/
Lemmas.v
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Lemmas.v
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Require Import Coq.Logic.FunctionalExtensionality.
Require Import Coq.Program.Equality.
Require Export Metalib.Metatheory.
Require Export Metalib.LibLNgen.
Require Export Stlc.Definitions.
(** NOTE: Auxiliary theorems are hidden in generated documentation.
In general, there is a [_rec] version of every lemma involving
[open] and [close]. *)
(* begin hide *)
(* *********************************************************************** *)
(** * Induction principles for nonterminals *)
Scheme typ_ind' := Induction for typ Sort Prop.
Definition typ_mutind :=
fun H1 H2 H3 =>
typ_ind' H1 H2 H3.
Scheme typ_rec' := Induction for typ Sort Set.
Definition typ_mutrec :=
fun H1 H2 H3 =>
typ_rec' H1 H2 H3.
Scheme exp_ind' := Induction for exp Sort Prop.
Definition exp_mutind :=
fun H1 H2 H3 H4 H5 =>
exp_ind' H1 H2 H3 H4 H5.
Scheme exp_rec' := Induction for exp Sort Set.
Definition exp_mutrec :=
fun H1 H2 H3 H4 H5 =>
exp_rec' H1 H2 H3 H4 H5.
(* end hide *)
(* *********************************************************************** *)
(** * Close *)
Fixpoint close_exp_wrt_exp_rec (n1 : nat) (x1 : var) (e1 : exp) {struct e1} : exp :=
match e1 with
| abs e2 => abs (close_exp_wrt_exp_rec (S n1) x1 e2)
| var_f x2 => if (x1 == x2) then (var_b n1) else (var_f x2)
| var_b n2 => if (lt_ge_dec n2 n1) then (var_b n2) else (var_b (S n2))
| app e2 e3 => app (close_exp_wrt_exp_rec n1 x1 e2) (close_exp_wrt_exp_rec n1 x1 e3)
end.
Definition close_exp_wrt_exp x1 e1 := close_exp_wrt_exp_rec 0 x1 e1.
(* *********************************************************************** *)
(** * Size *)
Fixpoint size_typ (T1 : typ) {struct T1} : nat :=
match T1 with
| typ_base => 1
| typ_arrow T2 T3 => 1 + (size_typ T2) + (size_typ T3)
end.
Fixpoint size_exp (e1 : exp) {struct e1} : nat :=
match e1 with
| abs e2 => 1 + (size_exp e2)
| var_f x1 => 1
| var_b n1 => 1
| app e2 e3 => 1 + (size_exp e2) + (size_exp e3)
end.
(* *********************************************************************** *)
(** * Degree *)
(** These define only an upper bound, not a strict upper bound. *)
Inductive degree_exp_wrt_exp : nat -> exp -> Prop :=
| degree_wrt_exp_abs : forall n1 e1,
degree_exp_wrt_exp (S n1) e1 ->
degree_exp_wrt_exp n1 (abs e1)
| degree_wrt_exp_var_f : forall n1 x1,
degree_exp_wrt_exp n1 (var_f x1)
| degree_wrt_exp_var_b : forall n1 n2,
lt n2 n1 ->
degree_exp_wrt_exp n1 (var_b n2)
| degree_wrt_exp_app : forall n1 e1 e2,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 e2 ->
degree_exp_wrt_exp n1 (app e1 e2).
Scheme degree_exp_wrt_exp_ind' := Induction for degree_exp_wrt_exp Sort Prop.
Definition degree_exp_wrt_exp_mutind :=
fun H1 H2 H3 H4 H5 =>
degree_exp_wrt_exp_ind' H1 H2 H3 H4 H5.
Hint Constructors degree_exp_wrt_exp : core lngen.
(* begin hide *)
(* *********************************************************************** *)
(** * Local closure (induction principles) *)
Scheme lc_exp_ind' := Induction for lc_exp Sort Prop.
Definition lc_exp_mutind :=
fun H1 H2 H3 H4 =>
lc_exp_ind' H1 H2 H3 H4.
Hint Constructors lc_exp : core lngen.
(* end hide *)
(* *********************************************************************** *)
(** * Body *)
Definition body_exp_wrt_exp e1 := forall x1, lc_exp (open_exp_wrt_exp e1 (var_f x1)).
Hint Unfold body_exp_wrt_exp.
(* *********************************************************************** *)
(** * Tactic support *)
(** Additional hint declarations. *)
Hint Resolve @plus_le_compat : lngen.
(** Redefine some tactics. *)
Ltac default_case_split ::=
first
[ progress destruct_notin
| progress destruct_sum
| progress safe_f_equal
].
(* *********************************************************************** *)
(** * Theorems about [size] *)
Ltac default_auto ::= auto with arith lngen; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma size_typ_min_mutual :
(forall T1, 1 <= size_typ T1).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_typ_min :
forall T1, 1 <= size_typ T1.
Proof.
pose proof size_typ_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_min : lngen.
(* begin hide *)
Lemma size_exp_min_mutual :
(forall e1, 1 <= size_exp e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_exp_min :
forall e1, 1 <= size_exp e1.
Proof.
pose proof size_exp_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_min : lngen.
(* begin hide *)
Lemma size_exp_close_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1,
size_exp (close_exp_wrt_exp_rec n1 x1 e1) = size_exp e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_exp_close_exp_wrt_exp_rec :
forall e1 x1 n1,
size_exp (close_exp_wrt_exp_rec n1 x1 e1) = size_exp e1.
Proof.
pose proof size_exp_close_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_close_exp_wrt_exp_rec : lngen.
Hint Rewrite size_exp_close_exp_wrt_exp_rec using solve [auto] : lngen.
(* end hide *)
Lemma size_exp_close_exp_wrt_exp :
forall e1 x1,
size_exp (close_exp_wrt_exp x1 e1) = size_exp e1.
Proof.
unfold close_exp_wrt_exp; default_simp.
Qed.
Hint Resolve size_exp_close_exp_wrt_exp : lngen.
Hint Rewrite size_exp_close_exp_wrt_exp using solve [auto] : lngen.
(* begin hide *)
Lemma size_exp_open_exp_wrt_exp_rec_mutual :
(forall e1 e2 n1,
size_exp e1 <= size_exp (open_exp_wrt_exp_rec n1 e2 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_exp_rec :
forall e1 e2 n1,
size_exp e1 <= size_exp (open_exp_wrt_exp_rec n1 e2 e1).
Proof.
pose proof size_exp_open_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_open_exp_wrt_exp_rec : lngen.
(* end hide *)
Lemma size_exp_open_exp_wrt_exp :
forall e1 e2,
size_exp e1 <= size_exp (open_exp_wrt_exp e1 e2).
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve size_exp_open_exp_wrt_exp : lngen.
(* begin hide *)
Lemma size_exp_open_exp_wrt_exp_rec_var_mutual :
(forall e1 x1 n1,
size_exp (open_exp_wrt_exp_rec n1 (var_f x1) e1) = size_exp e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_exp_rec_var :
forall e1 x1 n1,
size_exp (open_exp_wrt_exp_rec n1 (var_f x1) e1) = size_exp e1.
Proof.
pose proof size_exp_open_exp_wrt_exp_rec_var_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_open_exp_wrt_exp_rec_var : lngen.
Hint Rewrite size_exp_open_exp_wrt_exp_rec_var using solve [auto] : lngen.
(* end hide *)
Lemma size_exp_open_exp_wrt_exp_var :
forall e1 x1,
size_exp (open_exp_wrt_exp e1 (var_f x1)) = size_exp e1.
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve size_exp_open_exp_wrt_exp_var : lngen.
Hint Rewrite size_exp_open_exp_wrt_exp_var using solve [auto] : lngen.
(* *********************************************************************** *)
(** * Theorems about [degree] *)
Ltac default_auto ::= auto with lngen; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma degree_exp_wrt_exp_S_mutual :
(forall n1 e1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp (S n1) e1).
Proof.
apply_mutual_ind degree_exp_wrt_exp_mutind;
default_simp.
Qed.
(* end hide *)
Lemma degree_exp_wrt_exp_S :
forall n1 e1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp (S n1) e1.
Proof.
pose proof degree_exp_wrt_exp_S_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_exp_S : lngen.
Lemma degree_exp_wrt_exp_O :
forall n1 e1,
degree_exp_wrt_exp O e1 ->
degree_exp_wrt_exp n1 e1.
Proof.
induction n1; default_simp.
Qed.
Hint Resolve degree_exp_wrt_exp_O : lngen.
(* begin hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp (S n1) (close_exp_wrt_exp_rec n1 x1 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp_rec :
forall e1 x1 n1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp (S n1) (close_exp_wrt_exp_rec n1 x1 e1).
Proof.
pose proof degree_exp_wrt_exp_close_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_exp_close_exp_wrt_exp_rec : lngen.
(* end hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp :
forall e1 x1,
degree_exp_wrt_exp 0 e1 ->
degree_exp_wrt_exp 1 (close_exp_wrt_exp x1 e1).
Proof.
unfold close_exp_wrt_exp; default_simp.
Qed.
Hint Resolve degree_exp_wrt_exp_close_exp_wrt_exp : lngen.
(* begin hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp_rec_inv_mutual :
(forall e1 x1 n1,
degree_exp_wrt_exp (S n1) (close_exp_wrt_exp_rec n1 x1 e1) ->
degree_exp_wrt_exp n1 e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp_rec_inv :
forall e1 x1 n1,
degree_exp_wrt_exp (S n1) (close_exp_wrt_exp_rec n1 x1 e1) ->
degree_exp_wrt_exp n1 e1.
Proof.
pose proof degree_exp_wrt_exp_close_exp_wrt_exp_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_exp_wrt_exp_close_exp_wrt_exp_rec_inv : lngen.
(* end hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_exp_inv :
forall e1 x1,
degree_exp_wrt_exp 1 (close_exp_wrt_exp x1 e1) ->
degree_exp_wrt_exp 0 e1.
Proof.
unfold close_exp_wrt_exp; eauto with lngen.
Qed.
Hint Immediate degree_exp_wrt_exp_close_exp_wrt_exp_inv : lngen.
(* begin hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp_rec_mutual :
(forall e1 e2 n1,
degree_exp_wrt_exp (S n1) e1 ->
degree_exp_wrt_exp n1 e2 ->
degree_exp_wrt_exp n1 (open_exp_wrt_exp_rec n1 e2 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp_rec :
forall e1 e2 n1,
degree_exp_wrt_exp (S n1) e1 ->
degree_exp_wrt_exp n1 e2 ->
degree_exp_wrt_exp n1 (open_exp_wrt_exp_rec n1 e2 e1).
Proof.
pose proof degree_exp_wrt_exp_open_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_exp_open_exp_wrt_exp_rec : lngen.
(* end hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp :
forall e1 e2,
degree_exp_wrt_exp 1 e1 ->
degree_exp_wrt_exp 0 e2 ->
degree_exp_wrt_exp 0 (open_exp_wrt_exp e1 e2).
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve degree_exp_wrt_exp_open_exp_wrt_exp : lngen.
(* begin hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp_rec_inv_mutual :
(forall e1 e2 n1,
degree_exp_wrt_exp n1 (open_exp_wrt_exp_rec n1 e2 e1) ->
degree_exp_wrt_exp (S n1) e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp_rec_inv :
forall e1 e2 n1,
degree_exp_wrt_exp n1 (open_exp_wrt_exp_rec n1 e2 e1) ->
degree_exp_wrt_exp (S n1) e1.
Proof.
pose proof degree_exp_wrt_exp_open_exp_wrt_exp_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_exp_wrt_exp_open_exp_wrt_exp_rec_inv : lngen.
(* end hide *)
Lemma degree_exp_wrt_exp_open_exp_wrt_exp_inv :
forall e1 e2,
degree_exp_wrt_exp 0 (open_exp_wrt_exp e1 e2) ->
degree_exp_wrt_exp 1 e1.
Proof.
unfold open_exp_wrt_exp; eauto with lngen.
Qed.
Hint Immediate degree_exp_wrt_exp_open_exp_wrt_exp_inv : lngen.
(* *********************************************************************** *)
(** * Theorems about [open] and [close] *)
Ltac default_auto ::= auto with lngen brute_force; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma close_exp_wrt_exp_rec_inj_mutual :
(forall e1 e2 x1 n1,
close_exp_wrt_exp_rec n1 x1 e1 = close_exp_wrt_exp_rec n1 x1 e2 ->
e1 = e2).
Proof.
apply_mutual_ind exp_mutind;
intros; match goal with
| |- _ = ?term => destruct term
end;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma close_exp_wrt_exp_rec_inj :
forall e1 e2 x1 n1,
close_exp_wrt_exp_rec n1 x1 e1 = close_exp_wrt_exp_rec n1 x1 e2 ->
e1 = e2.
Proof.
pose proof close_exp_wrt_exp_rec_inj_mutual as H; intuition eauto.
Qed.
Hint Immediate close_exp_wrt_exp_rec_inj : lngen.
(* end hide *)
Lemma close_exp_wrt_exp_inj :
forall e1 e2 x1,
close_exp_wrt_exp x1 e1 = close_exp_wrt_exp x1 e2 ->
e1 = e2.
Proof.
unfold close_exp_wrt_exp; eauto with lngen.
Qed.
Hint Immediate close_exp_wrt_exp_inj : lngen.
(* begin hide *)
Lemma close_exp_wrt_exp_rec_open_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1,
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp_rec n1 x1 (open_exp_wrt_exp_rec n1 (var_f x1) e1) = e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma close_exp_wrt_exp_rec_open_exp_wrt_exp_rec :
forall e1 x1 n1,
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp_rec n1 x1 (open_exp_wrt_exp_rec n1 (var_f x1) e1) = e1.
Proof.
pose proof close_exp_wrt_exp_rec_open_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve close_exp_wrt_exp_rec_open_exp_wrt_exp_rec : lngen.
Hint Rewrite close_exp_wrt_exp_rec_open_exp_wrt_exp_rec using solve [auto] : lngen.
(* end hide *)
Lemma close_exp_wrt_exp_open_exp_wrt_exp :
forall e1 x1,
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp x1 (open_exp_wrt_exp e1 (var_f x1)) = e1.
Proof.
unfold close_exp_wrt_exp; unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve close_exp_wrt_exp_open_exp_wrt_exp : lngen.
Hint Rewrite close_exp_wrt_exp_open_exp_wrt_exp using solve [auto] : lngen.
(* begin hide *)
Lemma open_exp_wrt_exp_rec_close_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1,
open_exp_wrt_exp_rec n1 (var_f x1) (close_exp_wrt_exp_rec n1 x1 e1) = e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma open_exp_wrt_exp_rec_close_exp_wrt_exp_rec :
forall e1 x1 n1,
open_exp_wrt_exp_rec n1 (var_f x1) (close_exp_wrt_exp_rec n1 x1 e1) = e1.
Proof.
pose proof open_exp_wrt_exp_rec_close_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve open_exp_wrt_exp_rec_close_exp_wrt_exp_rec : lngen.
Hint Rewrite open_exp_wrt_exp_rec_close_exp_wrt_exp_rec using solve [auto] : lngen.
(* end hide *)
Lemma open_exp_wrt_exp_close_exp_wrt_exp :
forall e1 x1,
open_exp_wrt_exp (close_exp_wrt_exp x1 e1) (var_f x1) = e1.
Proof.
unfold close_exp_wrt_exp; unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve open_exp_wrt_exp_close_exp_wrt_exp : lngen.
Hint Rewrite open_exp_wrt_exp_close_exp_wrt_exp using solve [auto] : lngen.
(* begin hide *)
Lemma open_exp_wrt_exp_rec_inj_mutual :
(forall e2 e1 x1 n1,
x1 `notin` fv_exp e2 ->
x1 `notin` fv_exp e1 ->
open_exp_wrt_exp_rec n1 (var_f x1) e2 = open_exp_wrt_exp_rec n1 (var_f x1) e1 ->
e2 = e1).
Proof.
apply_mutual_ind exp_mutind;
intros; match goal with
| |- _ = ?term => destruct term
end;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma open_exp_wrt_exp_rec_inj :
forall e2 e1 x1 n1,
x1 `notin` fv_exp e2 ->
x1 `notin` fv_exp e1 ->
open_exp_wrt_exp_rec n1 (var_f x1) e2 = open_exp_wrt_exp_rec n1 (var_f x1) e1 ->
e2 = e1.
Proof.
pose proof open_exp_wrt_exp_rec_inj_mutual as H; intuition eauto.
Qed.
Hint Immediate open_exp_wrt_exp_rec_inj : lngen.
(* end hide *)
Lemma open_exp_wrt_exp_inj :
forall e2 e1 x1,
x1 `notin` fv_exp e2 ->
x1 `notin` fv_exp e1 ->
open_exp_wrt_exp e2 (var_f x1) = open_exp_wrt_exp e1 (var_f x1) ->
e2 = e1.
Proof.
unfold open_exp_wrt_exp; eauto with lngen.
Qed.
Hint Immediate open_exp_wrt_exp_inj : lngen.
(* *********************************************************************** *)
(** * Theorems about [lc] *)
Ltac default_auto ::= auto with lngen brute_force; tauto.
Ltac default_autorewrite ::= autorewrite with lngen.
(* begin hide *)
Lemma degree_exp_wrt_exp_of_lc_exp_mutual :
(forall e1,
lc_exp e1 ->
degree_exp_wrt_exp 0 e1).
Proof.
apply_mutual_ind lc_exp_mutind;
intros;
let x1 := fresh "x1" in pick_fresh x1;
repeat (match goal with
| H1 : _, H2 : _ |- _ => specialize H1 with H2
end);
default_simp; eauto with lngen.
Qed.
(* end hide *)
Lemma degree_exp_wrt_exp_of_lc_exp :
forall e1,
lc_exp e1 ->
degree_exp_wrt_exp 0 e1.
Proof.
pose proof degree_exp_wrt_exp_of_lc_exp_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_exp_of_lc_exp : lngen.
(* begin hide *)
Lemma lc_exp_of_degree_size_mutual :
forall i1,
(forall e1,
size_exp e1 = i1 ->
degree_exp_wrt_exp 0 e1 ->
lc_exp e1).
Proof.
intros i1; pattern i1; apply lt_wf_rec;
clear i1; intros i1 H1;
apply_mutual_ind exp_mutind;
default_simp;
(* non-trivial cases *)
constructor; default_simp; eapply_first_lt_hyp;
(* instantiate the size *)
match goal with
| |- _ = _ => reflexivity
| _ => idtac
end;
instantiate;
(* everything should be easy now *)
default_simp.
Qed.
(* end hide *)
Lemma lc_exp_of_degree :
forall e1,
degree_exp_wrt_exp 0 e1 ->
lc_exp e1.
Proof.
intros e1; intros;
pose proof (lc_exp_of_degree_size_mutual (size_exp e1));
intuition eauto.
Qed.
Hint Resolve lc_exp_of_degree : lngen.
Ltac typ_lc_exists_tac :=
repeat (match goal with
| H : _ |- _ =>
fail 1
end).
Ltac exp_lc_exists_tac :=
repeat (match goal with
| H : _ |- _ =>
let J1 := fresh in pose proof H as J1; apply degree_exp_wrt_exp_of_lc_exp in J1; clear H
end).
Lemma lc_abs_exists :
forall x1 e1,
lc_exp (open_exp_wrt_exp e1 (var_f x1)) ->
lc_exp (abs e1).
Proof.
intros; exp_lc_exists_tac; eauto with lngen.
Qed.
Hint Extern 1 (lc_exp (abs _)) =>
let x1 := fresh in
pick_fresh x1;
apply (lc_abs_exists x1).
Lemma lc_body_exp_wrt_exp :
forall e1 e2,
body_exp_wrt_exp e1 ->
lc_exp e2 ->
lc_exp (open_exp_wrt_exp e1 e2).
Proof.
unfold body_exp_wrt_exp;
default_simp;
let x1 := fresh "x" in
pick_fresh x1;
specialize_all x1;
exp_lc_exists_tac;
eauto with lngen.
Qed.
Hint Resolve lc_body_exp_wrt_exp : lngen.
Lemma lc_body_abs_1 :
forall e1,
lc_exp (abs e1) ->
body_exp_wrt_exp e1.
Proof.
default_simp.
Qed.
Hint Resolve lc_body_abs_1 : lngen.
(* *********************************************************************** *)
(** * More theorems about [open] and [close] *)
Ltac default_auto ::= auto with lngen; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma close_exp_wrt_exp_rec_degree_exp_wrt_exp_mutual :
(forall e1 x1 n1,
degree_exp_wrt_exp n1 e1 ->
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp_rec n1 x1 e1 = e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma close_exp_wrt_exp_rec_degree_exp_wrt_exp :
forall e1 x1 n1,
degree_exp_wrt_exp n1 e1 ->
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp_rec n1 x1 e1 = e1.
Proof.
pose proof close_exp_wrt_exp_rec_degree_exp_wrt_exp_mutual as H; intuition eauto.
Qed.
Hint Resolve close_exp_wrt_exp_rec_degree_exp_wrt_exp : lngen.
Hint Rewrite close_exp_wrt_exp_rec_degree_exp_wrt_exp using solve [auto] : lngen.
(* end hide *)
Lemma close_exp_wrt_exp_lc_exp :
forall e1 x1,
lc_exp e1 ->
x1 `notin` fv_exp e1 ->
close_exp_wrt_exp x1 e1 = e1.
Proof.
unfold close_exp_wrt_exp; default_simp.
Qed.
Hint Resolve close_exp_wrt_exp_lc_exp : lngen.
Hint Rewrite close_exp_wrt_exp_lc_exp using solve [auto] : lngen.
(* begin hide *)
Lemma open_exp_wrt_exp_rec_degree_exp_wrt_exp_mutual :
(forall e2 e1 n1,
degree_exp_wrt_exp n1 e2 ->
open_exp_wrt_exp_rec n1 e1 e2 = e2).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma open_exp_wrt_exp_rec_degree_exp_wrt_exp :
forall e2 e1 n1,
degree_exp_wrt_exp n1 e2 ->
open_exp_wrt_exp_rec n1 e1 e2 = e2.
Proof.
pose proof open_exp_wrt_exp_rec_degree_exp_wrt_exp_mutual as H; intuition eauto.
Qed.
Hint Resolve open_exp_wrt_exp_rec_degree_exp_wrt_exp : lngen.
Hint Rewrite open_exp_wrt_exp_rec_degree_exp_wrt_exp using solve [auto] : lngen.
(* end hide *)
Lemma open_exp_wrt_exp_lc_exp :
forall e2 e1,
lc_exp e2 ->
open_exp_wrt_exp e2 e1 = e2.
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve open_exp_wrt_exp_lc_exp : lngen.
Hint Rewrite open_exp_wrt_exp_lc_exp using solve [auto] : lngen.
(* *********************************************************************** *)
(** * Theorems about [fv] *)
Ltac default_auto ::= auto with set lngen; tauto.
Ltac default_autorewrite ::= autorewrite with lngen.
(* begin hide *)
Lemma fv_exp_close_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1,
fv_exp (close_exp_wrt_exp_rec n1 x1 e1) [=] remove x1 (fv_exp e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp; fsetdec.
Qed.
(* end hide *)
(* begin hide *)
Lemma fv_exp_close_exp_wrt_exp_rec :
forall e1 x1 n1,
fv_exp (close_exp_wrt_exp_rec n1 x1 e1) [=] remove x1 (fv_exp e1).
Proof.
pose proof fv_exp_close_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve fv_exp_close_exp_wrt_exp_rec : lngen.
Hint Rewrite fv_exp_close_exp_wrt_exp_rec using solve [auto] : lngen.
(* end hide *)
Lemma fv_exp_close_exp_wrt_exp :
forall e1 x1,
fv_exp (close_exp_wrt_exp x1 e1) [=] remove x1 (fv_exp e1).
Proof.
unfold close_exp_wrt_exp; default_simp.
Qed.
Hint Resolve fv_exp_close_exp_wrt_exp : lngen.
Hint Rewrite fv_exp_close_exp_wrt_exp using solve [auto] : lngen.
(* begin hide *)
Lemma fv_exp_open_exp_wrt_exp_rec_lower_mutual :
(forall e1 e2 n1,
fv_exp e1 [<=] fv_exp (open_exp_wrt_exp_rec n1 e2 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp; fsetdec.
Qed.
(* end hide *)
(* begin hide *)
Lemma fv_exp_open_exp_wrt_exp_rec_lower :
forall e1 e2 n1,
fv_exp e1 [<=] fv_exp (open_exp_wrt_exp_rec n1 e2 e1).
Proof.
pose proof fv_exp_open_exp_wrt_exp_rec_lower_mutual as H; intuition eauto.
Qed.
Hint Resolve fv_exp_open_exp_wrt_exp_rec_lower : lngen.
(* end hide *)
Lemma fv_exp_open_exp_wrt_exp_lower :
forall e1 e2,
fv_exp e1 [<=] fv_exp (open_exp_wrt_exp e1 e2).
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve fv_exp_open_exp_wrt_exp_lower : lngen.
(* begin hide *)
Lemma fv_exp_open_exp_wrt_exp_rec_upper_mutual :
(forall e1 e2 n1,
fv_exp (open_exp_wrt_exp_rec n1 e2 e1) [<=] fv_exp e2 `union` fv_exp e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp; fsetdec.
Qed.
(* end hide *)
(* begin hide *)
Lemma fv_exp_open_exp_wrt_exp_rec_upper :
forall e1 e2 n1,
fv_exp (open_exp_wrt_exp_rec n1 e2 e1) [<=] fv_exp e2 `union` fv_exp e1.
Proof.
pose proof fv_exp_open_exp_wrt_exp_rec_upper_mutual as H; intuition eauto.
Qed.
Hint Resolve fv_exp_open_exp_wrt_exp_rec_upper : lngen.
(* end hide *)
Lemma fv_exp_open_exp_wrt_exp_upper :
forall e1 e2,
fv_exp (open_exp_wrt_exp e1 e2) [<=] fv_exp e2 `union` fv_exp e1.
Proof.
unfold open_exp_wrt_exp; default_simp.
Qed.
Hint Resolve fv_exp_open_exp_wrt_exp_upper : lngen.
(* begin hide *)
Lemma fv_exp_subst_exp_fresh_mutual :
(forall e1 e2 x1,
x1 `notin` fv_exp e1 ->
fv_exp (subst_exp e2 x1 e1) [=] fv_exp e1).
Proof.
apply_mutual_ind exp_mutind;
default_simp; fsetdec.
Qed.
(* end hide *)
Lemma fv_exp_subst_exp_fresh :
forall e1 e2 x1,
x1 `notin` fv_exp e1 ->
fv_exp (subst_exp e2 x1 e1) [=] fv_exp e1.
Proof.
pose proof fv_exp_subst_exp_fresh_mutual as H; intuition eauto.
Qed.
Hint Resolve fv_exp_subst_exp_fresh : lngen.