Set Implicit Arguments.
Require Import List Metatheory STLC_Data_Definitions.
Fixpoint fv (t : trm) {struct t} : vars :=
match t with
| trm_bvar i j => {}
| trm_fvar x => {{x}}
| trm_abs t1 => (fv t1)
| trm_fix t1 => (fv t1)
| trm_app t1 t2 => (fv t1) \u (fv t2)
| trm_match t1 p1 e t2 => (fv t1) \u (fv e) \u (fv t2)
| trm_unit => {}
| trm_pair t1 t2 => (fv t1) \u (fv t2)
| trm_inj1 t1 => (fv t1)
| trm_inj2 t1 => (fv t1)
end.
Definition fv_list :=
List.fold_right (fun t acc => fv t \u acc) {}.
Lemma trm_def_fresh : fv trm_def = {}.
Proof.
auto.
Qed.
Fixpoint subst (z : var) (u : trm) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i j => trm_bvar i j
| trm_fvar x => if x == z then u else (trm_fvar x)
| trm_abs t1 => trm_abs (subst z u t1)
| trm_fix t1 => trm_fix (subst z u t1)
| trm_match t1 p1 e t2 => trm_match (subst z u t1) p1
(subst z u e)
(subst z u t2)
| trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2)
| trm_unit => trm_unit
| trm_pair t1 t2 => trm_pair (subst z u t1) (subst z u t2)
| trm_inj1 t1 => trm_inj1 (subst z u t1)
| trm_inj2 t1 => trm_inj2 (subst z u t1)
end.
Notation "[ z ~> u ] t" := (subst z u t) (at level 68).
Fixpoint substs (zs : list var) (us : list trm) (t : trm)
{struct zs} : trm :=
match zs, us with
| z::zs', u::us' => substs zs' us' ([z ~> u]t)
| _, _ => t
end.
Definition terms := list_for_n term.
Inductive typings (E : env) : list trm -> list typ -> Prop :=
| typings_nil : typings E nil nil
| typings_cons : forall ts Us t U,
typings E ts Us ->
typing E t U ->
typings E (t::ts) (U::Us).
Ltac gather_vars :=
let A := gather_vars_with (fun x : vars => x) in
let B := gather_vars_with (fun x : var => {{ x }}) in
let C := gather_vars_with (fun x : env => dom x) in
let D := gather_vars_with (fun x : trm => fv x) in
let E := gather_vars_with (fun x : list trm => fv_list x) in
constr:(A \u B \u C \u D \u E).
Ltac pick_fresh Y :=
let L := gather_vars in (pick_fresh_gen L Y).
Ltac pick_freshes n Y :=
let L := gather_vars in (pick_freshes_gen L n Y).
Tactic Notation "apply_fresh" constr(T) :=
apply_fresh_base_simple T gather_vars.
Tactic Notation "apply_fresh" "*" constr(T) :=
apply_fresh T; auto*.
Hint Constructors term value red typing typings.
Hint Extern 1 (_ \notin fv trm_def) =>
rewrite trm_def_fresh.
Hint Extern 1 (terms _ _) => split; auto.
Hint Extern 1 (fresh (dom (_ & _)) _ _) => simpl_env.
Lemma fv_list_map : forall ts1 ts2,
fv_list (ts1 ++ ts2) = fv_list ts1 \u fv_list ts2.
Proof.
induction ts1; simpl; intros.
rewrite* union_empty_l.
rewrite IHts1. rewrite* union_assoc.
Qed.
Lemma typings_concat : forall E ts1 Us1 ts2 Us2,
typings E ts1 Us1 ->
typings E ts2 Us2 ->
typings E (ts1++ts2) (Us1++Us2).
Proof.
induction ts1; introv Typ1 Typ2; inversions Typ1; simpl*.
Qed.
Lemma open_rec_term_core :forall t j v i u, i <> j ->
{j ~> v}t = {i ~> u}({j ~> v}t) -> t = {i ~> u}t.
Proof.
induction t; introv Neq Equ;
simpl in *; inversion* Equ; f_equal*.
case_nat*. case_nat*.
Qed.
Lemma open_rec_term : forall t u,
term t -> forall k, t = {k ~> u}t.
Proof.
induction 1; intros; simpl; f_equal*.
pick_fresh x. forward~ (H0 x) as K.
apply* (@open_rec_term_core t1 0 (trm_fvars (x::nil))).
pick_fresh x. pick_fresh f. forward~ (H0 x f) as K.
apply* (@open_rec_term_core t1 0 (trm_fvars (x::f::nil))).
pick_freshes (pat_arity p) xs. forward~ (H1 xs) as K.
apply* (@open_rec_term_core e 0 (trm_fvars xs)).
Qed.
Lemma subst_fresh : forall x t u,
x \notin fv t ->
[x ~> u] t = t.
Proof.
intros. induction t; simpls; f_equal*.
case_var*. notin_contradiction.
Qed.
Lemma subst_fresh_list : forall z u ts,
z \notin fv_list ts ->
ts = List.map (subst z u) ts.
Proof.
induction ts; simpl; intros Fr.
auto. f_equal. rewrite~ subst_fresh. auto.
Qed.
Lemma subst_fresh_trm_fvars : forall z u xs,
fresh ({{z}}) (length xs) xs ->
trm_fvars xs = List.map (subst z u) (trm_fvars xs).
Proof.
intros. apply subst_fresh_list.
induction xs; simpls. auto.
destruct H. notin_simpls; auto.
Qed.
Lemma substs_fresh : forall xs us t,
fresh (fv t) (length xs) xs ->
substs xs us t = t.
Proof.
induction xs; simpl; intros us t Fr.
auto. destruct us. auto.
inversions Fr. rewrite* subst_fresh.
Qed.
Lemma subst_open : forall x u t1 t2, term u ->
[x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ (List.map (subst x u) t2).
Proof.
intros. unfold open, opens. simpl. generalize 0.
induction t1; intros; simpl; f_equal*.
case_nat*. unfold trm_nth.
apply list_map_nth. apply* subst_fresh.
case_var*. apply* open_rec_term.
Qed.
Lemma subst_open_vars : forall x ys u t,
fresh {{x}} (length ys) ys ->
term u ->
([x ~> u]t) ^ ys = [x ~> u] (t ^ ys).
Proof.
introv Fr Tu. rewrite* subst_open.
unfold trm_fvars. f_equal.
induction ys; simpls. auto.
destruct Fr. case_var.
notin_contradiction. f_equal*.
Qed.
Lemma substs_intro_ind : forall t xs us vs,
fresh (fv t \u fv_list vs \u fv_list us) (length xs) xs ->
terms (length xs) us ->
terms (length vs) vs ->
t ^^ (vs ++ us) = substs xs us (t ^^ (vs ++ (trm_fvars xs))).
Proof.
induction xs; simpl; introv Fr Tu Tv;
destruct Tu; destruct us; try solve [ contradictions ].
auto.
inversions H0. inversions Fr. clear H0 Fr. simpls.
rewrite list_concat_right.
forward (IHxs us (vs++t0::nil)) as K; clear IHxs.
rewrite* fv_list_map.
auto.
split~. inversions Tv. apply* list_forall_concat.
rewrite K. clear K.
f_equal. rewrite~ subst_open. rewrite~ subst_fresh.
f_equal. rewrite map_app.
simpl. case_var; try solve [ contradictions* ].
rewrite <- list_concat_right.
f_equal. apply~ subst_fresh_list.
f_equal. apply* subst_fresh_trm_fvars.
Qed.
Lemma substs_intro : forall xs t us,
fresh (fv t \u fv_list us) (length xs) xs ->
terms (length xs) us ->
t ^^ us = substs xs us (t ^ xs).
Proof.
intros. apply* (@substs_intro_ind t xs us nil).
Qed.
Lemma subst_term : forall t z u,
term u -> term t -> term ([z ~> u]t).
Proof.
induction 2; simpl*.
case_var*.
apply_fresh term_abs. intros y Fr.
rewrite* subst_open_vars.
apply_fresh term_fix. intros y f Fr.
rewrite* subst_open_vars.
apply_fresh* term_match. intros ys Fr.
rewrite* subst_open_vars.
Qed.
Hint Resolve subst_term.
Lemma substs_terms : forall xs us t,
terms (length xs) us ->
term t ->
term (substs xs us t).
Proof.
induction xs; destruct us; introv TU TT; simpl; auto.
inversions TU. simpls. inversions H. inversions* H0.
Qed.
Lemma term_abs_to_body : forall t1,
term (trm_abs t1) -> bodys 1 t1.
Proof.
intros. unfold bodys. inversions* H.
exists L. intros.
destruct xs; simpl in H0; destruct H0.
destruct xs; simpl in H2; destruct H2. auto.
Qed.
Lemma body_to_term_abs : forall t1,
bodys 1 t1 -> term (trm_abs t1).
Proof.
destruct 1. apply_fresh* term_abs.
Qed.
Lemma term_fix_to_body : forall t1,
term (trm_fix t1) -> bodys 2 t1.
Proof.
intros. unfold bodys. inversions* H.
exists L. intros.
destruct xs; simpl in H0; destruct H0.
destruct xs; simpl in H2; destruct H2.
destruct xs; simpl in H3; destruct H3.
auto.
Qed.
Lemma body_to_term_fix : forall t1,
bodys 2 t1 -> term (trm_fix t1).
Proof.
destruct 1. apply_fresh* term_fix.
Qed.
Lemma term_match_to_body : forall t1 t2 e p,
term (trm_match t1 p e t2) -> bodys (pat_arity p) e.
Proof.
intros. unfold bodys. inversions* H.
Qed.
Lemma body_to_term_match : forall t1 t2 e p,
term t1 -> term t2 -> bodys (pat_arity p) e ->
term (trm_match t1 p e t2).
Proof.
destruct 3. apply_fresh* term_match.
Qed.
Hint Resolve body_to_term_abs term_abs_to_body
body_to_term_fix term_fix_to_body
body_to_term_match.
Hint Extern 1 (bodys (pat_arity ?p) ?e) =>
match goal with H: context [trm_match ?t1 p e ?t2] |- _ =>
apply (@term_match_to_body t1 t2) end.
Lemma open_terms : forall t us,
bodys (length us) t ->
terms (length us) us ->
term (t ^^ us).
Proof.
introv [L K] WT. pick_freshes (length us) xs. poses Fr' Fr.
rewrite (fresh_length _ _ _ Fr) in WT, Fr'.
rewrite* (@substs_intro xs). apply* substs_terms.
Qed.
Hint Resolve open_terms.
Lemma pat_match_terms : forall p t ts,
(pat_match p t) = Some ts -> term t ->
terms (pat_arity p) ts.
Proof.
induction p; simpl; introv EQ TT;
try solve [ inversions EQ; auto ];
destruct t; inversions EQ; inversions TT; auto*.
remember (pat_match p1 t1) as K1. symmetry in HeqK1.
remember (pat_match p2 t2) as K2. symmetry in HeqK2.
destruct K1 as [ts1|]; destruct K2 as [ts2|]; try discriminate.
inversions EQ. unfolds terms. apply* list_for_n_concat.
Qed.
Lemma typing_regular : forall E e T,
typing E e T -> ok E /\ term e.
Proof.
split; induction* H.
pick_fresh x. forward~ (H0 x) as K. inversions* K.
pick_fresh x. pick_fresh f. forward~ (H0 x f) as K.
inversions K. inversions* H3.
Qed.
Lemma value_regular : forall e,
value e -> term e.
Proof.
induction 1; auto*.
Qed.
Lemma red_regular : forall e e',
red e e' -> term e /\ term e'.
Proof.
induction 1; use value_regular.
splits*. forward~ (@pat_match_terms p t1 ts) as K.
rewrite (proj1 K) in H0, K. auto*.
Qed.
Hint Extern 1 (ok ?E) =>
match goal with
| H: typing E _ _ |- _ => apply (proj1 (typing_regular H))
end.
Hint Extern 1 (term ?t) =>
match goal with
| H: typing _ t _ |- _ => apply (proj2 (typing_regular H))
| H: red t _ |- _ => apply (proj1 (red_regular H))
| H: red _ t |- _ => apply (proj2 (red_regular H))
| H: value t |- _ => apply (value_regular H)
end.