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SystemF_inf.v
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SystemF_inf.v
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(* !!! WARNING: AUTO GENERATED. DO NOT MODIFY !!! *)
Require Import Coq.Logic.FunctionalExtensionality.
Require Import Coq.Program.Equality.
Require Export Metalib.Metatheory.
Require Export Metalib.LibLNgen.
Require Export Syntax_ott.
(** 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 ty_ind' := Induction for ty Sort Prop.
Definition ty_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 =>
ty_ind' H1 H2 H3 H4 H5 H6 H7 H8.
Scheme ty_rec' := Induction for ty Sort Set.
Definition ty_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 =>
ty_rec' H1 H2 H3 H4 H5 H6 H7 H8.
Scheme co_ind' := Induction for co Sort Prop.
Definition co_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 =>
co_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14.
Scheme co_rec' := Induction for co Sort Set.
Definition co_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 =>
co_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14.
Scheme exp_ind' := Induction for exp Sort Prop.
Definition exp_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
exp_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11.
Scheme exp_rec' := Induction for exp Sort Set.
Definition exp_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
exp_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11.
(* *********************************************************************** *)
(** * Close *)
Fixpoint close_ty_wrt_ty_rec (n1 : nat) (X1 : typvar) (T1 : ty) {struct T1} : ty :=
match T1 with
| ty_nat => ty_nat
| ty_unit => ty_unit
| ty_var_f X2 => if (X1 == X2) then (ty_var_b n1) else (ty_var_f X2)
| ty_var_b n2 => if (lt_ge_dec n2 n1) then (ty_var_b n2) else (ty_var_b (S n2))
| ty_arrow T2 T3 => ty_arrow (close_ty_wrt_ty_rec n1 X1 T2) (close_ty_wrt_ty_rec n1 X1 T3)
| ty_prod T2 T3 => ty_prod (close_ty_wrt_ty_rec n1 X1 T2) (close_ty_wrt_ty_rec n1 X1 T3)
| ty_all T2 => ty_all (close_ty_wrt_ty_rec (S n1) X1 T2)
end.
Definition close_ty_wrt_ty X1 T1 := close_ty_wrt_ty_rec 0 X1 T1.
Fixpoint close_exp_wrt_ty_rec (n1 : nat) (X1 : typvar) (e1 : exp) {struct e1} : exp :=
match e1 with
| exp_var_f x1 => exp_var_f x1
| exp_var_b n2 => exp_var_b n2
| exp_unit => exp_unit
| exp_lit i1 => exp_lit i1
| exp_abs e2 => exp_abs (close_exp_wrt_ty_rec n1 X1 e2)
| exp_app e2 e3 => exp_app (close_exp_wrt_ty_rec n1 X1 e2) (close_exp_wrt_ty_rec n1 X1 e3)
| exp_pair e2 e3 => exp_pair (close_exp_wrt_ty_rec n1 X1 e2) (close_exp_wrt_ty_rec n1 X1 e3)
| exp_capp c1 e2 => exp_capp c1 (close_exp_wrt_ty_rec n1 X1 e2)
| exp_tabs e2 => exp_tabs (close_exp_wrt_ty_rec (S n1) X1 e2)
| exp_tapp e2 T1 => exp_tapp (close_exp_wrt_ty_rec n1 X1 e2) (close_ty_wrt_ty_rec n1 X1 T1)
end.
Fixpoint close_exp_wrt_exp_rec (n1 : nat) (x1 : expvar) (e1 : exp) {struct e1} : exp :=
match e1 with
| exp_var_f x2 => if (x1 == x2) then (exp_var_b n1) else (exp_var_f x2)
| exp_var_b n2 => if (lt_ge_dec n2 n1) then (exp_var_b n2) else (exp_var_b (S n2))
| exp_unit => exp_unit
| exp_lit i1 => exp_lit i1
| exp_abs e2 => exp_abs (close_exp_wrt_exp_rec (S n1) x1 e2)
| exp_app e2 e3 => exp_app (close_exp_wrt_exp_rec n1 x1 e2) (close_exp_wrt_exp_rec n1 x1 e3)
| exp_pair e2 e3 => exp_pair (close_exp_wrt_exp_rec n1 x1 e2) (close_exp_wrt_exp_rec n1 x1 e3)
| exp_capp c1 e2 => exp_capp c1 (close_exp_wrt_exp_rec n1 x1 e2)
| exp_tabs e2 => exp_tabs (close_exp_wrt_exp_rec n1 x1 e2)
| exp_tapp e2 T1 => exp_tapp (close_exp_wrt_exp_rec n1 x1 e2) T1
end.
Definition close_exp_wrt_ty X1 e1 := close_exp_wrt_ty_rec 0 X1 e1.
Definition close_exp_wrt_exp x1 e1 := close_exp_wrt_exp_rec 0 x1 e1.
(* *********************************************************************** *)
(** * Size *)
Fixpoint size_ty (T1 : ty) {struct T1} : nat :=
match T1 with
| ty_nat => 1
| ty_unit => 1
| ty_var_f X1 => 1
| ty_var_b n1 => 1
| ty_arrow T2 T3 => 1 + (size_ty T2) + (size_ty T3)
| ty_prod T2 T3 => 1 + (size_ty T2) + (size_ty T3)
| ty_all T2 => 1 + (size_ty T2)
end.
Fixpoint size_co (c1 : co) {struct c1} : nat :=
match c1 with
| co_id => 1
| co_trans c2 c3 => 1 + (size_co c2) + (size_co c3)
| co_top => 1
| co_bot => 1
| co_arr c2 c3 => 1 + (size_co c2) + (size_co c3)
| co_pair c2 c3 => 1 + (size_co c2) + (size_co c3)
| co_proj1 => 1
| co_proj2 => 1
| co_forall c2 => 1 + (size_co c2)
| co_distArr => 1
| co_topArr => 1
| co_topAll => 1
| co_distPoly => 1
end.
Fixpoint size_exp (e1 : exp) {struct e1} : nat :=
match e1 with
| exp_var_f x1 => 1
| exp_var_b n1 => 1
| exp_unit => 1
| exp_lit i1 => 1
| exp_abs e2 => 1 + (size_exp e2)
| exp_app e2 e3 => 1 + (size_exp e2) + (size_exp e3)
| exp_pair e2 e3 => 1 + (size_exp e2) + (size_exp e3)
| exp_capp c1 e2 => 1 + (size_co c1) + (size_exp e2)
| exp_tabs e2 => 1 + (size_exp e2)
| exp_tapp e2 T1 => 1 + (size_exp e2) + (size_ty T1)
end.
(* *********************************************************************** *)
(** * Degree *)
(** These define only an upper bound, not a strict upper bound. *)
Inductive degree_ty_wrt_ty : nat -> ty -> Prop :=
| degree_wrt_ty_ty_nat : forall n1,
degree_ty_wrt_ty n1 (ty_nat)
| degree_wrt_ty_ty_unit : forall n1,
degree_ty_wrt_ty n1 (ty_unit)
| degree_wrt_ty_ty_var_f : forall n1 X1,
degree_ty_wrt_ty n1 (ty_var_f X1)
| degree_wrt_ty_ty_var_b : forall n1 n2,
lt n2 n1 ->
degree_ty_wrt_ty n1 (ty_var_b n2)
| degree_wrt_ty_ty_arrow : forall n1 T1 T2,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty n1 T2 ->
degree_ty_wrt_ty n1 (ty_arrow T1 T2)
| degree_wrt_ty_ty_prod : forall n1 T1 T2,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty n1 T2 ->
degree_ty_wrt_ty n1 (ty_prod T1 T2)
| degree_wrt_ty_ty_all : forall n1 T1,
degree_ty_wrt_ty (S n1) T1 ->
degree_ty_wrt_ty n1 (ty_all T1).
Scheme degree_ty_wrt_ty_ind' := Induction for degree_ty_wrt_ty Sort Prop.
Definition degree_ty_wrt_ty_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 =>
degree_ty_wrt_ty_ind' H1 H2 H3 H4 H5 H6 H7 H8.
Hint Constructors degree_ty_wrt_ty : core lngen.
Inductive degree_exp_wrt_ty : nat -> exp -> Prop :=
| degree_wrt_ty_exp_var_f : forall n1 x1,
degree_exp_wrt_ty n1 (exp_var_f x1)
| degree_wrt_ty_exp_var_b : forall n1 n2,
degree_exp_wrt_ty n1 (exp_var_b n2)
| degree_wrt_ty_exp_unit : forall n1,
degree_exp_wrt_ty n1 (exp_unit)
| degree_wrt_ty_exp_lit : forall n1 i1,
degree_exp_wrt_ty n1 (exp_lit i1)
| degree_wrt_ty_exp_abs : forall n1 e1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty n1 (exp_abs e1)
| degree_wrt_ty_exp_app : forall n1 e1 e2,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty n1 e2 ->
degree_exp_wrt_ty n1 (exp_app e1 e2)
| degree_wrt_ty_exp_pair : forall n1 e1 e2,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty n1 e2 ->
degree_exp_wrt_ty n1 (exp_pair e1 e2)
| degree_wrt_ty_exp_capp : forall n1 c1 e1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty n1 (exp_capp c1 e1)
| degree_wrt_ty_exp_tabs : forall n1 e1,
degree_exp_wrt_ty (S n1) e1 ->
degree_exp_wrt_ty n1 (exp_tabs e1)
| degree_wrt_ty_exp_tapp : forall n1 e1 T1,
degree_exp_wrt_ty n1 e1 ->
degree_ty_wrt_ty n1 T1 ->
degree_exp_wrt_ty n1 (exp_tapp e1 T1).
Inductive degree_exp_wrt_exp : nat -> exp -> Prop :=
| degree_wrt_exp_exp_var_f : forall n1 x1,
degree_exp_wrt_exp n1 (exp_var_f x1)
| degree_wrt_exp_exp_var_b : forall n1 n2,
lt n2 n1 ->
degree_exp_wrt_exp n1 (exp_var_b n2)
| degree_wrt_exp_exp_unit : forall n1,
degree_exp_wrt_exp n1 (exp_unit)
| degree_wrt_exp_exp_lit : forall n1 i1,
degree_exp_wrt_exp n1 (exp_lit i1)
| degree_wrt_exp_exp_abs : forall n1 e1,
degree_exp_wrt_exp (S n1) e1 ->
degree_exp_wrt_exp n1 (exp_abs e1)
| degree_wrt_exp_exp_app : forall n1 e1 e2,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 e2 ->
degree_exp_wrt_exp n1 (exp_app e1 e2)
| degree_wrt_exp_exp_pair : forall n1 e1 e2,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 e2 ->
degree_exp_wrt_exp n1 (exp_pair e1 e2)
| degree_wrt_exp_exp_capp : forall n1 c1 e1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 (exp_capp c1 e1)
| degree_wrt_exp_exp_tabs : forall n1 e1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 (exp_tabs e1)
| degree_wrt_exp_exp_tapp : forall n1 e1 T1,
degree_exp_wrt_exp n1 e1 ->
degree_exp_wrt_exp n1 (exp_tapp e1 T1).
Scheme degree_exp_wrt_ty_ind' := Induction for degree_exp_wrt_ty Sort Prop.
Definition degree_exp_wrt_ty_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
degree_exp_wrt_ty_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11.
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 H6 H7 H8 H9 H10 H11 =>
degree_exp_wrt_exp_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11.
Hint Constructors degree_exp_wrt_ty : core lngen.
Hint Constructors degree_exp_wrt_exp : core lngen.
(* *********************************************************************** *)
(** * Local closure (version in [Set], induction principles) *)
Inductive lc_set_ty : ty -> Set :=
| lc_set_ty_nat :
lc_set_ty (ty_nat)
| lc_set_ty_unit :
lc_set_ty (ty_unit)
| lc_set_ty_var_f : forall X1,
lc_set_ty (ty_var_f X1)
| lc_set_ty_arrow : forall T1 T2,
lc_set_ty T1 ->
lc_set_ty T2 ->
lc_set_ty (ty_arrow T1 T2)
| lc_set_ty_prod : forall T1 T2,
lc_set_ty T1 ->
lc_set_ty T2 ->
lc_set_ty (ty_prod T1 T2)
| lc_set_ty_all : forall T1,
(forall X1 : typvar, lc_set_ty (open_ty_wrt_ty T1 (ty_var_f X1))) ->
lc_set_ty (ty_all T1).
Scheme lc_ty_ind' := Induction for lc_ty Sort Prop.
Definition lc_ty_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 =>
lc_ty_ind' H1 H2 H3 H4 H5 H6 H7.
Scheme lc_set_ty_ind' := Induction for lc_set_ty Sort Prop.
Definition lc_set_ty_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 =>
lc_set_ty_ind' H1 H2 H3 H4 H5 H6 H7.
Scheme lc_set_ty_rec' := Induction for lc_set_ty Sort Set.
Definition lc_set_ty_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 =>
lc_set_ty_rec' H1 H2 H3 H4 H5 H6 H7.
Hint Constructors lc_ty : core lngen.
Hint Constructors lc_set_ty : core lngen.
Inductive lc_set_exp : exp -> Set :=
| lc_set_exp_var_f : forall x1,
lc_set_exp (exp_var_f x1)
| lc_set_exp_unit :
lc_set_exp (exp_unit)
| lc_set_exp_lit : forall i1,
lc_set_exp (exp_lit i1)
| lc_set_exp_abs : forall e1,
(forall x1 : expvar, lc_set_exp (open_exp_wrt_exp e1 (exp_var_f x1))) ->
lc_set_exp (exp_abs e1)
| lc_set_exp_app : forall e1 e2,
lc_set_exp e1 ->
lc_set_exp e2 ->
lc_set_exp (exp_app e1 e2)
| lc_set_exp_pair : forall e1 e2,
lc_set_exp e1 ->
lc_set_exp e2 ->
lc_set_exp (exp_pair e1 e2)
| lc_set_exp_capp : forall c1 e1,
lc_set_exp e1 ->
lc_set_exp (exp_capp c1 e1)
| lc_set_exp_tabs : forall e1,
(forall X1 : typvar, lc_set_exp (open_exp_wrt_ty e1 (ty_var_f X1))) ->
lc_set_exp (exp_tabs e1)
| lc_set_exp_tapp : forall e1 T1,
lc_set_exp e1 ->
lc_set_ty T1 ->
lc_set_exp (exp_tapp e1 T1).
Scheme lc_exp_ind' := Induction for lc_exp Sort Prop.
Definition lc_exp_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
lc_exp_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Scheme lc_set_exp_ind' := Induction for lc_set_exp Sort Prop.
Definition lc_set_exp_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
lc_set_exp_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Scheme lc_set_exp_rec' := Induction for lc_set_exp Sort Set.
Definition lc_set_exp_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 =>
lc_set_exp_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10.
Hint Constructors lc_exp : core lngen.
Hint Constructors lc_set_exp : core lngen.
(* *********************************************************************** *)
(** * Body *)
Definition body_ty_wrt_ty T1 := forall X1, lc_ty (open_ty_wrt_ty T1 (ty_var_f X1)).
Hint Unfold body_ty_wrt_ty.
Definition body_exp_wrt_ty e1 := forall X1, lc_exp (open_exp_wrt_ty e1 (ty_var_f X1)).
Definition body_exp_wrt_exp e1 := forall x1, lc_exp (open_exp_wrt_exp e1 (exp_var_f x1)).
Hint Unfold body_exp_wrt_ty.
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_ty_min_mutual :
(forall T1, 1 <= size_ty T1).
Proof.
apply_mutual_ind ty_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_ty_min :
forall T1, 1 <= size_ty T1.
Proof.
pose proof size_ty_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_ty_min : lngen.
(* begin hide *)
Lemma size_co_min_mutual :
(forall c1, 1 <= size_co c1).
Proof.
apply_mutual_ind co_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_co_min :
forall c1, 1 <= size_co c1.
Proof.
pose proof size_co_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_co_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_ty_close_ty_wrt_ty_rec_mutual :
(forall T1 X1 n1,
size_ty (close_ty_wrt_ty_rec n1 X1 T1) = size_ty T1).
Proof.
apply_mutual_ind ty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_ty_close_ty_wrt_ty_rec :
forall T1 X1 n1,
size_ty (close_ty_wrt_ty_rec n1 X1 T1) = size_ty T1.
Proof.
pose proof size_ty_close_ty_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_ty_close_ty_wrt_ty_rec : lngen.
Hint Rewrite size_ty_close_ty_wrt_ty_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_exp_close_exp_wrt_ty_rec_mutual :
(forall e1 X1 n1,
size_exp (close_exp_wrt_ty_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_ty_rec :
forall e1 X1 n1,
size_exp (close_exp_wrt_ty_rec n1 X1 e1) = size_exp e1.
Proof.
pose proof size_exp_close_exp_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_close_exp_wrt_ty_rec : lngen.
Hint Rewrite size_exp_close_exp_wrt_ty_rec using solve [auto] : lngen.
(* end hide *)
(* 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_ty_close_ty_wrt_ty :
forall T1 X1,
size_ty (close_ty_wrt_ty X1 T1) = size_ty T1.
Proof.
unfold close_ty_wrt_ty; default_simp.
Qed.
Hint Resolve size_ty_close_ty_wrt_ty : lngen.
Hint Rewrite size_ty_close_ty_wrt_ty using solve [auto] : lngen.
Lemma size_exp_close_exp_wrt_ty :
forall e1 X1,
size_exp (close_exp_wrt_ty X1 e1) = size_exp e1.
Proof.
unfold close_exp_wrt_ty; default_simp.
Qed.
Hint Resolve size_exp_close_exp_wrt_ty : lngen.
Hint Rewrite size_exp_close_exp_wrt_ty using solve [auto] : lngen.
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_ty_open_ty_wrt_ty_rec_mutual :
(forall T1 T2 n1,
size_ty T1 <= size_ty (open_ty_wrt_ty_rec n1 T2 T1)).
Proof.
apply_mutual_ind ty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_ty_open_ty_wrt_ty_rec :
forall T1 T2 n1,
size_ty T1 <= size_ty (open_ty_wrt_ty_rec n1 T2 T1).
Proof.
pose proof size_ty_open_ty_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_ty_open_ty_wrt_ty_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_ty_rec_mutual :
(forall e1 T1 n1,
size_exp e1 <= size_exp (open_exp_wrt_ty_rec n1 T1 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_ty_rec :
forall e1 T1 n1,
size_exp e1 <= size_exp (open_exp_wrt_ty_rec n1 T1 e1).
Proof.
pose proof size_exp_open_exp_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_open_exp_wrt_ty_rec : lngen.
(* end hide *)
(* 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_ty_open_ty_wrt_ty :
forall T1 T2,
size_ty T1 <= size_ty (open_ty_wrt_ty T1 T2).
Proof.
unfold open_ty_wrt_ty; default_simp.
Qed.
Hint Resolve size_ty_open_ty_wrt_ty : lngen.
Lemma size_exp_open_exp_wrt_ty :
forall e1 T1,
size_exp e1 <= size_exp (open_exp_wrt_ty e1 T1).
Proof.
unfold open_exp_wrt_ty; default_simp.
Qed.
Hint Resolve size_exp_open_exp_wrt_ty : lngen.
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_ty_open_ty_wrt_ty_rec_var_mutual :
(forall T1 X1 n1,
size_ty (open_ty_wrt_ty_rec n1 (ty_var_f X1) T1) = size_ty T1).
Proof.
apply_mutual_ind ty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_ty_open_ty_wrt_ty_rec_var :
forall T1 X1 n1,
size_ty (open_ty_wrt_ty_rec n1 (ty_var_f X1) T1) = size_ty T1.
Proof.
pose proof size_ty_open_ty_wrt_ty_rec_var_mutual as H; intuition eauto.
Qed.
Hint Resolve size_ty_open_ty_wrt_ty_rec_var : lngen.
Hint Rewrite size_ty_open_ty_wrt_ty_rec_var using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_ty_rec_var_mutual :
(forall e1 X1 n1,
size_exp (open_exp_wrt_ty_rec n1 (ty_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_ty_rec_var :
forall e1 X1 n1,
size_exp (open_exp_wrt_ty_rec n1 (ty_var_f X1) e1) = size_exp e1.
Proof.
pose proof size_exp_open_exp_wrt_ty_rec_var_mutual as H; intuition eauto.
Qed.
Hint Resolve size_exp_open_exp_wrt_ty_rec_var : lngen.
Hint Rewrite size_exp_open_exp_wrt_ty_rec_var using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_exp_open_exp_wrt_exp_rec_var_mutual :
(forall e1 x1 n1,
size_exp (open_exp_wrt_exp_rec n1 (exp_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 (exp_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_ty_open_ty_wrt_ty_var :
forall T1 X1,
size_ty (open_ty_wrt_ty T1 (ty_var_f X1)) = size_ty T1.
Proof.
unfold open_ty_wrt_ty; default_simp.
Qed.
Hint Resolve size_ty_open_ty_wrt_ty_var : lngen.
Hint Rewrite size_ty_open_ty_wrt_ty_var using solve [auto] : lngen.
Lemma size_exp_open_exp_wrt_ty_var :
forall e1 X1,
size_exp (open_exp_wrt_ty e1 (ty_var_f X1)) = size_exp e1.
Proof.
unfold open_exp_wrt_ty; default_simp.
Qed.
Hint Resolve size_exp_open_exp_wrt_ty_var : lngen.
Hint Rewrite size_exp_open_exp_wrt_ty_var using solve [auto] : lngen.
Lemma size_exp_open_exp_wrt_exp_var :
forall e1 x1,
size_exp (open_exp_wrt_exp e1 (exp_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_ty_wrt_ty_S_mutual :
(forall n1 T1,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty (S n1) T1).
Proof.
apply_mutual_ind degree_ty_wrt_ty_mutind;
default_simp.
Qed.
(* end hide *)
Lemma degree_ty_wrt_ty_S :
forall n1 T1,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty (S n1) T1.
Proof.
pose proof degree_ty_wrt_ty_S_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_ty_wrt_ty_S : lngen.
(* begin hide *)
Lemma degree_exp_wrt_ty_S_mutual :
(forall n1 e1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty (S n1) e1).
Proof.
apply_mutual_ind degree_exp_wrt_ty_mutind;
default_simp.
Qed.
(* end hide *)
Lemma degree_exp_wrt_ty_S :
forall n1 e1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty (S n1) e1.
Proof.
pose proof degree_exp_wrt_ty_S_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_ty_S : lngen.
(* 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_ty_wrt_ty_O :
forall n1 T1,
degree_ty_wrt_ty O T1 ->
degree_ty_wrt_ty n1 T1.
Proof.
induction n1; default_simp.
Qed.
Hint Resolve degree_ty_wrt_ty_O : lngen.
Lemma degree_exp_wrt_ty_O :
forall n1 e1,
degree_exp_wrt_ty O e1 ->
degree_exp_wrt_ty n1 e1.
Proof.
induction n1; default_simp.
Qed.
Hint Resolve degree_exp_wrt_ty_O : 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_ty_wrt_ty_close_ty_wrt_ty_rec_mutual :
(forall T1 X1 n1,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty (S n1) (close_ty_wrt_ty_rec n1 X1 T1)).
Proof.
apply_mutual_ind ty_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_ty_wrt_ty_close_ty_wrt_ty_rec :
forall T1 X1 n1,
degree_ty_wrt_ty n1 T1 ->
degree_ty_wrt_ty (S n1) (close_ty_wrt_ty_rec n1 X1 T1).
Proof.
pose proof degree_ty_wrt_ty_close_ty_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_ty_wrt_ty_close_ty_wrt_ty_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_ty_close_exp_wrt_ty_rec_mutual :
(forall e1 X1 n1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty (S n1) (close_exp_wrt_ty_rec n1 X1 e1)).
Proof.
apply_mutual_ind exp_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_ty_close_exp_wrt_ty_rec :
forall e1 X1 n1,
degree_exp_wrt_ty n1 e1 ->
degree_exp_wrt_ty (S n1) (close_exp_wrt_ty_rec n1 X1 e1).
Proof.
pose proof degree_exp_wrt_ty_close_exp_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_ty_close_exp_wrt_ty_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_ty_close_exp_wrt_exp_rec_mutual :
(forall e1 x1 n1 n2,
degree_exp_wrt_ty n2 e1 ->
degree_exp_wrt_ty n2 (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_ty_close_exp_wrt_exp_rec :
forall e1 x1 n1 n2,
degree_exp_wrt_ty n2 e1 ->
degree_exp_wrt_ty n2 (close_exp_wrt_exp_rec n1 x1 e1).
Proof.
pose proof degree_exp_wrt_ty_close_exp_wrt_exp_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_ty_close_exp_wrt_exp_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma degree_exp_wrt_exp_close_exp_wrt_ty_rec_mutual :
(forall e1 X1 n1 n2,
degree_exp_wrt_exp n2 e1 ->
degree_exp_wrt_exp n2 (close_exp_wrt_ty_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_ty_rec :
forall e1 X1 n1 n2,
degree_exp_wrt_exp n2 e1 ->
degree_exp_wrt_exp n2 (close_exp_wrt_ty_rec n1 X1 e1).
Proof.
pose proof degree_exp_wrt_exp_close_exp_wrt_ty_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve degree_exp_wrt_exp_close_exp_wrt_ty_rec : lngen.
(* end hide *)
(* 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 *)