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Infrastructure.v
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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 varref_ind' := Induction for varref Sort Prop.
Definition varref_mutind :=
fun H1 H2 H3 =>
varref_ind' H1 H2 H3.
Scheme varref_rec' := Induction for varref Sort Set.
Definition varref_mutrec :=
fun H1 H2 H3 =>
varref_rec' H1 H2 H3.
Scheme typ_ind' := Induction for typ Sort Prop
with dec_ind' := Induction for dec Sort Prop.
Definition typ_dec_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(conj (typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(dec_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Scheme typ_rec' := Induction for typ Sort Set
with dec_rec' := Induction for dec Sort Set.
Definition typ_dec_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(pair (typ_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(dec_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Scheme def_ind' := Induction for def Sort Prop
with defs_ind' := Induction for defs Sort Prop
with val_ind' := Induction for val Sort Prop
with trm_ind' := Induction for trm Sort Prop.
Definition def_defs_val_trm_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(conj (def_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (defs_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (val_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(trm_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))))).
Scheme def_rec' := Induction for def Sort Set
with defs_rec' := Induction for defs Sort Set
with val_rec' := Induction for val Sort Set
with trm_rec' := Induction for trm Sort Set.
Definition def_defs_val_trm_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(pair ((pair ((pair (def_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(defs_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))
(val_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))
(trm_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)).
(* *********************************************************************** *)
(** * Close *)
Fixpoint close_varref_wrt_varref_rec (n1 : nat) (x1 : termvar) (v1 : varref) {struct v1} : varref :=
match v1 with
| var_termvar_f x2 => if (x1 == x2) then (var_termvar_b n1) else (var_termvar_f x2)
| var_termvar_b n2 => if (lt_ge_dec n2 n1) then (var_termvar_b n2) else (var_termvar_b (S n2))
end.
Definition close_varref_wrt_varref x1 v1 := close_varref_wrt_varref_rec 0 x1 v1.
Fixpoint close_typ_wrt_varref_rec (n1 : nat) (x1 : termvar) (T1 : typ) {struct T1} : typ :=
match T1 with
| typ_all T2 T3 => typ_all (close_typ_wrt_varref_rec n1 x1 T2) (close_typ_wrt_varref_rec (S n1) x1 T3)
| typ_bnd T2 => typ_bnd (close_typ_wrt_varref_rec (S n1) x1 T2)
| typ_dec dec1 => typ_dec (close_dec_wrt_varref_rec n1 x1 dec1)
| typ_sel v1 A1 => typ_sel (close_varref_wrt_varref_rec n1 x1 v1) A1
| typ_and T2 T3 => typ_and (close_typ_wrt_varref_rec n1 x1 T2) (close_typ_wrt_varref_rec n1 x1 T3)
| typ_top => typ_top
| typ_bot => typ_bot
end
with close_dec_wrt_varref_rec (n1 : nat) (x1 : termvar) (dec1 : dec) {struct dec1} : dec :=
match dec1 with
| dec_trm a1 T1 => dec_trm a1 (close_typ_wrt_varref_rec n1 x1 T1)
| dec_typ A1 T1 T2 => dec_typ A1 (close_typ_wrt_varref_rec n1 x1 T1) (close_typ_wrt_varref_rec n1 x1 T2)
end.
Definition close_typ_wrt_varref x1 T1 := close_typ_wrt_varref_rec 0 x1 T1.
Definition close_dec_wrt_varref x1 dec1 := close_dec_wrt_varref_rec 0 x1 dec1.
Fixpoint close_def_wrt_varref_rec (n1 : nat) (x1 : termvar) (d1 : def) {struct d1} : def :=
match d1 with
| def_trm a1 t1 => def_trm a1 (close_trm_wrt_varref_rec n1 x1 t1)
| def_typ A1 T1 => def_typ A1 (close_typ_wrt_varref_rec n1 x1 T1)
end
with close_defs_wrt_varref_rec (n1 : nat) (x1 : termvar) (defs1 : defs) {struct defs1} : defs :=
match defs1 with
| defs_nil => defs_nil
| defs_cons d1 defs2 => defs_cons (close_def_wrt_varref_rec n1 x1 d1) (close_defs_wrt_varref_rec n1 x1 defs2)
end
with close_val_wrt_varref_rec (n1 : nat) (x1 : termvar) (val1 : val) {struct val1} : val :=
match val1 with
| val_new T1 defs1 => val_new (close_typ_wrt_varref_rec n1 x1 T1) (close_defs_wrt_varref_rec (S n1) x1 defs1)
| val_lambda T1 t1 => val_lambda (close_typ_wrt_varref_rec n1 x1 T1) (close_trm_wrt_varref_rec (S n1) x1 t1)
end
with close_trm_wrt_varref_rec (n1 : nat) (x1 : termvar) (t1 : trm) {struct t1} : trm :=
match t1 with
| trm_var v1 => trm_var (close_varref_wrt_varref_rec n1 x1 v1)
| trm_val val1 => trm_val (close_val_wrt_varref_rec n1 x1 val1)
| trm_sel v1 a1 => trm_sel (close_varref_wrt_varref_rec n1 x1 v1) a1
| trm_app v1 v2 => trm_app (close_varref_wrt_varref_rec n1 x1 v1) (close_varref_wrt_varref_rec n1 x1 v2)
| trm_let t2 t3 => trm_let (close_trm_wrt_varref_rec n1 x1 t2) (close_trm_wrt_varref_rec (S n1) x1 t3)
end.
Definition close_def_wrt_varref x1 d1 := close_def_wrt_varref_rec 0 x1 d1.
Definition close_defs_wrt_varref x1 defs1 := close_defs_wrt_varref_rec 0 x1 defs1.
Definition close_val_wrt_varref x1 val1 := close_val_wrt_varref_rec 0 x1 val1.
Definition close_trm_wrt_varref x1 t1 := close_trm_wrt_varref_rec 0 x1 t1.
(* *********************************************************************** *)
(** * Size *)
Fixpoint size_varref (v1 : varref) {struct v1} : nat :=
match v1 with
| var_termvar_f x1 => 1
| var_termvar_b n1 => 1
end.
Fixpoint size_typ (T1 : typ) {struct T1} : nat :=
match T1 with
| typ_all T2 T3 => 1 + (size_typ T2) + (size_typ T3)
| typ_bnd T2 => 1 + (size_typ T2)
| typ_dec dec1 => 1 + (size_dec dec1)
| typ_sel v1 A1 => 1 + (size_varref v1)
| typ_and T2 T3 => 1 + (size_typ T2) + (size_typ T3)
| typ_top => 1
| typ_bot => 1
end
with size_dec (dec1 : dec) {struct dec1} : nat :=
match dec1 with
| dec_trm a1 T1 => 1 + (size_typ T1)
| dec_typ A1 T1 T2 => 1 + (size_typ T1) + (size_typ T2)
end.
Fixpoint size_def (d1 : def) {struct d1} : nat :=
match d1 with
| def_trm a1 t1 => 1 + (size_trm t1)
| def_typ A1 T1 => 1 + (size_typ T1)
end
with size_defs (defs1 : defs) {struct defs1} : nat :=
match defs1 with
| defs_nil => 1
| defs_cons d1 defs2 => 1 + (size_def d1) + (size_defs defs2)
end
with size_val (val1 : val) {struct val1} : nat :=
match val1 with
| val_new T1 defs1 => 1 + (size_typ T1) + (size_defs defs1)
| val_lambda T1 t1 => 1 + (size_typ T1) + (size_trm t1)
end
with size_trm (t1 : trm) {struct t1} : nat :=
match t1 with
| trm_var v1 => 1 + (size_varref v1)
| trm_val val1 => 1 + (size_val val1)
| trm_sel v1 a1 => 1 + (size_varref v1)
| trm_app v1 v2 => 1 + (size_varref v1) + (size_varref v2)
| trm_let t2 t3 => 1 + (size_trm t2) + (size_trm t3)
end.
(* *********************************************************************** *)
(** * Degree *)
(** These define only an upper bound, not a strict upper bound. *)
Inductive degree_varref_wrt_varref : nat -> varref -> Prop :=
| degree_wrt_varref_var_termvar_f : forall n1 x1,
degree_varref_wrt_varref n1 (var_termvar_f x1)
| degree_wrt_varref_var_termvar_b : forall n1 n2,
lt n2 n1 ->
degree_varref_wrt_varref n1 (var_termvar_b n2).
Scheme degree_varref_wrt_varref_ind' := Induction for degree_varref_wrt_varref Sort Prop.
Definition degree_varref_wrt_varref_mutind :=
fun H1 H2 H3 =>
degree_varref_wrt_varref_ind' H1 H2 H3.
Hint Constructors degree_varref_wrt_varref : core lngen.
Inductive degree_typ_wrt_varref : nat -> typ -> Prop :=
| degree_wrt_varref_typ_all : forall n1 T1 T2,
degree_typ_wrt_varref n1 T1 ->
degree_typ_wrt_varref (S n1) T2 ->
degree_typ_wrt_varref n1 (typ_all T1 T2)
| degree_wrt_varref_typ_bnd : forall n1 T1,
degree_typ_wrt_varref (S n1) T1 ->
degree_typ_wrt_varref n1 (typ_bnd T1)
| degree_wrt_varref_typ_dec : forall n1 dec1,
degree_dec_wrt_varref n1 dec1 ->
degree_typ_wrt_varref n1 (typ_dec dec1)
| degree_wrt_varref_typ_sel : forall n1 v1 A1,
degree_varref_wrt_varref n1 v1 ->
degree_typ_wrt_varref n1 (typ_sel v1 A1)
| degree_wrt_varref_typ_and : forall n1 T1 T2,
degree_typ_wrt_varref n1 T1 ->
degree_typ_wrt_varref n1 T2 ->
degree_typ_wrt_varref n1 (typ_and T1 T2)
| degree_wrt_varref_typ_top : forall n1,
degree_typ_wrt_varref n1 (typ_top)
| degree_wrt_varref_typ_bot : forall n1,
degree_typ_wrt_varref n1 (typ_bot)
with degree_dec_wrt_varref : nat -> dec -> Prop :=
| degree_wrt_varref_dec_trm : forall n1 a1 T1,
degree_typ_wrt_varref n1 T1 ->
degree_dec_wrt_varref n1 (dec_trm a1 T1)
| degree_wrt_varref_dec_typ : forall n1 A1 T1 T2,
degree_typ_wrt_varref n1 T1 ->
degree_typ_wrt_varref n1 T2 ->
degree_dec_wrt_varref n1 (dec_typ A1 T1 T2).
Scheme degree_typ_wrt_varref_ind' := Induction for degree_typ_wrt_varref Sort Prop
with degree_dec_wrt_varref_ind' := Induction for degree_dec_wrt_varref Sort Prop.
Definition degree_typ_wrt_varref_degree_dec_wrt_varref_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(conj (degree_typ_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(degree_dec_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Hint Constructors degree_typ_wrt_varref : core lngen.
Hint Constructors degree_dec_wrt_varref : core lngen.
Inductive degree_def_wrt_varref : nat -> def -> Prop :=
| degree_wrt_varref_def_trm : forall n1 a1 t1,
degree_trm_wrt_varref n1 t1 ->
degree_def_wrt_varref n1 (def_trm a1 t1)
| degree_wrt_varref_def_typ : forall n1 A1 T1,
degree_typ_wrt_varref n1 T1 ->
degree_def_wrt_varref n1 (def_typ A1 T1)
with degree_defs_wrt_varref : nat -> defs -> Prop :=
| degree_wrt_varref_defs_nil : forall n1,
degree_defs_wrt_varref n1 (defs_nil)
| degree_wrt_varref_defs_cons : forall n1 d1 defs1,
degree_def_wrt_varref n1 d1 ->
degree_defs_wrt_varref n1 defs1 ->
degree_defs_wrt_varref n1 (defs_cons d1 defs1)
with degree_val_wrt_varref : nat -> val -> Prop :=
| degree_wrt_varref_val_new : forall n1 T1 defs1,
degree_typ_wrt_varref n1 T1 ->
degree_defs_wrt_varref (S n1) defs1 ->
degree_val_wrt_varref n1 (val_new T1 defs1)
| degree_wrt_varref_val_lambda : forall n1 T1 t1,
degree_typ_wrt_varref n1 T1 ->
degree_trm_wrt_varref (S n1) t1 ->
degree_val_wrt_varref n1 (val_lambda T1 t1)
with degree_trm_wrt_varref : nat -> trm -> Prop :=
| degree_wrt_varref_trm_var : forall n1 v1,
degree_varref_wrt_varref n1 v1 ->
degree_trm_wrt_varref n1 (trm_var v1)
| degree_wrt_varref_trm_val : forall n1 val1,
degree_val_wrt_varref n1 val1 ->
degree_trm_wrt_varref n1 (trm_val val1)
| degree_wrt_varref_trm_sel : forall n1 v1 a1,
degree_varref_wrt_varref n1 v1 ->
degree_trm_wrt_varref n1 (trm_sel v1 a1)
| degree_wrt_varref_trm_app : forall n1 v1 v2,
degree_varref_wrt_varref n1 v1 ->
degree_varref_wrt_varref n1 v2 ->
degree_trm_wrt_varref n1 (trm_app v1 v2)
| degree_wrt_varref_trm_let : forall n1 t1 t2,
degree_trm_wrt_varref n1 t1 ->
degree_trm_wrt_varref (S n1) t2 ->
degree_trm_wrt_varref n1 (trm_let t1 t2).
Scheme degree_def_wrt_varref_ind' := Induction for degree_def_wrt_varref Sort Prop
with degree_defs_wrt_varref_ind' := Induction for degree_defs_wrt_varref Sort Prop
with degree_val_wrt_varref_ind' := Induction for degree_val_wrt_varref Sort Prop
with degree_trm_wrt_varref_ind' := Induction for degree_trm_wrt_varref Sort Prop.
Definition degree_def_wrt_varref_degree_defs_wrt_varref_degree_val_wrt_varref_degree_trm_wrt_varref_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(conj (degree_def_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (degree_defs_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (degree_val_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(degree_trm_wrt_varref_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))))).
Hint Constructors degree_def_wrt_varref : core lngen.
Hint Constructors degree_defs_wrt_varref : core lngen.
Hint Constructors degree_val_wrt_varref : core lngen.
Hint Constructors degree_trm_wrt_varref : core lngen.
(* *********************************************************************** *)
(** * Local closure (version in [Set], induction principles) *)
Inductive lc_set_varref : varref -> Set :=
| lc_set_var_termvar_f : forall x1,
lc_set_varref (var_termvar_f x1).
Scheme lc_varref_ind' := Induction for lc_varref Sort Prop.
Definition lc_varref_mutind :=
fun H1 H2 =>
lc_varref_ind' H1 H2.
Scheme lc_set_varref_ind' := Induction for lc_set_varref Sort Prop.
Definition lc_set_varref_mutind :=
fun H1 H2 =>
lc_set_varref_ind' H1 H2.
Scheme lc_set_varref_rec' := Induction for lc_set_varref Sort Set.
Definition lc_set_varref_mutrec :=
fun H1 H2 =>
lc_set_varref_rec' H1 H2.
Hint Constructors lc_varref : core lngen.
Hint Constructors lc_set_varref : core lngen.
Inductive lc_set_typ : typ -> Set :=
| lc_set_typ_all : forall T1 T2,
lc_set_typ T1 ->
(forall x1 : termvar, lc_set_typ (open_typ_wrt_varref T2 (var_termvar_f x1))) ->
lc_set_typ (typ_all T1 T2)
| lc_set_typ_bnd : forall T1,
(forall x1 : termvar, lc_set_typ (open_typ_wrt_varref T1 (var_termvar_f x1))) ->
lc_set_typ (typ_bnd T1)
| lc_set_typ_dec : forall dec1,
lc_set_dec dec1 ->
lc_set_typ (typ_dec dec1)
| lc_set_typ_sel : forall v1 A1,
lc_set_varref v1 ->
lc_set_typ (typ_sel v1 A1)
| lc_set_typ_and : forall T1 T2,
lc_set_typ T1 ->
lc_set_typ T2 ->
lc_set_typ (typ_and T1 T2)
| lc_set_typ_top :
lc_set_typ (typ_top)
| lc_set_typ_bot :
lc_set_typ (typ_bot)
with lc_set_dec : dec -> Set :=
| lc_set_dec_trm : forall a1 T1,
lc_set_typ T1 ->
lc_set_dec (dec_trm a1 T1)
| lc_set_dec_typ : forall A1 T1 T2,
lc_set_typ T1 ->
lc_set_typ T2 ->
lc_set_dec (dec_typ A1 T1 T2).
Scheme lc_typ_ind' := Induction for lc_typ Sort Prop
with lc_dec_ind' := Induction for lc_dec Sort Prop.
Definition lc_typ_lc_dec_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(conj (lc_typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(lc_dec_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Scheme lc_set_typ_ind' := Induction for lc_set_typ Sort Prop
with lc_set_dec_ind' := Induction for lc_set_dec Sort Prop.
Definition lc_set_typ_lc_set_dec_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(conj (lc_set_typ_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(lc_set_dec_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Scheme lc_set_typ_rec' := Induction for lc_set_typ Sort Set
with lc_set_dec_rec' := Induction for lc_set_dec Sort Set.
Definition lc_set_typ_lc_set_dec_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 =>
(pair (lc_set_typ_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)
(lc_set_dec_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11)).
Hint Constructors lc_typ : core lngen.
Hint Constructors lc_dec : core lngen.
Hint Constructors lc_set_typ : core lngen.
Hint Constructors lc_set_dec : core lngen.
Inductive lc_set_def : def -> Set :=
| lc_set_def_trm : forall a1 t1,
lc_set_trm t1 ->
lc_set_def (def_trm a1 t1)
| lc_set_def_typ : forall A1 T1,
lc_set_typ T1 ->
lc_set_def (def_typ A1 T1)
with lc_set_defs : defs -> Set :=
| lc_set_defs_nil :
lc_set_defs (defs_nil)
| lc_set_defs_cons : forall d1 defs1,
lc_set_def d1 ->
lc_set_defs defs1 ->
lc_set_defs (defs_cons d1 defs1)
with lc_set_val : val -> Set :=
| lc_set_val_new : forall T1 defs1,
lc_set_typ T1 ->
(forall x1 : termvar, lc_set_defs (open_defs_wrt_varref defs1 (var_termvar_f x1))) ->
lc_set_val (val_new T1 defs1)
| lc_set_val_lambda : forall T1 t1,
lc_set_typ T1 ->
(forall x1 : termvar, lc_set_trm (open_trm_wrt_varref t1 (var_termvar_f x1))) ->
lc_set_val (val_lambda T1 t1)
with lc_set_trm : trm -> Set :=
| lc_set_trm_var : forall v1,
lc_set_varref v1 ->
lc_set_trm (trm_var v1)
| lc_set_trm_val : forall val1,
lc_set_val val1 ->
lc_set_trm (trm_val val1)
| lc_set_trm_sel : forall v1 a1,
lc_set_varref v1 ->
lc_set_trm (trm_sel v1 a1)
| lc_set_trm_app : forall v1 v2,
lc_set_varref v1 ->
lc_set_varref v2 ->
lc_set_trm (trm_app v1 v2)
| lc_set_trm_let : forall t1 t2,
lc_set_trm t1 ->
(forall x1 : termvar, lc_set_trm (open_trm_wrt_varref t2 (var_termvar_f x1))) ->
lc_set_trm (trm_let t1 t2).
Scheme lc_def_ind' := Induction for lc_def Sort Prop
with lc_defs_ind' := Induction for lc_defs Sort Prop
with lc_val_ind' := Induction for lc_val Sort Prop
with lc_trm_ind' := Induction for lc_trm Sort Prop.
Definition lc_def_lc_defs_lc_val_lc_trm_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(conj (lc_def_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (lc_defs_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (lc_val_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(lc_trm_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))))).
Scheme lc_set_def_ind' := Induction for lc_set_def Sort Prop
with lc_set_defs_ind' := Induction for lc_set_defs Sort Prop
with lc_set_val_ind' := Induction for lc_set_val Sort Prop
with lc_set_trm_ind' := Induction for lc_set_trm Sort Prop.
Definition lc_set_def_lc_set_defs_lc_set_val_lc_set_trm_mutind :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(conj (lc_set_def_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (lc_set_defs_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
((conj (lc_set_val_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(lc_set_trm_ind' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))))).
Scheme lc_set_def_rec' := Induction for lc_set_def Sort Set
with lc_set_defs_rec' := Induction for lc_set_defs Sort Set
with lc_set_val_rec' := Induction for lc_set_val Sort Set
with lc_set_trm_rec' := Induction for lc_set_trm Sort Set.
Definition lc_set_def_lc_set_defs_lc_set_val_lc_set_trm_mutrec :=
fun H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 =>
(pair ((pair ((pair (lc_set_def_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)
(lc_set_defs_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))
(lc_set_val_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)))
(lc_set_trm_rec' H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15)).
Hint Constructors lc_def : core lngen.
Hint Constructors lc_defs : core lngen.
Hint Constructors lc_val : core lngen.
Hint Constructors lc_trm : core lngen.
Hint Constructors lc_set_def : core lngen.
Hint Constructors lc_set_defs : core lngen.
Hint Constructors lc_set_val : core lngen.
Hint Constructors lc_set_trm : core lngen.
(* *********************************************************************** *)
(** * Body *)
Definition body_varref_wrt_varref v1 := forall x1, lc_varref (open_varref_wrt_varref v1 (var_termvar_f x1)).
Hint Unfold body_varref_wrt_varref.
Definition body_typ_wrt_varref T1 := forall x1, lc_typ (open_typ_wrt_varref T1 (var_termvar_f x1)).
Definition body_dec_wrt_varref dec1 := forall x1, lc_dec (open_dec_wrt_varref dec1 (var_termvar_f x1)).
Hint Unfold body_typ_wrt_varref.
Hint Unfold body_dec_wrt_varref.
Definition body_def_wrt_varref d1 := forall x1, lc_def (open_def_wrt_varref d1 (var_termvar_f x1)).
Definition body_defs_wrt_varref defs1 := forall x1, lc_defs (open_defs_wrt_varref defs1 (var_termvar_f x1)).
Definition body_val_wrt_varref val1 := forall x1, lc_val (open_val_wrt_varref val1 (var_termvar_f x1)).
Definition body_trm_wrt_varref t1 := forall x1, lc_trm (open_trm_wrt_varref t1 (var_termvar_f x1)).
Hint Unfold body_def_wrt_varref.
Hint Unfold body_defs_wrt_varref.
Hint Unfold body_val_wrt_varref.
Hint Unfold body_trm_wrt_varref.
(* *********************************************************************** *)
(** * 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_varref_min_mutual :
(forall v1, 1 <= size_varref v1).
Proof.
apply_mutual_ind varref_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_varref_min :
forall v1, 1 <= size_varref v1.
Proof.
pose proof size_varref_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_varref_min : lngen.
(* begin hide *)
Lemma size_typ_min_size_dec_min_mutual :
(forall T1, 1 <= size_typ T1) /\
(forall dec1, 1 <= size_dec dec1).
Proof.
apply_mutual_ind typ_dec_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_typ_min :
forall T1, 1 <= size_typ T1.
Proof.
pose proof size_typ_min_size_dec_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_min : lngen.
Lemma size_dec_min :
forall dec1, 1 <= size_dec dec1.
Proof.
pose proof size_typ_min_size_dec_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_dec_min : lngen.
(* begin hide *)
Lemma size_def_min_size_defs_min_size_val_min_size_trm_min_mutual :
(forall d1, 1 <= size_def d1) /\
(forall defs1, 1 <= size_defs defs1) /\
(forall val1, 1 <= size_val val1) /\
(forall t1, 1 <= size_trm t1).
Proof.
apply_mutual_ind def_defs_val_trm_mutind;
default_simp.
Qed.
(* end hide *)
Lemma size_def_min :
forall d1, 1 <= size_def d1.
Proof.
pose proof size_def_min_size_defs_min_size_val_min_size_trm_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_def_min : lngen.
Lemma size_defs_min :
forall defs1, 1 <= size_defs defs1.
Proof.
pose proof size_def_min_size_defs_min_size_val_min_size_trm_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_defs_min : lngen.
Lemma size_val_min :
forall val1, 1 <= size_val val1.
Proof.
pose proof size_def_min_size_defs_min_size_val_min_size_trm_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_val_min : lngen.
Lemma size_trm_min :
forall t1, 1 <= size_trm t1.
Proof.
pose proof size_def_min_size_defs_min_size_val_min_size_trm_min_mutual as H; intuition eauto.
Qed.
Hint Resolve size_trm_min : lngen.
(* begin hide *)
Lemma size_varref_close_varref_wrt_varref_rec_mutual :
(forall v1 x1 n1,
size_varref (close_varref_wrt_varref_rec n1 x1 v1) = size_varref v1).
Proof.
apply_mutual_ind varref_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_varref_close_varref_wrt_varref_rec :
forall v1 x1 n1,
size_varref (close_varref_wrt_varref_rec n1 x1 v1) = size_varref v1.
Proof.
pose proof size_varref_close_varref_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_varref_close_varref_wrt_varref_rec : lngen.
Hint Rewrite size_varref_close_varref_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_typ_close_typ_wrt_varref_rec_size_dec_close_dec_wrt_varref_rec_mutual :
(forall T1 x1 n1,
size_typ (close_typ_wrt_varref_rec n1 x1 T1) = size_typ T1) /\
(forall dec1 x1 n1,
size_dec (close_dec_wrt_varref_rec n1 x1 dec1) = size_dec dec1).
Proof.
apply_mutual_ind typ_dec_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_typ_close_typ_wrt_varref_rec :
forall T1 x1 n1,
size_typ (close_typ_wrt_varref_rec n1 x1 T1) = size_typ T1.
Proof.
pose proof size_typ_close_typ_wrt_varref_rec_size_dec_close_dec_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_close_typ_wrt_varref_rec : lngen.
Hint Rewrite size_typ_close_typ_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_dec_close_dec_wrt_varref_rec :
forall dec1 x1 n1,
size_dec (close_dec_wrt_varref_rec n1 x1 dec1) = size_dec dec1.
Proof.
pose proof size_typ_close_typ_wrt_varref_rec_size_dec_close_dec_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_dec_close_dec_wrt_varref_rec : lngen.
Hint Rewrite size_dec_close_dec_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_def_close_def_wrt_varref_rec_size_defs_close_defs_wrt_varref_rec_size_val_close_val_wrt_varref_rec_size_trm_close_trm_wrt_varref_rec_mutual :
(forall d1 x1 n1,
size_def (close_def_wrt_varref_rec n1 x1 d1) = size_def d1) /\
(forall defs1 x1 n1,
size_defs (close_defs_wrt_varref_rec n1 x1 defs1) = size_defs defs1) /\
(forall val1 x1 n1,
size_val (close_val_wrt_varref_rec n1 x1 val1) = size_val val1) /\
(forall t1 x1 n1,
size_trm (close_trm_wrt_varref_rec n1 x1 t1) = size_trm t1).
Proof.
apply_mutual_ind def_defs_val_trm_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_def_close_def_wrt_varref_rec :
forall d1 x1 n1,
size_def (close_def_wrt_varref_rec n1 x1 d1) = size_def d1.
Proof.
pose proof size_def_close_def_wrt_varref_rec_size_defs_close_defs_wrt_varref_rec_size_val_close_val_wrt_varref_rec_size_trm_close_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_def_close_def_wrt_varref_rec : lngen.
Hint Rewrite size_def_close_def_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_defs_close_defs_wrt_varref_rec :
forall defs1 x1 n1,
size_defs (close_defs_wrt_varref_rec n1 x1 defs1) = size_defs defs1.
Proof.
pose proof size_def_close_def_wrt_varref_rec_size_defs_close_defs_wrt_varref_rec_size_val_close_val_wrt_varref_rec_size_trm_close_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_defs_close_defs_wrt_varref_rec : lngen.
Hint Rewrite size_defs_close_defs_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_val_close_val_wrt_varref_rec :
forall val1 x1 n1,
size_val (close_val_wrt_varref_rec n1 x1 val1) = size_val val1.
Proof.
pose proof size_def_close_def_wrt_varref_rec_size_defs_close_defs_wrt_varref_rec_size_val_close_val_wrt_varref_rec_size_trm_close_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_val_close_val_wrt_varref_rec : lngen.
Hint Rewrite size_val_close_val_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
(* begin hide *)
Lemma size_trm_close_trm_wrt_varref_rec :
forall t1 x1 n1,
size_trm (close_trm_wrt_varref_rec n1 x1 t1) = size_trm t1.
Proof.
pose proof size_def_close_def_wrt_varref_rec_size_defs_close_defs_wrt_varref_rec_size_val_close_val_wrt_varref_rec_size_trm_close_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_trm_close_trm_wrt_varref_rec : lngen.
Hint Rewrite size_trm_close_trm_wrt_varref_rec using solve [auto] : lngen.
(* end hide *)
Lemma size_varref_close_varref_wrt_varref :
forall v1 x1,
size_varref (close_varref_wrt_varref x1 v1) = size_varref v1.
Proof.
unfold close_varref_wrt_varref; default_simp.
Qed.
Hint Resolve size_varref_close_varref_wrt_varref : lngen.
Hint Rewrite size_varref_close_varref_wrt_varref using solve [auto] : lngen.
Lemma size_typ_close_typ_wrt_varref :
forall T1 x1,
size_typ (close_typ_wrt_varref x1 T1) = size_typ T1.
Proof.
unfold close_typ_wrt_varref; default_simp.
Qed.
Hint Resolve size_typ_close_typ_wrt_varref : lngen.
Hint Rewrite size_typ_close_typ_wrt_varref using solve [auto] : lngen.
Lemma size_dec_close_dec_wrt_varref :
forall dec1 x1,
size_dec (close_dec_wrt_varref x1 dec1) = size_dec dec1.
Proof.
unfold close_dec_wrt_varref; default_simp.
Qed.
Hint Resolve size_dec_close_dec_wrt_varref : lngen.
Hint Rewrite size_dec_close_dec_wrt_varref using solve [auto] : lngen.
Lemma size_def_close_def_wrt_varref :
forall d1 x1,
size_def (close_def_wrt_varref x1 d1) = size_def d1.
Proof.
unfold close_def_wrt_varref; default_simp.
Qed.
Hint Resolve size_def_close_def_wrt_varref : lngen.
Hint Rewrite size_def_close_def_wrt_varref using solve [auto] : lngen.
Lemma size_defs_close_defs_wrt_varref :
forall defs1 x1,
size_defs (close_defs_wrt_varref x1 defs1) = size_defs defs1.
Proof.
unfold close_defs_wrt_varref; default_simp.
Qed.
Hint Resolve size_defs_close_defs_wrt_varref : lngen.
Hint Rewrite size_defs_close_defs_wrt_varref using solve [auto] : lngen.
Lemma size_val_close_val_wrt_varref :
forall val1 x1,
size_val (close_val_wrt_varref x1 val1) = size_val val1.
Proof.
unfold close_val_wrt_varref; default_simp.
Qed.
Hint Resolve size_val_close_val_wrt_varref : lngen.
Hint Rewrite size_val_close_val_wrt_varref using solve [auto] : lngen.
Lemma size_trm_close_trm_wrt_varref :
forall t1 x1,
size_trm (close_trm_wrt_varref x1 t1) = size_trm t1.
Proof.
unfold close_trm_wrt_varref; default_simp.
Qed.
Hint Resolve size_trm_close_trm_wrt_varref : lngen.
Hint Rewrite size_trm_close_trm_wrt_varref using solve [auto] : lngen.
(* begin hide *)
Lemma size_varref_open_varref_wrt_varref_rec_mutual :
(forall v1 v2 n1,
size_varref v1 <= size_varref (open_varref_wrt_varref_rec n1 v2 v1)).
Proof.
apply_mutual_ind varref_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_varref_open_varref_wrt_varref_rec :
forall v1 v2 n1,
size_varref v1 <= size_varref (open_varref_wrt_varref_rec n1 v2 v1).
Proof.
pose proof size_varref_open_varref_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_varref_open_varref_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_typ_open_typ_wrt_varref_rec_size_dec_open_dec_wrt_varref_rec_mutual :
(forall T1 v1 n1,
size_typ T1 <= size_typ (open_typ_wrt_varref_rec n1 v1 T1)) /\
(forall dec1 v1 n1,
size_dec dec1 <= size_dec (open_dec_wrt_varref_rec n1 v1 dec1)).
Proof.
apply_mutual_ind typ_dec_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_typ_open_typ_wrt_varref_rec :
forall T1 v1 n1,
size_typ T1 <= size_typ (open_typ_wrt_varref_rec n1 v1 T1).
Proof.
pose proof size_typ_open_typ_wrt_varref_rec_size_dec_open_dec_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_typ_open_typ_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_dec_open_dec_wrt_varref_rec :
forall dec1 v1 n1,
size_dec dec1 <= size_dec (open_dec_wrt_varref_rec n1 v1 dec1).
Proof.
pose proof size_typ_open_typ_wrt_varref_rec_size_dec_open_dec_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_dec_open_dec_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_def_open_def_wrt_varref_rec_size_defs_open_defs_wrt_varref_rec_size_val_open_val_wrt_varref_rec_size_trm_open_trm_wrt_varref_rec_mutual :
(forall d1 v1 n1,
size_def d1 <= size_def (open_def_wrt_varref_rec n1 v1 d1)) /\
(forall defs1 v1 n1,
size_defs defs1 <= size_defs (open_defs_wrt_varref_rec n1 v1 defs1)) /\
(forall val1 v1 n1,
size_val val1 <= size_val (open_val_wrt_varref_rec n1 v1 val1)) /\
(forall t1 v1 n1,
size_trm t1 <= size_trm (open_trm_wrt_varref_rec n1 v1 t1)).
Proof.
apply_mutual_ind def_defs_val_trm_mutind;
default_simp.
Qed.
(* end hide *)
(* begin hide *)
Lemma size_def_open_def_wrt_varref_rec :
forall d1 v1 n1,
size_def d1 <= size_def (open_def_wrt_varref_rec n1 v1 d1).
Proof.
pose proof size_def_open_def_wrt_varref_rec_size_defs_open_defs_wrt_varref_rec_size_val_open_val_wrt_varref_rec_size_trm_open_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_def_open_def_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_defs_open_defs_wrt_varref_rec :
forall defs1 v1 n1,
size_defs defs1 <= size_defs (open_defs_wrt_varref_rec n1 v1 defs1).
Proof.
pose proof size_def_open_def_wrt_varref_rec_size_defs_open_defs_wrt_varref_rec_size_val_open_val_wrt_varref_rec_size_trm_open_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_defs_open_defs_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_val_open_val_wrt_varref_rec :
forall val1 v1 n1,
size_val val1 <= size_val (open_val_wrt_varref_rec n1 v1 val1).
Proof.
pose proof size_def_open_def_wrt_varref_rec_size_defs_open_defs_wrt_varref_rec_size_val_open_val_wrt_varref_rec_size_trm_open_trm_wrt_varref_rec_mutual as H; intuition eauto.
Qed.
Hint Resolve size_val_open_val_wrt_varref_rec : lngen.
(* end hide *)
(* begin hide *)
Lemma size_trm_open_trm_wrt_varref_rec :
forall t1 v1 n1,
size_trm t1 <= size_trm (open_trm_wrt_varref_rec n1 v1 t1).