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LN_Lemmas.v
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Set Warnings "-non-recursive,-deprecated-hint-without-locality,-deprecated,-fragile-hint-constr".
Require Import Coq.Arith.Wf_nat.
Require Import Coq.Logic.FunctionalExtensionality.
Require Import Coq.Program.Equality.
Require Export Metalib.Metatheory.
Require Export Metalib.LibLNgen.
Require Export Definitions.
(** NOTE: Auxiliary theorems are hidden in generated documentation.
In general, there is a [_rec] version of every lemma involving
[open] and [close]. *)
(* *********************************************************************** *)
(** * Induction principles for nonterminals *)
Scheme l_ind' := Induction for l Sort Prop.
Definition l_mutind :=
fun H1 H2 H3 H4 =>
l_ind' H1 H2 H3 H4.
Scheme l_rec' := Induction for l Sort Set.
Definition l_mutrec :=
fun H1 H2 H3 H4 =>
l_rec' H1 H2 H3 H4.
Scheme typ_ind' := Induction for typ Sort Prop.
Definition typ_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Scheme typ_rec' := Induction for typ Sort Set.
Definition typ_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
typ_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Scheme Fty_ind' := Induction for Fty Sort Prop.
Definition Fty_mutind :=
fun H1 H2 H3 =>
Fty_ind' H1 H2 H3.
Scheme Fty_rec' := Induction for Fty Sort Set.
Definition Fty_mutrec :=
fun H1 H2 H3 =>
Fty_rec' H1 H2 H3.
(* *********************************************************************** *)
(** * Close *)
Fixpoint close_typ_wrt_typ_rec (n1 : nat) (X1 : typevar) (A1 : typ) {struct A1} : typ :=
match A1 with
| t_tvar_f X2 => if (X1 == X2) then (t_tvar_b n1) else (t_tvar_f X2)
| t_tvar_b n2 => if (lt_ge_dec n2 n1) then (t_tvar_b n2) else (t_tvar_b (S n2))
| t_rcd l1 A2 => t_rcd l1 (close_typ_wrt_typ_rec n1 X1 A2)
| t_and A2 A3 => t_and (close_typ_wrt_typ_rec n1 X1 A2) (close_typ_wrt_typ_rec n1 X1 A3)
| t_or A2 A3 => t_or (close_typ_wrt_typ_rec n1 X1 A2) (close_typ_wrt_typ_rec n1 X1 A3)
| t_arrow A2 B1 => t_arrow (close_typ_wrt_typ_rec n1 X1 A2) (close_typ_wrt_typ_rec n1 X1 B1)
| t_forall B1 => t_forall (close_typ_wrt_typ_rec (S n1) X1 B1)
| t_top => t_top
| t_bot => t_bot
end.
Definition close_typ_wrt_typ X1 A1 := close_typ_wrt_typ_rec 0 X1 A1.
Fixpoint close_Fty_wrt_typ_rec (n1 : nat) (X1 : typevar) (Fty1 : Fty) {struct Fty1} : Fty :=
match Fty1 with
| fty_StackArg A1 => fty_StackArg (close_typ_wrt_typ_rec n1 X1 A1)
| fty_StackTyArg A1 => fty_StackTyArg (close_typ_wrt_typ_rec n1 X1 A1)
end.
Definition close_Fty_wrt_typ X1 Fty1 := close_Fty_wrt_typ_rec 0 X1 Fty1.
(* *********************************************************************** *)
(** * Size *)
Fixpoint size_l (l1 : l) {struct l1} : nat :=
match l1 with
| lbl_TagIndex i1 => 1
| lbl_TagLeft => 1
| lbl_TagRight => 1
end.
Fixpoint size_typ (A1 : typ) {struct A1} : nat :=
match A1 with
| t_tvar_f X1 => 1
| t_tvar_b n1 => 1
| t_rcd l1 A2 => 1 + (size_l l1) + (size_typ A2)
| t_and A2 A3 => 1 + (size_typ A2) + (size_typ A3)
| t_or A2 A3 => 1 + (size_typ A2) + (size_typ A3)
| t_arrow A2 B1 => 1 + (size_typ A2) + (size_typ B1)
| t_forall B1 => 1 + (size_typ B1)
| t_top => 1
| t_bot => 1
end.
Fixpoint size_Fty (Fty1 : Fty) {struct Fty1} : nat :=
match Fty1 with
| fty_StackArg A1 => 1 + (size_typ A1)
| fty_StackTyArg A1 => 1 + (size_typ A1)
end.
(* *********************************************************************** *)
(** * Degree *)
(** These define only an upper bound, not a strict upper bound. *)
Inductive degree_typ_wrt_typ : nat -> typ -> Prop :=
| degree_wrt_typ_t_tvar_f : forall n1 X1,
degree_typ_wrt_typ n1 (t_tvar_f X1)
| degree_wrt_typ_t_tvar_b : forall n1 n2,
lt n2 n1 ->
degree_typ_wrt_typ n1 (t_tvar_b n2)
| degree_wrt_typ_t_rcd : forall n1 l1 A1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ n1 (t_rcd l1 A1)
| degree_wrt_typ_t_and : forall n1 A1 A2,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ n1 A2 ->
degree_typ_wrt_typ n1 (t_and A1 A2)
| degree_wrt_typ_t_or : forall n1 A1 A2,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ n1 A2 ->
degree_typ_wrt_typ n1 (t_or A1 A2)
| degree_wrt_typ_t_arrow : forall n1 A1 B1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ n1 B1 ->
degree_typ_wrt_typ n1 (t_arrow A1 B1)
| degree_wrt_typ_t_forall : forall n1 B1,
degree_typ_wrt_typ (S n1) B1 ->
degree_typ_wrt_typ n1 (t_forall B1)
| degree_wrt_typ_t_top : forall n1,
degree_typ_wrt_typ n1 (t_top)
| degree_wrt_typ_t_bot : forall n1,
degree_typ_wrt_typ n1 (t_bot).
Scheme degree_typ_wrt_typ_ind' := Induction for degree_typ_wrt_typ Sort Prop.
Definition degree_typ_wrt_typ_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
degree_typ_wrt_typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Hint Constructors degree_typ_wrt_typ : core lngen.
Inductive degree_Fty_wrt_typ : nat -> Fty -> Prop :=
| degree_wrt_typ_fty_StackArg : forall n1 A1,
degree_typ_wrt_typ n1 A1 ->
degree_Fty_wrt_typ n1 (fty_StackArg A1)
| degree_wrt_typ_fty_StackTyArg : forall n1 A1,
degree_typ_wrt_typ n1 A1 ->
degree_Fty_wrt_typ n1 (fty_StackTyArg A1).
Scheme degree_Fty_wrt_typ_ind' := Induction for degree_Fty_wrt_typ Sort Prop.
Definition degree_Fty_wrt_typ_mutind :=
fun H1 H2 H3 =>
degree_Fty_wrt_typ_ind' H1 H2 H3.
Hint Constructors degree_Fty_wrt_typ : core lngen.
(* *********************************************************************** *)
(** * Local closure (version in [Set], induction principles) *)
Inductive lc_set_typ : typ -> Set :=
| lc_set_t_tvar_f : forall X1,
lc_set_typ (t_tvar_f X1)
| lc_set_t_rcd : forall l1 A1,
lc_set_typ A1 ->
lc_set_typ (t_rcd l1 A1)
| lc_set_t_and : forall A1 A2,
lc_set_typ A1 ->
lc_set_typ A2 ->
lc_set_typ (t_and A1 A2)
| lc_set_t_or : forall A1 A2,
lc_set_typ A1 ->
lc_set_typ A2 ->
lc_set_typ (t_or A1 A2)
| lc_set_t_arrow : forall A1 B1,
lc_set_typ A1 ->
lc_set_typ B1 ->
lc_set_typ (t_arrow A1 B1)
| lc_set_t_forall : forall B1,
(forall X1 : typevar, lc_set_typ (open_typ_wrt_typ B1 (t_tvar_f X1))) ->
lc_set_typ (t_forall B1)
| lc_set_t_top :
lc_set_typ (t_top)
| lc_set_t_bot :
lc_set_typ (t_bot).
Scheme lc_typ_ind' := Induction for lc_typ Sort Prop.
Definition lc_typ_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 =>
lc_typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9.
Scheme lc_set_typ_ind' := Induction for lc_set_typ Sort Prop.
Definition lc_set_typ_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 =>
lc_set_typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9.
Scheme lc_set_typ_rec' := Induction for lc_set_typ Sort Set.
Definition lc_set_typ_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 =>
lc_set_typ_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9.
Hint Constructors lc_typ : core lngen.
Hint Constructors lc_set_typ : core lngen.
Inductive lc_set_Fty : Fty -> Set :=
| lc_set_fty_StackArg : forall A1,
lc_set_typ A1 ->
lc_set_Fty (fty_StackArg A1)
| lc_set_fty_StackTyArg : forall A1,
lc_set_typ A1 ->
lc_set_Fty (fty_StackTyArg A1).
Scheme lc_Fty_ind' := Induction for lc_Fty Sort Prop.
Definition lc_Fty_mutind :=
fun H1 H2 H3 =>
lc_Fty_ind' H1 H2 H3.
Scheme lc_set_Fty_ind' := Induction for lc_set_Fty Sort Prop.
Definition lc_set_Fty_mutind :=
fun H1 H2 H3 =>
lc_set_Fty_ind' H1 H2 H3.
Scheme lc_set_Fty_rec' := Induction for lc_set_Fty Sort Set.
Definition lc_set_Fty_mutrec :=
fun H1 H2 H3 =>
lc_set_Fty_rec' H1 H2 H3.
Hint Constructors lc_Fty : core lngen.
Hint Constructors lc_set_Fty : core lngen.
(* *********************************************************************** *)
(** * Body *)
Definition body_typ_wrt_typ A1 := forall X1, lc_typ (open_typ_wrt_typ A1 (t_tvar_f X1)).
Hint Unfold body_typ_wrt_typ : core.
Definition body_Fty_wrt_typ Fty1 := forall X1, lc_Fty (open_Fty_wrt_typ Fty1 (t_tvar_f X1)).
Hint Unfold body_Fty_wrt_typ : core.
(* *********************************************************************** *)
(** * 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_l_min_mutual :
(forall l1, 1 <= size_l l1).
Proof.
apply_mutual_ind l_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_l_min :
forall l1, 1 <= size_l l1.
Proof.
pose proof size_l_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_l_min : lngen.
(* begin hide *)
Lemma size_typ_min_mutual :
(forall A1, 1 <= size_typ A1).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_typ_min :
forall A1, 1 <= size_typ A1.
Proof.
pose proof size_typ_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_min : lngen.
(* begin hide *)
Lemma size_Fty_min_mutual :
(forall Fty1, 1 <= size_Fty Fty1).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_Fty_min :
forall Fty1, 1 <= size_Fty Fty1.
Proof.
pose proof size_Fty_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_Fty_min : lngen.
(* begin hide *)
Lemma size_typ_close_typ_wrt_typ_rec_mutual :
(forall A1 X1 n1,
size_typ (close_typ_wrt_typ_rec n1 X1 A1) = size_typ A1).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_typ_close_typ_wrt_typ_rec :
forall A1 X1 n1,
size_typ (close_typ_wrt_typ_rec n1 X1 A1) = size_typ A1.
Proof.
pose proof size_typ_close_typ_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_close_typ_wrt_typ_rec : lngen.
Hint Rewrite size_typ_close_typ_wrt_typ_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_Fty_close_Fty_wrt_typ_rec_mutual :
(forall Fty1 X1 n1,
size_Fty (close_Fty_wrt_typ_rec n1 X1 Fty1) = size_Fty Fty1).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_Fty_close_Fty_wrt_typ_rec :
forall Fty1 X1 n1,
size_Fty (close_Fty_wrt_typ_rec n1 X1 Fty1) = size_Fty Fty1.
Proof.
pose proof size_Fty_close_Fty_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_Fty_close_Fty_wrt_typ_rec : lngen.
Hint Rewrite size_Fty_close_Fty_wrt_typ_rec using solve [auto] : lngen.
(* end hide *)
Lemma size_typ_close_typ_wrt_typ :
forall A1 X1,
size_typ (close_typ_wrt_typ X1 A1) = size_typ A1.
Proof.
unfold close_typ_wrt_typ; default_simp.
Qed.
Hint Resolve size_typ_close_typ_wrt_typ : lngen.
Hint Rewrite size_typ_close_typ_wrt_typ using solve [auto] : lngen.
Lemma size_Fty_close_Fty_wrt_typ :
forall Fty1 X1,
size_Fty (close_Fty_wrt_typ X1 Fty1) = size_Fty Fty1.
Proof.
unfold close_Fty_wrt_typ; default_simp.
Qed.
Hint Resolve size_Fty_close_Fty_wrt_typ : lngen.
Hint Rewrite size_Fty_close_Fty_wrt_typ using solve [auto] : lngen.
(* begin hide *)
Lemma size_typ_open_typ_wrt_typ_rec_mutual :
(forall A1 A2 n1,
size_typ A1 <= size_typ (open_typ_wrt_typ_rec n1 A2 A1)).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_typ_open_typ_wrt_typ_rec :
forall A1 A2 n1,
size_typ A1 <= size_typ (open_typ_wrt_typ_rec n1 A2 A1).
Proof.
pose proof size_typ_open_typ_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_open_typ_wrt_typ_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_Fty_open_Fty_wrt_typ_rec_mutual :
(forall Fty1 A1 n1,
size_Fty Fty1 <= size_Fty (open_Fty_wrt_typ_rec n1 A1 Fty1)).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_Fty_open_Fty_wrt_typ_rec :
forall Fty1 A1 n1,
size_Fty Fty1 <= size_Fty (open_Fty_wrt_typ_rec n1 A1 Fty1).
Proof.
pose proof size_Fty_open_Fty_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_Fty_open_Fty_wrt_typ_rec : lngen.
(* end hide *)
Lemma size_typ_open_typ_wrt_typ :
forall A1 A2,
size_typ A1 <= size_typ (open_typ_wrt_typ A1 A2).
Proof.
unfold open_typ_wrt_typ; default_simp.
Qed.
Hint Resolve size_typ_open_typ_wrt_typ : lngen.
Lemma size_Fty_open_Fty_wrt_typ :
forall Fty1 A1,
size_Fty Fty1 <= size_Fty (open_Fty_wrt_typ Fty1 A1).
Proof.
unfold open_Fty_wrt_typ; default_simp.
Qed.
Hint Resolve size_Fty_open_Fty_wrt_typ : lngen.
(* begin hide *)
Lemma size_typ_open_typ_wrt_typ_rec_var_mutual :
(forall A1 X1 n1,
size_typ (open_typ_wrt_typ_rec n1 (t_tvar_f X1) A1) = size_typ A1).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_typ_open_typ_wrt_typ_rec_var :
forall A1 X1 n1,
size_typ (open_typ_wrt_typ_rec n1 (t_tvar_f X1) A1) = size_typ A1.
Proof.
pose proof size_typ_open_typ_wrt_typ_rec_var_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_open_typ_wrt_typ_rec_var : lngen.
Hint Rewrite size_typ_open_typ_wrt_typ_rec_var using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_Fty_open_Fty_wrt_typ_rec_var_mutual :
(forall Fty1 X1 n1,
size_Fty (open_Fty_wrt_typ_rec n1 (t_tvar_f X1) Fty1) = size_Fty Fty1).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_Fty_open_Fty_wrt_typ_rec_var :
forall Fty1 X1 n1,
size_Fty (open_Fty_wrt_typ_rec n1 (t_tvar_f X1) Fty1) = size_Fty Fty1.
Proof.
pose proof size_Fty_open_Fty_wrt_typ_rec_var_mutual as H; intuition eauto.
Qed.
Hint Resolve size_Fty_open_Fty_wrt_typ_rec_var : lngen.
Hint Rewrite size_Fty_open_Fty_wrt_typ_rec_var using solve [auto] : lngen.
(* end hide *)
Lemma size_typ_open_typ_wrt_typ_var :
forall A1 X1,
size_typ (open_typ_wrt_typ A1 (t_tvar_f X1)) = size_typ A1.
Proof.
unfold open_typ_wrt_typ; default_simp.
Qed.
Hint Resolve size_typ_open_typ_wrt_typ_var : lngen.
Hint Rewrite size_typ_open_typ_wrt_typ_var using solve [auto] : lngen.
Lemma size_Fty_open_Fty_wrt_typ_var :
forall Fty1 X1,
size_Fty (open_Fty_wrt_typ Fty1 (t_tvar_f X1)) = size_Fty Fty1.
Proof.
unfold open_Fty_wrt_typ; default_simp.
Qed.
Hint Resolve size_Fty_open_Fty_wrt_typ_var : lngen.
Hint Rewrite size_Fty_open_Fty_wrt_typ_var using solve [auto] : lngen.
(* *********************************************************************** *)
(** * Theorems about [degree] *)
Ltac default_auto ::= auto with lngen; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma degree_typ_wrt_typ_S_mutual :
(forall n1 A1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ (S n1) A1).
Proof.
apply_mutual_ind degree_typ_wrt_typ_mutind;
default_simp.
Qed.
(* end hide *)
Lemma degree_typ_wrt_typ_S :
forall n1 A1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ (S n1) A1.
Proof.
pose proof degree_typ_wrt_typ_S_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_typ_wrt_typ_S : lngen.
(* begin hide *)
Lemma degree_Fty_wrt_typ_S_mutual :
(forall n1 Fty1,
degree_Fty_wrt_typ n1 Fty1 ->
degree_Fty_wrt_typ (S n1) Fty1).
Proof.
apply_mutual_ind degree_Fty_wrt_typ_mutind;
default_simp.
Qed.
(* end hide *)
Lemma degree_Fty_wrt_typ_S :
forall n1 Fty1,
degree_Fty_wrt_typ n1 Fty1 ->
degree_Fty_wrt_typ (S n1) Fty1.
Proof.
pose proof degree_Fty_wrt_typ_S_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_Fty_wrt_typ_S : lngen.
Lemma degree_typ_wrt_typ_O :
forall n1 A1,
degree_typ_wrt_typ O A1 ->
degree_typ_wrt_typ n1 A1.
Proof.
induction n1; default_simp.
Qed.
Hint Resolve degree_typ_wrt_typ_O : lngen.
Lemma degree_Fty_wrt_typ_O :
forall n1 Fty1,
degree_Fty_wrt_typ O Fty1 ->
degree_Fty_wrt_typ n1 Fty1.
Proof.
induction n1; default_simp.
Qed.
Hint Resolve degree_Fty_wrt_typ_O : lngen.
(* begin hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ_rec_mutual :
(forall A1 X1 n1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ (S n1) (close_typ_wrt_typ_rec n1 X1 A1)).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ_rec :
forall A1 X1 n1,
degree_typ_wrt_typ n1 A1 ->
degree_typ_wrt_typ (S n1) (close_typ_wrt_typ_rec n1 X1 A1).
Proof.
pose proof degree_typ_wrt_typ_close_typ_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_typ_wrt_typ_close_typ_wrt_typ_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_mutual :
(forall Fty1 X1 n1,
degree_Fty_wrt_typ n1 Fty1 ->
degree_Fty_wrt_typ (S n1) (close_Fty_wrt_typ_rec n1 X1 Fty1)).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ_rec :
forall Fty1 X1 n1,
degree_Fty_wrt_typ n1 Fty1 ->
degree_Fty_wrt_typ (S n1) (close_Fty_wrt_typ_rec n1 X1 Fty1).
Proof.
pose proof degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_Fty_wrt_typ_close_Fty_wrt_typ_rec : lngen.
(* end hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ :
forall A1 X1,
degree_typ_wrt_typ 0 A1 ->
degree_typ_wrt_typ 1 (close_typ_wrt_typ X1 A1).
Proof.
unfold close_typ_wrt_typ; default_simp.
Qed.
Hint Resolve degree_typ_wrt_typ_close_typ_wrt_typ : lngen.
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ :
forall Fty1 X1,
degree_Fty_wrt_typ 0 Fty1 ->
degree_Fty_wrt_typ 1 (close_Fty_wrt_typ X1 Fty1).
Proof.
unfold close_Fty_wrt_typ; default_simp.
Qed.
Hint Resolve degree_Fty_wrt_typ_close_Fty_wrt_typ : lngen.
(* begin hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ_rec_inv_mutual :
(forall A1 X1 n1,
degree_typ_wrt_typ (S n1) (close_typ_wrt_typ_rec n1 X1 A1) ->
degree_typ_wrt_typ n1 A1).
Proof.
apply_mutual_ind typ_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ_rec_inv :
forall A1 X1 n1,
degree_typ_wrt_typ (S n1) (close_typ_wrt_typ_rec n1 X1 A1) ->
degree_typ_wrt_typ n1 A1.
Proof.
pose proof degree_typ_wrt_typ_close_typ_wrt_typ_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_typ_wrt_typ_close_typ_wrt_typ_rec_inv : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_inv_mutual :
(forall Fty1 X1 n1,
degree_Fty_wrt_typ (S n1) (close_Fty_wrt_typ_rec n1 X1 Fty1) ->
degree_Fty_wrt_typ n1 Fty1).
Proof.
apply_mutual_ind Fty_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_inv :
forall Fty1 X1 n1,
degree_Fty_wrt_typ (S n1) (close_Fty_wrt_typ_rec n1 X1 Fty1) ->
degree_Fty_wrt_typ n1 Fty1.
Proof.
pose proof degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_Fty_wrt_typ_close_Fty_wrt_typ_rec_inv : lngen.
(* end hide *)
Lemma degree_typ_wrt_typ_close_typ_wrt_typ_inv :
forall A1 X1,
degree_typ_wrt_typ 1 (close_typ_wrt_typ X1 A1) ->
degree_typ_wrt_typ 0 A1.
Proof.
unfold close_typ_wrt_typ; eauto with lngen.
Qed.
Hint Immediate degree_typ_wrt_typ_close_typ_wrt_typ_inv : lngen.
Lemma degree_Fty_wrt_typ_close_Fty_wrt_typ_inv :
forall Fty1 X1,
degree_Fty_wrt_typ 1 (close_Fty_wrt_typ X1 Fty1) ->
degree_Fty_wrt_typ 0 Fty1.
Proof.
unfold close_Fty_wrt_typ; eauto with lngen.
Qed.
Hint Immediate degree_Fty_wrt_typ_close_Fty_wrt_typ_inv : lngen.
(* begin hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ_rec_mutual :
(forall A1 A2 n1,
degree_typ_wrt_typ (S n1) A1 ->
degree_typ_wrt_typ n1 A2 ->
degree_typ_wrt_typ n1 (open_typ_wrt_typ_rec n1 A2 A1)).
Proof.
apply_mutual_ind typ_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ_rec :
forall A1 A2 n1,
degree_typ_wrt_typ (S n1) A1 ->
degree_typ_wrt_typ n1 A2 ->
degree_typ_wrt_typ n1 (open_typ_wrt_typ_rec n1 A2 A1).
Proof.
pose proof degree_typ_wrt_typ_open_typ_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_typ_wrt_typ_open_typ_wrt_typ_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_mutual :
(forall Fty1 A1 n1,
degree_Fty_wrt_typ (S n1) Fty1 ->
degree_typ_wrt_typ n1 A1 ->
degree_Fty_wrt_typ n1 (open_Fty_wrt_typ_rec n1 A1 Fty1)).
Proof.
apply_mutual_ind Fty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ_rec :
forall Fty1 A1 n1,
degree_Fty_wrt_typ (S n1) Fty1 ->
degree_typ_wrt_typ n1 A1 ->
degree_Fty_wrt_typ n1 (open_Fty_wrt_typ_rec n1 A1 Fty1).
Proof.
pose proof degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_Fty_wrt_typ_open_Fty_wrt_typ_rec : lngen.
(* end hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ :
forall A1 A2,
degree_typ_wrt_typ 1 A1 ->
degree_typ_wrt_typ 0 A2 ->
degree_typ_wrt_typ 0 (open_typ_wrt_typ A1 A2).
Proof.
unfold open_typ_wrt_typ; default_simp.
Qed.
Hint Resolve degree_typ_wrt_typ_open_typ_wrt_typ : lngen.
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ :
forall Fty1 A1,
degree_Fty_wrt_typ 1 Fty1 ->
degree_typ_wrt_typ 0 A1 ->
degree_Fty_wrt_typ 0 (open_Fty_wrt_typ Fty1 A1).
Proof.
unfold open_Fty_wrt_typ; default_simp.
Qed.
Hint Resolve degree_Fty_wrt_typ_open_Fty_wrt_typ : lngen.
(* begin hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ_rec_inv_mutual :
(forall A1 A2 n1,
degree_typ_wrt_typ n1 (open_typ_wrt_typ_rec n1 A2 A1) ->
degree_typ_wrt_typ (S n1) A1).
Proof.
apply_mutual_ind typ_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ_rec_inv :
forall A1 A2 n1,
degree_typ_wrt_typ n1 (open_typ_wrt_typ_rec n1 A2 A1) ->
degree_typ_wrt_typ (S n1) A1.
Proof.
pose proof degree_typ_wrt_typ_open_typ_wrt_typ_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_typ_wrt_typ_open_typ_wrt_typ_rec_inv : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_inv_mutual :
(forall Fty1 A1 n1,
degree_Fty_wrt_typ n1 (open_Fty_wrt_typ_rec n1 A1 Fty1) ->
degree_Fty_wrt_typ (S n1) Fty1).
Proof.
apply_mutual_ind Fty_mutind;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_inv :
forall Fty1 A1 n1,
degree_Fty_wrt_typ n1 (open_Fty_wrt_typ_rec n1 A1 Fty1) ->
degree_Fty_wrt_typ (S n1) Fty1.
Proof.
pose proof degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_inv_mutual as H; intuition eauto.
Qed.
Hint Immediate degree_Fty_wrt_typ_open_Fty_wrt_typ_rec_inv : lngen.
(* end hide *)
Lemma degree_typ_wrt_typ_open_typ_wrt_typ_inv :
forall A1 A2,
degree_typ_wrt_typ 0 (open_typ_wrt_typ A1 A2) ->
degree_typ_wrt_typ 1 A1.
Proof.
unfold open_typ_wrt_typ; eauto with lngen.
Qed.
Hint Immediate degree_typ_wrt_typ_open_typ_wrt_typ_inv : lngen.
Lemma degree_Fty_wrt_typ_open_Fty_wrt_typ_inv :
forall Fty1 A1,
degree_Fty_wrt_typ 0 (open_Fty_wrt_typ Fty1 A1) ->
degree_Fty_wrt_typ 1 Fty1.
Proof.
unfold open_Fty_wrt_typ; eauto with lngen.
Qed.
Hint Immediate degree_Fty_wrt_typ_open_Fty_wrt_typ_inv : lngen.
(* *********************************************************************** *)
(** * Theorems about [open] and [close] *)
Ltac default_auto ::= auto with lngen brute_force; tauto.
Ltac default_autorewrite ::= fail.
(* begin hide *)
Lemma close_typ_wrt_typ_rec_inj_mutual :
(forall A1 A2 X1 n1,
close_typ_wrt_typ_rec n1 X1 A1 = close_typ_wrt_typ_rec n1 X1 A2 ->
A1 = A2).
Proof.
apply_mutual_ind typ_mutind;
intros; match goal with
| |- _ = ?term => destruct term
end;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma close_typ_wrt_typ_rec_inj :
forall A1 A2 X1 n1,
close_typ_wrt_typ_rec n1 X1 A1 = close_typ_wrt_typ_rec n1 X1 A2 ->
A1 = A2.
Proof.
pose proof close_typ_wrt_typ_rec_inj_mutual as H; intuition eauto.
Qed.
Hint Immediate close_typ_wrt_typ_rec_inj : lngen.
(* end hide *)
(* begin hide *)
Lemma close_Fty_wrt_typ_rec_inj_mutual :
(forall Fty1 Fty2 X1 n1,
close_Fty_wrt_typ_rec n1 X1 Fty1 = close_Fty_wrt_typ_rec n1 X1 Fty2 ->
Fty1 = Fty2).
Proof.
apply_mutual_ind Fty_mutind;
intros; match goal with
| |- _ = ?term => destruct term
end;
default_simp; eauto with lngen.
Qed.
(* end hide *)
(* begin hide *)
Lemma close_Fty_wrt_typ_rec_inj :
forall Fty1 Fty2 X1 n1,
close_Fty_wrt_typ_rec n1 X1 Fty1 = close_Fty_wrt_typ_rec n1 X1 Fty2 ->
Fty1 = Fty2.
Proof.
pose proof close_Fty_wrt_typ_rec_inj_mutual as H; intuition eauto.
Qed.
Hint Immediate close_Fty_wrt_typ_rec_inj : lngen.
(* end hide *)