σ)) (S ix)
        reflexivity.
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: ix < 1
Hbound: ix `bound in` 1 = true
ret ix = ((ret 0) ⋅ (rename (+1) ∘ σ)) ix apply bound_1 in Hbound.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
ix: nat
Hord: ix < 1
Hbound: ix = 0
ret ix = ((ret 0) ⋅ (rename (+1) ∘ σ)) ix
      now subst.
  Qed.

  Lemma local__sub_policy_repr: forall depth ix σ,
      local__sub σ (depth, ix) = iterate depth up__sub σ ix.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (depth ix : nat) (σ : nat -> T nat),
local__sub σ (depth, ix) = iterate depth up__sub σ ix
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (depth ix : nat) (σ : nat -> T nat),
local__sub σ (depth, ix) = iterate depth up__sub σ ix
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
σ: nat -> T nat
local__sub σ (depth, ix) = iterate depth up__sub σ ix
    unfold local__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
σ: nat -> T nat
lift__sub depth σ ix = iterate depth up__sub σ ix
    rewrite lift__sub_iter.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
σ: nat -> T nat
iterate depth (lift__sub 1) σ ix =
iterate depth up__sub σ ix
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
depth, ix: nat
σ: nat -> T nat
lift__sub 1 = up__sub
    apply up_spec.
  Qed.

  Lemma local__sub_preincr_1 (σ: nat -> T nat):
    (local__sub σ) ⦿ 1 =
      local__sub (up__sub σ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
local__sub σ ⦿ 1 = local__sub (up__sub σ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
local__sub σ ⦿ 1 = local__sub (up__sub σ)
    rewrite local__sub_preincr.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
local__sub (lift__sub 1 σ) = local__sub (up__sub σ)
    rewrite up_spec.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
local__sub (up__sub σ) = local__sub (up__sub σ)
    reflexivity.
  Qed.

  Definition bind_with_policy `{Bindd nat T U}
    (policy: (nat -> T nat) -> (nat -> T nat)) (σ: nat -> T nat): U nat -> U nat :=
    bindd (fun '(depth, ix) => iterate depth policy σ ix).

  Lemma subst_policy_repr (σ: nat -> T nat):
    subst σ = bind_with_policy up__sub σ.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
subst σ = bind_with_policy up__sub σ
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
subst σ = bind_with_policy up__sub σ
    unfold subst, bind_with_policy.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
bindd (local__sub σ) =
bindd (fun '(depth, ix) => iterate depth up__sub σ ix)
    fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
local__sub σ =
(fun '(depth, ix) => iterate depth up__sub σ ix)
    ext [depth ix].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
depth, ix: nat
local__sub σ (depth, ix) = iterate depth up__sub σ ix
    rewrite local__sub_policy_repr.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
depth, ix: nat
iterate depth up__sub σ ix =
iterate depth up__sub σ ix
    reflexivity.
  Qed.

  Lemma iterate_up__sub_unfold (σ: nat -> T nat) (n: nat):
    iterate (S n) up__sub σ = ret 0 ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
iterate (S n) up__sub σ =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
iterate (S n) up__sub σ =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ)
    rewrite iterate_rw2A.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
up__sub (iterate n up__sub σ) =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ)
    rewrite up__sub_unfold.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n: nat
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ) =
(ret 0) ⋅ (subst (ret ∘ (+1)) ∘ iterate n up__sub σ)
    reflexivity.
  Qed.

  (** ** Characterizing occurrence *)
  (********************************************************************)
  Lemma ind_local_sub1: forall (σ: nat -> T nat) n1 ix1,
      ix1 `bound in` n1 = true ->
                   (0, ix1) ∈d local__sub σ (n1, ix1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n1 ix1 : nat),
ix1 `bound in` n1 = true ->
(0, ix1) ∈d local__sub σ (n1, ix1)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n1 ix1 : nat),
ix1 `bound in` n1 = true ->
(0, ix1) ∈d local__sub σ (n1, ix1)
    introv H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n1, ix1: nat
H: ix1 `bound in` n1 = true
(0, ix1) ∈d local__sub σ (n1, ix1)
    unfold local__sub, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n1, ix1: nat
H: ix1 `bound in` n1 = true
(0, ix1)
∈d (if ix1 `bound in` n1
    then ret ix1
    else rename (+n1) (σ (ix1 - n1)))
    bound_induction_in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n1, ix1: nat
H: true = true
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
(0, ix1) ∈d ret ix1
    rewrite ind_ret_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n1, ix1: nat
H: true = true
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
0 = Ƶ /\ ix1 = ix1
    auto.
  Qed.

  Lemma ind_local_sub_iff: forall (σ: nat -> T nat) n2 ix2 n1 ix1,
      (n2, ix2) ∈d local__sub σ (n1, ix1) ->
      (ix1 >= n1 /\
         exists l1: nat,
           (n2, l1) ∈d σ (ix1 - n1) /\
             ix2 = (if l1 `bound in` n2 then l1 else n2 + (+n1) (l1 - n2)))
      \/ (ix1 < n1 /\ n2 = 0 /\ ix2 = ix1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n2 ix2 n1 ix1 : nat),
(n2, ix2) ∈d local__sub σ (n1, ix1) ->
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (n2 ix2 n1 ix1 : nat),
(n2, ix2) ∈d local__sub σ (n1, ix1) ->
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d local__sub σ (n1, ix1)
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
    unfold local__sub, lift__sub in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2)
∈d (if ix1 `bound in` n1
    then ret ix1
    else rename (+n1) (σ (ix1 - n1)))
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
    bound_induction_in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d rename (+n1) (σ (ix1 - n1))
Hord: ix1 >= n1
Hbound: ix1 `bound in` n1 = false
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d ret ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
    {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d rename (+n1) (σ (ix1 - n1))
Hord: ix1 >= n1
Hbound: ix1 `bound in` n1 = false
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1 rewrite ind_rename_iff in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: exists l1 : nat,
  (n2, l1) ∈d σ (ix1 - n1) /\
  ix2 =
  (if l1 `bound in` n2
   then l1
   else n2 + (+n1) (l1 - n2))
Hord: ix1 >= n1
Hbound: ix1 `bound in` n1 = false
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
      tauto. }
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d ret ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
    {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d ret ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 >= n1 /\
(exists l1 : nat,
   (n2, l1) ∈d σ (ix1 - n1) /\
   ix2 =
   (if l1 `bound in` n2
    then l1
    else n2 + (+n1) (l1 - n2))) \/
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1 right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: (n2, ix2) ∈d ret ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1 rewrite ind_ret_iff in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n2, ix2, n1, ix1: nat
H: n2 = Ƶ /\ ix2 = ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 < n1 /\ n2 = 0 /\ ix2 = ix1
      inversion H; subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
n1, ix1: nat
H: Ƶ = Ƶ /\ ix1 = ix1
Hord: ix1 < n1
Hbound: ix1 `bound in` n1 = true
ix1 < n1 /\ Ƶ = 0 /\ ix1 = ix1 intuition.
    }
  Qed.

  Lemma ind_subst_iff: forall l n (t:U nat) σ,
      (n, l) ∈d subst σ t <-> exists n1 n2 l1,
        (n1, l1) ∈d t /\ (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (l n : nat) (t : U nat) (σ : nat -> T nat),
(n, l) ∈d subst σ t <->
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (l n : nat) (t : U nat) (σ : nat -> T nat),
(n, l) ∈d subst σ t <->
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d subst σ t <->
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) unfold subst in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d bindd (local__sub σ) t <->
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
    now rewrite ind_bindd_iff'.
  Qed.

  Lemma ind_subst_iff2: forall l n (t:U nat) σ,
      (n, l) ∈d subst σ t <->
        ((n, l) ∈d t /\ (l `bound in` n = true))
        \/ exists n1 n2 l1,
          (n1, l1) ∈d t /\
            (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (l n : nat) (t : U nat) (σ : nat -> T nat),
(n, l) ∈d subst σ t <->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (l n : nat) (t : U nat) (σ : nat -> T nat),
(n, l) ∈d subst σ t <->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d subst σ t <->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) unfold subst in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d bindd (local__sub σ) t <->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
    rewrite ind_bindd_iff'.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(exists w1 w2 a : nat,
   (w1, a) ∈d t /\
   (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2) <->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
    split.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(exists w1 w2 a : nat,
   (w1, a) ∈d t /\
   (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2) ->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) ->
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(exists w1 w2 a : nat,
   (w1, a) ∈d t /\
   (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2) ->
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
      destruct H as [n1 [n2 [a [H1 [H2 H3]]]]].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d local__sub σ (n1, a)
H3: n = n1 ● n2
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
      assert (H2copy: (n2, l) ∈d local__sub σ (n1, a)) by apply H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d local__sub σ (n1, a)
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
      unfold local__sub, lift__sub in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l)
∈d (if a `bound in` n1
    then ret a
    else rename (+n1) (σ (a - n1)))
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
      bound_induction_in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d rename (+n1) (σ (a - n1))
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a >= n1
Hbound: a `bound in` n1 = false
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d ret a
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a < n1
Hbound: a `bound in` n1 = true
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d rename (+n1) (σ (a - n1))
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a >= n1
Hbound: a `bound in` n1 = false
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) right.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d rename (+n1) (σ (a - n1))
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a >= n1
Hbound: a `bound in` n1 = false
exists n1 n2 l1 : nat,
  (n1, l1) ∈d t /\
  (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2
        exists n1 n2 a.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d rename (+n1) (σ (a - n1))
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a >= n1
Hbound: a `bound in` n1 = false
(n1, a) ∈d t /\
(n2, l) ∈d local__sub σ (n1, a) /\ n = n1 + n2 auto.
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: (n2, l) ∈d ret a
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a < n1
Hbound: a `bound in` n1 = true
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) rewrite ind_ret_iff in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
n1, n2, a: nat
H1: (n1, a) ∈d t
H2: n2 = Ƶ /\ l = a
H3: n = n1 ● n2
H2copy: (n2, l) ∈d local__sub σ (n1, a)
Hord: a < n1
Hbound: a `bound in` n1 = true
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
        inversion H2; subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U nat
σ: nat -> T nat
n1, a: nat
H1: (n1, a) ∈d t
H2: Ƶ = Ƶ /\ a = a
H2copy: (Ƶ, a) ∈d local__sub σ (n1, a)
Hord: a < n1
Hbound: a `bound in` n1 = true
(n1 ● Ƶ, a) ∈d t /\ a `bound in` (n1 ● Ƶ) = true \/
(exists n2 n3 l1 : nat,
   (n2, l1) ∈d t /\
   (n3, a) ∈d local__sub σ (n2, l1) /\
   n1 ● Ƶ = n2 + n3)
        simpl_monoid.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: U nat
σ: nat -> T nat
n1, a: nat
H1: (n1, a) ∈d t
H2: Ƶ = Ƶ /\ a = a
H2copy: (Ƶ, a) ∈d local__sub σ (n1, a)
Hord: a < n1
Hbound: a `bound in` n1 = true
(n1, a) ∈d t /\ a `bound in` n1 = true \/
(exists n2 n3 l1 : nat,
   (n2, l1) ∈d t /\
   (n3, a) ∈d local__sub σ (n2, l1) /\ n1 = n2 + n3) now left.
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
(n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2) ->
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2 intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true \/
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2)
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2 destruct H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: exists n1 n2 l1 : nat,
  (n1, l1) ∈d t /\
  (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2 exists n 0 l.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(n, l) ∈d t /\
(0, l) ∈d local__sub σ (n, l) /\ n = n ● 0 splits.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(n, l) ∈d t
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(0, l) ∈d local__sub σ (n, l)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
n = n ● 0
        {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(n, l) ∈d t tauto. }
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(0, l) ∈d local__sub σ (n, l)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
n = n ● 0
        {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(0, l) ∈d local__sub σ (n, l) unfold local__sub, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
(0, l)
∈d (if l `bound in` n
    then ret l
    else rename (+n) (σ (l - n)))
          bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ true = true
Hord: l < n
Hbound: l `bound in` n = true
(0, l) ∈d ret l now rewrite ind_ret_iff. }
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
n = n ● 0
        {
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: (n, l) ∈d t /\ l `bound in` n = true
n = n ● 0 easy. }
      +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
l, n: nat
t: U nat
σ: nat -> T nat
H: exists n1 n2 l1 : nat,
  (n1, l1) ∈d t /\
  (n2, l) ∈d local__sub σ (n1, l1) /\ n = n1 + n2
exists w1 w2 a : nat,
  (w1, a) ∈d t /\
  (w2, l) ∈d local__sub σ (w1, a) /\ n = w1 ● w2 assumption.
  Qed.

End theory.

(** * Notations *)
(**********************************************************************)
Module Notations.
  Notation "↑" := S.
  Notation "'⇑'" := up__sub.
  Notation "'⇑__ren'" := up__ren.
  Notation "f ';' g" := (kc1 g f) (at level 30).
  Infix "⋅" := (scons) (at level 10).
  Notation "( + x )" := (lift x) (format "( + x )").
End Notations.

Import Notations.

(** * Other lemmas *)
(**********************************************************************)
Section theory.

  Context
    `{ret_inst: Return T}
    `{Map_T_inst: Map T}
    `{Mapd_T_inst: Mapd nat T}
    `{Traverse_T_inst: Traverse T}
    `{Bind_T_inst: Bind T T}
    `{Mapdt_T_inst: Mapdt nat T}
    `{Bindd_T_inst: Bindd nat T T}
    `{Bindt_T_inst: Bindt T T}
    `{Binddt_T_inst: Binddt nat T T}
    `{! Compat_Map_Binddt nat T T}
    `{! Compat_Mapd_Binddt nat T T}
    `{! Compat_Traverse_Binddt nat T T}
    `{! Compat_Bind_Binddt nat T T}
    `{! Compat_Mapdt_Binddt nat T T}
    `{! Compat_Bindd_Binddt nat T T}
    `{! Compat_Bindt_Binddt nat T T}
    `{Monad_inst: ! DecoratedTraversableMonad nat T}.

  Context
    `{Map_U_inst: Map U}
    `{Mapd_U_inst: Mapd nat U}
    `{Traverse_U_inst: Traverse U}
    `{Bind_U_inst: Bind T U}
    `{Mapdt_U_inst: Mapdt nat U}
    `{Bindd_U_inst: Bindd nat T U}
    `{Bindt_U_inst: Bindt T U}
    `{Binddt_U_inst: Binddt nat T U}
    `{! Compat_Map_Binddt nat T U}
    `{! Compat_Mapd_Binddt nat T U}
    `{! Compat_Traverse_Binddt nat T U}
    `{! Compat_Bind_Binddt nat T U}
    `{! Compat_Mapdt_Binddt nat T U}
    `{! Compat_Bindd_Binddt nat T U}
    `{! Compat_Bindt_Binddt nat T U}
    `{Module_inst: ! DecoratedTraversableRightPreModule nat T U
    (unit := Monoid_unit_zero)
    (op := Monoid_op_plus)}.

  Lemma cl_at_loc_dec {gap}:
    decidable_pred (fun p: nat * nat => cl_at_loc gap p).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
decidable_pred (fun p : nat * nat => cl_at_loc gap p)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
decidable_pred (fun p : nat * nat => cl_at_loc gap p)
    unfold decidable_pred.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
forall a : nat * nat,
Decidable.decidable (cl_at_loc gap a)
    intros [p n].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, p, n: nat
Decidable.decidable (cl_at_loc gap (p, n))
    unfold cl_at_loc.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, p, n: nat
Decidable.decidable (bound_within gap n p)
    unfold bound_within.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, p, n: nat
Decidable.decidable (n < p + gap)
    unfold Decidable.decidable.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap, p, n: nat
n < p + gap \/ ~ n < p + gap
    lia.
  Qed.

  Lemma cl_at_spec (gap: nat) (t: U nat):
    cl_at gap t = forall (d: nat) (n: nat), (d, n) ∈d t -> cl_at_loc gap (d, n).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
cl_at gap t =
(forall d n : nat, (d, n) ∈d t -> cl_at_loc gap (d, n))
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
cl_at gap t =
(forall d n : nat, (d, n) ∈d t -> cl_at_loc gap (d, n))
    unfold cl_at.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
Forall_ctx (cl_at_loc gap) t =
(forall d n : nat, (d, n) ∈d t -> cl_at_loc gap (d, n))
    apply propositional_extensionality.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
Forall_ctx (cl_at_loc gap) t <->
(forall d n : nat, (d, n) ∈d t -> cl_at_loc gap (d, n))
    rewrite forall_ctx_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
(forall e a : nat, (e, a) ∈d t -> cl_at_loc gap (e, a)) <->
(forall d n : nat, (d, n) ∈d t -> cl_at_loc gap (d, n))
    reflexivity.
  Qed.

  Lemma cl_at_decidable (gap: nat) (t: U nat):
    cl_at gap t \/ ~ cl_at gap t.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
cl_at gap t \/ ~ cl_at gap t
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
cl_at gap t \/ ~ cl_at gap t
    apply decidable_Forall_ctx.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
decidable_pred (cl_at_loc gap)
    apply cl_at_loc_dec.
  Qed.

  Lemma cl_at_spec_not (gap: nat) (t: U nat):
    (~ cl_at gap t) = exists (d: nat) (n: nat), (d, n) ∈d t /\ (~ cl_at_loc gap (d, n)).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
(~ cl_at gap t) =
(exists d n : nat,
   (d, n) ∈d t /\ ~ cl_at_loc gap (d, n))
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
(~ cl_at gap t) =
(exists d n : nat,
   (d, n) ∈d t /\ ~ cl_at_loc gap (d, n))
    unfold cl_at.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
(~ Forall_ctx (cl_at_loc gap) t) =
(exists d n : nat,
   (d, n) ∈d t /\ ~ cl_at_loc gap (d, n))
    apply propositional_extensionality.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
~ Forall_ctx (cl_at_loc gap) t <->
(exists d n : nat,
   (d, n) ∈d t /\ ~ cl_at_loc gap (d, n))
    rewrite not_forall_ctx_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
(exists e a : nat,
   (e, a) ∈d t /\ ~ cl_at_loc gap (e, a)) <->
(exists d n : nat,
   (d, n) ∈d t /\ ~ cl_at_loc gap (d, n))
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
decidable_pred (cl_at_loc gap)
    reflexivity.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
gap: nat
t: U nat
decidable_pred (cl_at_loc gap)
    apply cl_at_loc_dec.
  Qed.

  Lemma closed_at_sub1: forall (σ: nat -> T nat) (k d1 i1: nat),
      (forall (n: nat), cl_at k (σ n)) ->
      cl_at (k + d1) (lift__sub d1 σ i1).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (k d1 i1 : nat),
(forall n : nat, cl_at k (σ n)) ->
cl_at (k + d1) (lift__sub d1 σ i1)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (σ : nat -> T nat) (k d1 i1 : nat),
(forall n : nat, cl_at k (σ n)) ->
cl_at (k + d1) (lift__sub d1 σ i1)
    introv Hpremise.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n : nat, cl_at k (σ n)
cl_at (k + d1) (lift__sub d1 σ i1)
    unfold cl_at, cl_at_loc in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n : nat,
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in
   bound_within k ix depth) 
  (σ n)
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in
   bound_within (k + d1) ix depth) (lift__sub d1 σ i1)
    rewrite forall_ctx_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n : nat,
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in
   bound_within k ix depth) 
  (σ n)
forall e a : nat,
(e, a) ∈d lift__sub d1 σ i1 ->
bound_within (k + d1) a e
    setoid_rewrite forall_ctx_iff in Hpremise.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> bound_within k a e
forall e a : nat,
(e, a) ∈d lift__sub d1 σ i1 ->
bound_within (k + d1) a e
    unfold bound_within in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
forall e a : nat,
(e, a) ∈d lift__sub d1 σ i1 -> a < e + (k + d1)
    unfold lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
forall e a : nat,
(e, a)
∈d (if i1 `bound in` d1
    then ret i1
    else rename (+d1) (σ (i1 - d1))) ->
a < e + (k + d1)
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
forall e a : nat,
(e, a) ∈d rename (+d1) (σ (i1 - d1)) ->
a < e + (k + d1)
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
forall e a : nat, (e, a) ∈d ret i1 -> a < e + (k + d1)
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
forall e a : nat,
(e, a) ∈d rename (+d1) (σ (i1 - d1)) ->
a < e + (k + d1) intros e a.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, a: nat
(e, a) ∈d rename (+d1) (σ (i1 - d1)) ->
a < e + (k + d1)
      rewrite ind_rename_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, a: nat
(exists l1 : nat,
   (e, l1) ∈d σ (i1 - d1) /\
   a =
   (if l1 `bound in` e then l1 else e + (+d1) (l1 - e))) ->
a < e + (k + d1)
      intros [l1 [Hin Heq]].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, a, l1: nat
Hin: (e, l1) ∈d σ (i1 - d1)
Heq: a =
(if l1 `bound in` e
 then l1
 else e + (+d1) (l1 - e))
a < e + (k + d1)
      subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, l1: nat
Hin: (e, l1) ∈d σ (i1 - d1)
(if l1 `bound in` e then l1 else e + (+d1) (l1 - e)) <
e + (k + d1)
      bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, l1: nat
Hin: (e, l1) ∈d σ (i1 - d1)
Hord0: l1 >= e
Hbound0: l1 `bound in` e = false
e + (+d1) (l1 - e) < e + (k + d1)
      cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
e, l1: nat
Hin: (e, l1) ∈d σ (i1 - d1)
Hord0: l1 >= e
Hbound0: l1 `bound in` e = false
e + (d1 + (l1 - e)) < e + (k + d1)
      specialize (Hpremise  (i1 - d1) e l1 Hin).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1, e, l1: nat
Hpremise: l1 < e + k
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
Hin: (e, l1) ∈d σ (i1 - d1)
Hord0: l1 >= e
Hbound0: l1 `bound in` e = false
e + (d1 + (l1 - e)) < e + (k + d1)
      lia.
    -
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
forall e a : nat, (e, a) ∈d ret i1 -> a < e + (k + d1) setoid_rewrite ind_ret_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
forall e a : nat, e = Ƶ /\ a = i1 -> a < e + (k + d1)
      unfold transparent tcs.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
forall e a : nat, e = 0 /\ a = i1 -> a < e + (k + d1)
      intros e a [HeZ Hai1].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
e, a: nat
HeZ: e = 0
Hai1: a = i1
a < e + (k + d1)
      subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
σ: nat -> T nat
k, d1, i1: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
i1 < 0 + (k + d1) lia.
  Qed.

  Lemma closed_at_sub2:
    forall (t: T nat) (σ: nat -> T nat) (k: nat),
      (forall (n: nat), cl_at k (σ n)) ->
      cl_at k (subst σ t).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T nat) (σ : nat -> T nat) (k : nat),
(forall n : nat, cl_at k (σ n)) -> cl_at k (subst σ t)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall (t : T nat) (σ : nat -> T nat) (k : nat),
(forall n : nat, cl_at k (σ n)) -> cl_at k (subst σ t)
    introv Hpremise.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n : nat, cl_at k (σ n)
cl_at k (subst σ t)
    unfold cl_at in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n : nat,
Forall_ctx (cl_at_loc k) (σ n)
Forall_ctx (cl_at_loc k) (subst σ t)
    unfold cl_at_loc in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n : nat,
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in
   bound_within k ix depth) 
  (σ n)
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in bound_within k ix depth)
  (subst σ t)
    unfold bound_within in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n : nat,
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in ix < depth + k)
  (σ n)
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in ix < depth + k) (subst σ t)
    rewrite forall_ctx_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n : nat,
Forall_ctx
  (fun p : nat * nat =>
   let (depth, ix) := p in ix < depth + k)
  (σ n)
forall e a : nat, (e, a) ∈d subst σ t -> a < e + k
    setoid_rewrite forall_ctx_iff in Hpremise.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
forall e a : nat, (e, a) ∈d subst σ t -> a < e + k
    setoid_rewrite ind_subst_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
forall e a : nat,
(exists n1 n2 l1 : nat,
   (n1, l1) ∈d t /\
   (n2, a) ∈d local__sub σ (n1, l1) /\ e = n1 + n2) ->
a < e + k
    intros e a Hin.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
e, a: nat
Hin: exists n1 n2 l1 : nat,
  (n1, l1) ∈d t /\
  (n2, a) ∈d local__sub σ (n1, l1) /\
  e = n1 + n2
a < e + k
    destruct Hin as [d1 [d2 [i1 [H1 [H2 H3]]]]].
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
e, a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d local__sub σ (d1, i1)
H3: e = d1 + d2
a < e + k
    subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d local__sub σ (d1, i1)
a < d1 + d2 + k
    unfold local__sub, lift__sub in *.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a)
∈d (if i1 `bound in` d1
    then ret i1
    else rename (+d1) (σ (i1 - d1)))
a < d1 + d2 + k
    bound_induction_in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d rename (+d1) (σ (i1 - d1))
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
a < d1 + d2 + k
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d ret i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
a < d1 + d2 + k
    +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d rename (+d1) (σ (i1 - d1))
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
a < d1 + d2 + k rewrite ind_rename_iff in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: exists l1 : nat,
  (d2, l1) ∈d σ (i1 - d1) /\
  a =
  (if l1 `bound in` d2
   then l1
   else d2 + (+d1) (l1 - d2))
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
a < d1 + d2 + k
      preprocess.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
d1, d2, i1: nat
H1: (d1, i1) ∈d t
x: nat
H: (d2, x) ∈d σ (i1 - d1)
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
(if x `bound in` d2 then x else d2 + (+d1) (x - d2)) <
d1 + d2 + k
      unfold lift.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
d1, d2, i1: nat
H1: (d1, i1) ∈d t
x: nat
H: (d2, x) ∈d σ (i1 - d1)
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
(if x `bound in` d2 then x else d2 + (d1 + (x - d2))) <
d1 + d2 + k
      specialize (Hpremise (i1 -  d1) d2 x H).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k, d2, x: nat
Hpremise: x < d2 + k
d1, i1: nat
H1: (d1, i1) ∈d t
H: (d2, x) ∈d σ (i1 - d1)
Hord: i1 >= d1
Hbound: i1 `bound in` d1 = false
(if x `bound in` d2 then x else d2 + (d1 + (x - d2))) <
d1 + d2 + k
      bound_induction.
    +
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: (d2, a) ∈d ret i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
a < d1 + d2 + k rewrite ind_ret_iff in H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: d2 = Ƶ /\ a = i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
a < d1 + d2 + k
      inversion H2.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
a, d1, d2, i1: nat
H1: (d1, i1) ∈d t
H2: d2 = Ƶ /\ a = i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
H: d2 = Ƶ
H0: a = i1
a < d1 + d2 + k subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
d1, i1: nat
H1: (d1, i1) ∈d t
H2: Ƶ = Ƶ /\ i1 = i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
i1 < d1 + Ƶ + k
      unfold transparent tcs.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
t: T nat
σ: nat -> T nat
k: nat
Hpremise: forall n e a : nat,
(e, a) ∈d σ n -> a < e + k
d1, i1: nat
H1: (d1, i1) ∈d t
H2: Ƶ = Ƶ /\ i1 = i1
Hord: i1 < d1
Hbound: i1 `bound in` d1 = true
i1 < d1 + 0 + k lia.
  Qed.

  Lemma not_closed_at_ret:
    forall n, ~ (cl_at n (@ret T _ nat n)).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, ~ cl_at n (ret n)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, ~ cl_at n (ret n)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
~ cl_at n (ret n)
    introv H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: cl_at n (ret n)
False
    unfold cl_at in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: Forall_ctx (cl_at_loc n) (ret n)
False
    rewrite forall_ctx_iff in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: forall e a : nat,
(e, a) ∈d ret n -> cl_at_loc n (e, a)
False
    specialize (H 0 n).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: (0, n) ∈d ret n -> cl_at_loc n (0, n)
False
    rewrite ind_ret_iff in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: 0 = Ƶ /\ n = n -> cl_at_loc n (0, n)
False
    specialize (H ltac:(auto)).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: cl_at_loc n (0, n)
False
    unfold cl_at_loc in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: bound_within n n 0
False
    unfold bound_within in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: n < 0 + n
False
    lia.
  Qed.

  Lemma closed_at_ret:
    forall n, cl_at (S n) (@ret T _ nat n).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, cl_at (↑ n) (ret n)
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
forall n : nat, cl_at (↑ n) (ret n)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
cl_at (↑ n) (ret n)
    unfold cl_at.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
Forall_ctx (cl_at_loc (↑ n)) (ret n)
    rewrite forall_ctx_iff.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
forall e a : nat,
(e, a) ∈d ret n -> cl_at_loc (↑ n) (e, a)
    intros.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, e, a: nat
H: (e, a) ∈d ret n
cl_at_loc (↑ n) (e, a)
    rewrite ind_ret_iff in H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, e, a: nat
H: e = Ƶ /\ a = n
cl_at_loc (↑ n) (e, a)
    inversion H.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n, e, a: nat
H: e = Ƶ /\ a = n
H0: e = Ƶ
H1: a = n
cl_at_loc (↑ n) (e, a) subst.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: Ƶ = Ƶ /\ n = n
cl_at_loc (↑ n) (Ƶ, n) cbn.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: Ƶ = Ƶ /\ n = n
bound_within (↑ n) n Ƶ
    unfold bound_within.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: Ƶ = Ƶ /\ n = n
n < Ƶ + ↑ n
    cbv.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
n: nat
H: Ƶ = Ƶ /\ n = n
↑ n <= ↑ n
    lia.
  Qed.

  Lemma subst_pointwise (k: nat) (σ1 σ2: nat -> T nat) (t: T nat):
    cl_at k t ->
    (forall i, i < k -> σ1 i = σ2 i) ->
    subst σ1 t = subst σ2 t.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
cl_at k t ->
(forall i : nat, i < k -> σ1 i = σ2 i) ->
subst σ1 t = subst σ2 t
  Proof.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
cl_at k t ->
(forall i : nat, i < k -> σ1 i = σ2 i) ->
subst σ1 t = subst σ2 t
    unfold subst, cl_at.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
Forall_ctx (cl_at_loc k) t ->
(forall i : nat, i < k -> σ1 i = σ2 i) ->
bindd (local__sub σ1) t = bindd (local__sub σ2) t
    intros Hclosed Hpw.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
Hclosed: Forall_ctx (cl_at_loc k) t
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
bindd (local__sub σ1) t = bindd (local__sub σ2) t
    apply bindd_respectful.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
Hclosed: Forall_ctx (cl_at_loc k) t
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
forall w a : nat,
(w, a) ∈d t ->
local__sub σ1 (w, a) = local__sub σ2 (w, a)
    intros w a Hin.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
Hclosed: Forall_ctx (cl_at_loc k) t
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
w, a: nat
Hin: (w, a) ∈d t
local__sub σ1 (w, a) = local__sub σ2 (w, a)
    rewrite forall_ctx_iff in Hclosed.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
Hclosed: forall e a : nat,
(e, a) ∈d t -> cl_at_loc k (e, a)
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
w, a: nat
Hin: (w, a) ∈d t
local__sub σ1 (w, a) = local__sub σ2 (w, a)
    specialize (Hclosed w a Hin).
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: cl_at_loc k (w, a)
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
local__sub σ1 (w, a) = local__sub σ2 (w, a)
    unfold cl_at_loc, bound_within in Hclosed.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: a < w + k
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
local__sub σ1 (w, a) = local__sub σ2 (w, a)
    unfold local__sub, lift__sub.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: a < w + k
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
(if a `bound in` w
 then ret a
 else rename (+w) (σ1 (a - w))) =
(if a `bound in` w
 then ret a
 else rename (+w) (σ2 (a - w)))
    bound_induction.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: a < w + k
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
Hord: a >= w
Hbound: a `bound in` w = false
rename (+w) (σ1 (a - w)) = rename (+w) (σ2 (a - w)) fequal.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: a < w + k
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
Hord: a >= w
Hbound: a `bound in` w = false
σ1 (a - w) = σ2 (a - w)
    apply Hpw.
T: Type -> Type
ret_inst: Return T
Map_T_inst: Map T
Mapd_T_inst: Mapd nat T
Traverse_T_inst: Traverse T
Bind_T_inst: Bind T T
Mapdt_T_inst: Mapdt nat T
Bindd_T_inst: Bindd nat T T
Bindt_T_inst: Bindt T T
Binddt_T_inst: Binddt nat T T
Compat_Map_Binddt0: Compat_Map_Binddt nat T T
Compat_Mapd_Binddt0: Compat_Mapd_Binddt nat T T
Compat_Traverse_Binddt0: Compat_Traverse_Binddt nat T
  T
Compat_Bind_Binddt0: Compat_Bind_Binddt nat T T
Compat_Mapdt_Binddt0: Compat_Mapdt_Binddt nat T T
Compat_Bindd_Binddt0: Compat_Bindd_Binddt nat T T
Compat_Bindt_Binddt0: Compat_Bindt_Binddt nat T T
Monad_inst: DecoratedTraversableMonad nat T
U: Type -> Type
Map_U_inst: Map U
Mapd_U_inst: Mapd nat U
Traverse_U_inst: Traverse U
Bind_U_inst: Bind T U
Mapdt_U_inst: Mapdt nat U
Bindd_U_inst: Bindd nat T U
Bindt_U_inst: Bindt T U
Binddt_U_inst: Binddt nat T U
Compat_Map_Binddt1: Compat_Map_Binddt nat T U
Compat_Mapd_Binddt1: Compat_Mapd_Binddt nat T U
Compat_Traverse_Binddt1: Compat_Traverse_Binddt nat T
  U
Compat_Bind_Binddt1: Compat_Bind_Binddt nat T U
Compat_Mapdt_Binddt1: Compat_Mapdt_Binddt nat T U
Compat_Bindd_Binddt1: Compat_Bindd_Binddt nat T U
Compat_Bindt_Binddt1: Compat_Bindt_Binddt nat T U
Module_inst: DecoratedTraversableRightPreModule nat T
  U
k: nat
σ1, σ2: nat -> T nat
t: T nat
w, a: nat
Hclosed: a < w + k
Hpw: forall i : nat, i < k -> σ1 i = σ2 i
Hin: (w, a) ∈d t
Hord: a >= w
Hbound: a `bound in` w = false
a - w < k lia.
  Qed.

End theory.