Set Implicit Arguments.
Require Import Metatheory CoC_Definitions.
Fixpoint fv (t : trm) {struct t} : vars :=
match t with
| trm_bvar i => {}
| trm_fvar x => {{x}}
| trm_type n => {}
| trm_app t1 t2 => (fv t1) \u (fv t2)
| trm_abs t1 t2 => (fv t1) \u (fv t2)
| trm_prod t1 t2 => (fv t1) \u (fv t2)
end.
Fixpoint subst (z : var) (u : trm) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i => trm_bvar i
| trm_fvar x => if x == z then u else (trm_fvar x)
| trm_type n => trm_type n
| trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2)
| trm_abs t1 t2 => trm_abs (subst z u t1) (subst z u t2)
| trm_prod t1 t2 => trm_prod (subst z u t1) (subst z u t2)
end.
Notation "[ z ~> u ] t" := (subst z u t) (at level 68).
Fixpoint close_var_rec (k : nat) (z : var) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i => trm_bvar i
| trm_fvar x => if x == z then (trm_bvar k) else (trm_fvar x)
| trm_type n => trm_type n
| trm_app t1 t2 => trm_app (close_var_rec k z t1) (close_var_rec k z t2)
| trm_abs t1 t2 => trm_abs (close_var_rec k z t1) (close_var_rec (S k) z t2)
| trm_prod t1 t2 => trm_prod (close_var_rec k z t1) (close_var_rec (S k) z t2)
end.
Definition close_var z t := close_var_rec 0 z t.
Definition contains_terms E :=
forall x U, binds x U E -> term U.
Definition simulated (R1 R2 : relation) :=
forall (t t' : trm), R1 t t' -> R2 t t'.
Infix "simulated_by" := simulated (at level 69).
Definition red_regular (R : relation) :=
forall t t', R t t' -> term t /\ term t'.
Definition red_refl (R : relation) :=
forall t, term t -> R t t.
Definition red_in (R : relation) :=
forall t x u u', term t -> R u u' ->
R ([x ~> u]t) ([x ~> u']t).
Definition red_all (R : relation) :=
forall x t t', R t t' ->
forall u u', R u u' ->
R ([x~>u]t) ([x~>u']t').
Definition red_out (R : relation) :=
forall x u t t', term u -> R t t' ->
R ([x~>u]t) ([x~>u]t').
Definition red_rename (R : relation) :=
forall x t t' y,
R (t ^ x) (t' ^ x) ->
x \notin (fv t) -> x \notin (fv t') ->
R (t ^ y) (t' ^ y).
Definition red_through (R : relation) :=
forall x t1 t2 u1 u2,
x \notin (fv t1) -> x \notin (fv u1) ->
R (t1 ^ x) (u1 ^ x) -> R t2 u2 ->
R (t1 ^^ t2) (u1 ^^ u2).
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 : trm => fv x) in
let D := gather_vars_with (fun x : env => dom x) in
constr:(A \u B \u C \u D).
Ltac pick_fresh X :=
let L := gather_vars in (pick_fresh_gen L X).
Tactic Notation "apply_fresh" constr(T) "as" ident(x) :=
apply_fresh_base T gather_vars x.
Tactic Notation "apply_fresh" "*" constr(T) "as" ident(x) :=
apply_fresh T as x; auto*.
Ltac exists_fresh :=
let L := gather_vars_with (fun x : vars => x) in exists L.
Scheme typing_induct := Induction for typing Sort Prop
with wf_induct := Induction for wf Sort Prop.
Hint Constructors beta star_ equiv_ less typing wf.
Hint Unfold conv.
Section SubstProperties.
Hint Constructors term.
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_open : forall x u t1 t2, term u ->
[x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ ([x ~> u]t2).
Proof.
intros. unfold open. generalize 0.
induction t1; intros; simpl; f_equal*.
case_nat*. case_var*. apply* open_rec_term.
Qed.
Lemma subst_open_var : forall x y u e, y <> x -> term u ->
([x ~> u]e) ^ y = [x ~> u] (e ^ y).
Proof.
introv Neq Wu. rewrite* subst_open.
simpl. case_var*.
Qed.
Lemma subst_intro : forall x t u,
x \notin (fv t) -> term u ->
t ^^ u = [x ~> u](t ^ x).
Proof.
introv Fr Wu. rewrite* subst_open.
rewrite* subst_fresh. simpl. case_var*.
Qed.
End SubstProperties.
Ltac cross :=
rewrite subst_open_var; try cross.
Tactic Notation "cross" "*" :=
cross; auto*.
Hint Constructors term.
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 as y. rewrite* subst_open_var.
apply_fresh* term_prod as y. rewrite* subst_open_var.
Qed.
Lemma open_term : forall t u,
body t -> term u -> term (t ^^ u).
Proof.
intros. destruct H. pick_fresh y.
rewrite* (@subst_intro y). apply* subst_term.
Qed.
Hint Resolve subst_term open_term.
Lemma term_abs1 : forall t2 t1,
term (trm_abs t1 t2) -> term t1.
Proof.
intros. inversion* H.
Qed.
Lemma body_abs2 : forall t1 t2,
term (trm_abs t1 t2) -> body t2.
Proof.
intros. unfold body. inversion* H.
Qed.
Lemma body_prod2 : forall t1 t2,
term (trm_prod t1 t2) -> body t2.
Proof.
intros. unfold body. inversion* H.
Qed.
Hint Extern 1 (term ?t) =>
match goal with H: term (trm_abs t ?t2) |- _ =>
apply (@term_abs1 t2) end.
Hint Extern 1 (body ?t) =>
match goal with
| H: context [trm_abs ?t1 t] |- _ => apply (@body_abs2 t1)
| H: context [trm_prod ?t1 t] |- _ => apply (@body_prod2 t1)
| H: context [t ^ _] |- _ =>
let x := fresh in let L := gather_vars in
exists L; intros x Fr; destructi (H x); auto
end.
Lemma term_abs_prove : forall t1 t2,
term t1 -> body t2 -> term (trm_abs t1 t2).
Proof.
intros. apply_fresh* term_abs as x.
Qed.
Lemma term_prod_prove : forall t1 t2,
term t1 -> body t2 -> term (trm_prod t1 t2).
Proof.
intros. apply_fresh* term_prod as x.
Qed.
Hint Resolve term_abs_prove term_prod_prove.
Lemma body_prove : forall L t,
(forall x, x \notin L -> term (t ^ x)) -> body t.
Proof.
intros. exists* L.
Qed.
Hint Extern 1 (body ?t) =>
match goal with
| H: forall _, _ \notin ?L -> term (t ^ _) |- _ =>
apply (@body_prove L)
end.
Lemma body_subst : forall x t u,
term u -> body t -> body ([x ~> u]t).
Proof.
introv. intros Wu [L Bt].
exists ({{x}} \u L). intros y Fr. cross*.
Qed.
Hint Resolve body_subst.
Section PrimProperties.
Hint Constructors term.
Lemma open_var_inj : forall x t1 t2,
x \notin (fv t1) -> x \notin (fv t2) ->
(t1 ^ x = t2 ^ x) -> (t1 = t2).
Proof.
intros x t1. unfold open. generalize 0.
induction t1; intro k; destruct t2; simpl; intros; inversion H1;
try solve [ f_equal*
| do 2 try case_nat; inversions* H1; try notin_contradiction ].
Qed.
Lemma close_var_fresh : forall x t,
x \notin fv (close_var x t).
Proof.
introv. unfold close_var. generalize 0.
induction t; intros k; simpls; notin_simpl; auto.
case_var; simpl*.
Qed.
Lemma close_var_body : forall x t,
term t -> body (close_var x t).
Proof.
introv W. exists {{x}}. intros y Fr.
unfold open, close_var. generalize 0. gen y.
induction W; intros y Fr k; simpls.
case_var; simpl*. case_nat*.
auto*.
auto*.
apply_fresh* term_abs as z.
unfolds open. rewrite* close_var_rec_open.
apply_fresh* term_prod as z.
unfolds open. rewrite* close_var_rec_open.
Qed.
Lemma close_var_open : forall x t,
term t -> t = (close_var x t) ^ x.
Proof.
introv W. unfold close_var, open. generalize 0.
induction W; intros k; simpls; f_equal*.
case_var*. simpl. case_nat*.
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
unfolds open. rewrite* close_var_rec_open.
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
unfolds open. rewrite* close_var_rec_open.
Qed.
Lemma close_var_spec : forall t x, term t ->
exists u, t = u ^ x /\ body u /\ x \notin (fv u).
Proof.
intros. exists (close_var x t). split3.
apply* close_var_open.
apply* close_var_body.
apply* close_var_fresh.
Qed.
End PrimProperties.
Lemma red_regular_beta : red_regular beta.
Proof.
introz. induction* H.
Qed.
Lemma red_regular_beta_star : red_regular (beta star).
introz. induction* H. apply* red_regular_beta.
Qed.
Lemma red_regular_conv : red_regular conv.
Proof.
introz. induction* H. apply* red_regular_beta.
Qed.
Hint Extern 1 (term ?t) => match goal with
| H: beta t _ |- _ => apply (proj1 (red_regular_beta H))
| H: beta _ t |- _ => apply (proj2 (red_regular_beta H))
| H: (beta star) t _ |- _ => apply (proj1 (red_regular_beta_star H))
| H: (beta star) _ t |- _ => apply (proj2 (red_regular_beta_star H))
| H: conv t _ |- _ => apply (proj1 (red_regular_conv H))
| H: conv _ t |- _ => apply (proj2 (red_regular_conv H))
end.
Hint Extern 1 (term (trm_abs ([?x ~> ?u]?t1) ([?x ~> ?u]?t2))) =>
match goal with H: term (trm_abs t1 t2) |- _ =>
unsimpl ([x ~> u](trm_abs t1 t2)) end.
Lemma red_regular_less : red_regular less.
Proof.
introz. induction* H.
Qed.
Hint Extern 1 (term ?t) => match goal with
| H: less t _ |- _ => apply (proj1 (red_regular_less H))
| H: less _ t |- _ => apply (proj2 (red_regular_less H))
end.
Lemma regular_typing : forall E t T, typing E t T ->
(wf E /\ ok E /\ contains_terms E /\ term t /\ term T).
Proof.
apply typing_induct with
(P0 := fun E (_ : wf E) => ok E /\ contains_terms E);
unfolds contains_terms; intros; splits*.
apply_fresh* term_prod as y. forward* (H0 y).
apply_fresh* term_abs as y.
pick_fresh y. forward~ (H0 y) as K. destructs K; auto*.
forward* (H0 y).
intros. inversion H.
intros. unfolds binds. destructs H. simpls.
case_var*. inversions* H0.
Qed.
Hint Extern 1 (term ?t) => match goal with
| H: typing _ t _ |- _ => apply (proj32 (proj33 (regular_typing H)))
| H: typing _ _ t |- _ => apply (proj33 (proj33 (regular_typing H)))
end.
Lemma ok_from_wf : forall E, wf E -> ok E.
Proof.
induction 1. auto. use (regular_typing H0).
Qed.
Hint Extern 1 (ok ?E) => match goal with
| H: wf E |- _ => apply (ok_from_wf H)
end.
Hint Extern 1 (wf ?E) => match goal with
| H: typing E _ _ |- _ => apply (proj1 (regular_typing H))
end.
Lemma term_from_binds_in_wf : forall x E U,
wf E -> binds x U E -> term U.
Proof.
unfold binds. introv W Has. induction E as [|(y,a)]; simpls.
contradictions.
inversions W. case_var*. inversions* Has.
Qed.
Hint Extern 1 (term ?t) => match goal with
H: binds ?x t ?E |- _ => apply (@term_from_binds_in_wf x E)
end.
Lemma wf_left : forall E F,
wf (E & F) -> wf E.
Proof.
intros. induction F as [|(y,a)]; env_fix.
auto.
rewrite <- concat_assoc in H.
inversions H. env_fix.
use (proj1 (regular_typing H4)).
Qed.
Implicit Arguments wf_left [E F].
Lemma fv_open_var : forall y x t,
x <> y -> x \notin fv (t ^ y) -> x \notin fv t.
Proof.
introv Neq. unfold open. generalize 0.
induction t; simpl; intros; try notin_solve.
puts (IHt1 n). puts (IHt2 n). notin_simpl; auto.
puts (IHt1 n). puts (IHt2 (S n)). notin_simpl; auto.
puts (IHt1 n). puts (IHt2 (S n)). notin_simpl; auto.
Qed.
Lemma typing_fresh : forall E T i x,
typing E T (trm_type i) -> x # E -> x \notin fv T.
Proof.
introv Typ.
induction Typ; simpls; intros.
auto.
rewrite notin_singleton. intro. subst. apply* binds_fresh.
pick_fresh y. notin_simpl. auto. apply* (@fv_open_var y).
pick_fresh y. puts (IHTyp H1). notin_simpl. auto. apply* (@fv_open_var y).
puts (IHTyp1 H). puts (IHTyp2 H). auto.
auto.
Qed.
Lemma notin_fv_from_wf : forall E F x V,
wf (E & x ~ V & F) -> x \notin fv V.
Proof.
intros.
poses W (wf_left H).
inversions W.
apply* typing_fresh.
Qed.
Lemma notin_fv_from_binds : forall x y U E,
wf E -> binds y U E -> x # E -> x \notin fv U.
Proof.
induction E as [|(z,a)]; simpl; intros; env_fix.
inversions H0.
inversions H0. case_var.
inversions H3. inversions H. apply* typing_fresh.
inversions H. apply* IHE.
Qed.
Lemma notin_fv_from_binds' : forall E F x V y U,
wf (E & x ~ V & F) -> binds y U E -> x \notin fv U.
Proof.
intros. poses W (wf_left H). inversions W.
poses W' (wf_left W). apply* notin_fv_from_binds.
Qed.
Hint Extern 1 (?x \notin fv ?V) =>
match goal with H: wf (?E & x ~ V & ?F) |- _ =>
apply (@notin_fv_from_wf E F) end.
Hint Extern 1 (?x \notin fv ?U) =>
match goal with H: wf (?E & x ~ ?V & ?F), B: binds ?y U ?E |- _ =>
apply (@notin_fv_from_binds' E F x V y) end.
|