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nano4-8.v
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nano4-8.v
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(* Full safety for STLC *)
(* values well-typed with respect to runtime environment *)
(* inversion lemma structure *)
(* subtyping (in addition to nano2.v) *)
(* singleton types (in addtion to nano4-1.v *)
(* full proof for singleton types! *)
(* added stp_vary rule *)
(* TODO: type members *)
(* TODO: dependent application *)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Le.
Module STLC.
Definition id := nat.
Inductive ty : Type :=
| TBool : ty
| TFun : ty -> ty -> ty
| TVar : id -> ty
| TVarB : id -> ty
| TMem : ty -> ty -> ty (* intro *)
| TSel : ty -> ty (* elim *)
| TBot : ty
| TTop : ty
.
Inductive tm : Type :=
| ttrue : tm
| tfalse : tm
| tvar : id -> tm
| tapp : tm -> tm -> tm (* f(x) *)
| tabs : tm -> tm (* \f x.y *)
.
Inductive vl : Type :=
| vbool : bool -> vl
| vabs : list vl -> tm -> vl
| vty : list vl -> ty -> vl
.
Definition venv := list vl.
Definition tenv := list ty.
Hint Unfold venv.
Hint Unfold tenv.
Fixpoint index {X : Type} (n : id) (l : list X) : option X :=
match l with
| [] => None
| a :: l' => if beq_nat n (length l') then Some a else index n l'
end.
(* closed j k means normal variables < j, bound variables < k *)
Inductive closed: nat -> nat -> ty -> Prop :=
| cl_bool: forall j k,
closed j k TBool
| cl_fun: forall j k T1 T2,
closed j k T1 ->
closed j k T2 ->
closed j k (TFun T1 T2)
| cl_var: forall j k x,
j > x ->
closed j k (TVar x)
| cl_varB: forall j k x,
k > x ->
closed j k (TVarB x)
| cl_mem: forall j k T1 T2,
closed j k T1 ->
closed j k T2 ->
closed j k (TMem T1 T2)
| cl_sel: forall j k T1,
closed j k T1 ->
closed j k (TSel T1)
| cl_bot: forall j k,
closed j k TBot
| cl_top: forall j k,
closed j k TTop
.
Fixpoint open (k: nat) (u: nat) (T: ty) { struct T }: ty :=
match T with
| TVar x => TVar x (* free var remains free. functional, so we can't check for conflict *)
| TVarB x => if beq_nat k x then TVar u else TVarB x
(* | TBind T2 => TBind (open_rec (S k) u T2) *)
| TTop => TTop
| TBot => TBot
| TBool => TBool
| TSel T1 => TSel (open k u T1)
| TFun T1 T2 => TFun (open k u T1) (open k u T2)
| TMem T1 T2 => TMem (open k u T1) (open k u T2)
end.
Inductive real: ty -> Prop :=
| real_var: forall x,
real (TVar x)
| real_mem: forall T,
real (TMem T T)
.
Inductive stp: tenv -> ty -> ty -> Prop :=
| stp_bool: forall G1,
stp G1 TBool TBool
| stp_fun: forall G1 T1 T2 T3 T4,
stp G1 T3 T1 ->
stp G1 T2 T4 ->
stp G1 (TFun T1 T2) (TFun T3 T4)
| stp_varx: forall G1 x,
x < length G1 ->
stp G1 (TVar x) (TVar x)
| stp_vary: forall G1 x y,
index x G1 = Some (TVar y) ->
y < length G1 ->
stp G1 (TVar y) (TVar x)
| stp_var1: forall G1 x T1,
index x G1 = Some T1 ->
closed (length G1) 0 T1 ->
stp G1 (TVar x) T1
| stp_mem: forall G1 T1 T2 T3 T4,
stp G1 T3 T1 ->
stp G1 T2 T4 ->
stp G1 (TMem T1 T2) (TMem T3 T4)
| stp_sel: forall G1 T1 T2,
stp G1 T1 T2 ->
stp G1 T2 T1 ->
stp G1 (TSel T1) (TSel T2)
| stp_red: forall G1 T1 T2,
stp G1 T1 (TMem TBot T2) ->
stp G1 (TSel T1) T2
| stp_red2: forall G1 T1 T2 T0,
real T0 ->
stp G1 T0 T2 -> (* this is the main trick ... *)
stp G1 T2 (TMem T1 TTop) ->
stp G1 T1 (TSel T2)
.
Inductive has_type : tenv -> tm -> ty -> Prop :=
| t_true: forall env,
has_type env ttrue TBool
| t_false: forall env,
has_type env tfalse TBool
| t_varx: forall x env T1,
index x env = Some T1 -> (* could use closed *)
has_type env (tvar x) (TVar x)
(*| t_var: forall x env T1,
index x env = Some T1 ->
has_type env (tvar x) T1*)
| t_app: forall env f x T1 T2,
has_type env f (TFun T1 T2) ->
has_type env x T1 ->
has_type env (tapp f x) T2
| t_abs: forall env y T1 T2,
has_type (T1::(TFun T1 T2)::env) y T2 ->
closed (length env) 0 (TFun T1 T2) ->
has_type env (tabs y) (TFun T1 T2)
| t_sub: forall env x T1 T2,
has_type env x T1 ->
stp env T1 T2 ->
has_type env x T2
.
Inductive stp2: bool -> venv -> ty -> venv -> ty -> nat -> Prop :=
| stp2_bot: forall G1 G2 T n1,
closed (length G2) 0 T ->
stp2 true G1 TBot G2 T (S n1)
| stp2_top: forall G1 G2 T n1,
closed (length G1) 0 T ->
stp2 true G1 T G2 TTop (S n1)
| stp2_bool: forall G1 G2 n1,
stp2 true G1 TBool G2 TBool (S n1)
| stp2_fun: forall G1 G2 T1 T2 T3 T4 n1 n2,
stp2 false G2 T3 G1 T1 n1 ->
stp2 false G1 T2 G2 T4 n2 ->
stp2 true G1 (TFun T1 T2) G2 (TFun T3 T4) (S (n1+n2))
| stp2_varx: forall G1 G2 x1 x2 v n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 true G1 (TVar x1) G2 (TVar x2) (S n1)
| stp2_var1: forall G1 GX TX G2 x T2 v n1,
index x G1 = Some v ->
val_type0 GX v TX -> (* slack for: val_type G2 v T2 *)
stp2 true GX TX G2 T2 n1 ->
stp2 true G1 (TVar x) G2 T2 (S n1)
| stp2_mem: forall G1 G2 T1 T2 T3 T4 n1 n2,
stp2 false G2 T3 G1 T1 n2 ->
stp2 true G1 T2 G2 T4 n1 ->
stp2 true G1 (TMem T1 T2) G2 (TMem T3 T4) (S (n1+n2))
| stp2_sel: forall G1 G2 T1 T2 n1 n2,
stp2 false G2 T2 G1 T1 n2 ->
stp2 true G1 T1 G2 T2 n1 ->
stp2 true G1 (TSel T1) G2 (TSel T2) (S (n1+n2))
| stp2_red: forall G1 G2 T1 T2 n1,
stp2 true G1 T1 G2 (TMem TBot T2) n1 ->
stp2 true G1 (TSel T1) G2 T2 (S n1)
| stp2_red2: forall G1 G2 GX TX T1 T2 GY T0 n0 n1 n2 n3,
(* long form of:
stp2 true GY (TVar y) G2 T2 n0 ->
stp2 true G2 T2 G1 (TMem T1 TTop) n1 -> *)
real T0 ->
stp2 true GY T0 G2 T2 n0 ->
stp2 false G2 T2 GX (TMem TX TTop) n1 ->
stp2 true GY T0 GX (TMem TX TTop) n3 ->
stp2 false G1 T1 GX TX n2 ->
stp2 true G1 T1 G2 (TSel T2) (S (n0+n1+n2+n3))
(*
x < T^x
----------------
x < { z => T^z }
x < { z => T^z }
----------------
x < T^x
*)
| stp2_wrapf: forall G1 G2 T1 T2 n1,
stp2 true G1 T1 G2 T2 n1 ->
stp2 false G1 T1 G2 T2 (S n1)
| stp2_transf: forall G1 G2 G3 T1 T2 T3 n1 n2,
stp2 true G1 T1 G2 T2 n1 ->
stp2 false G2 T2 G3 T3 n2 ->
stp2 false G1 T1 G3 T3 (S (n1+n2))
with wf_env : venv -> tenv -> Prop :=
| wfe_nil : wf_env nil nil
| wfe_cons : forall v t vs ts,
val_type (v::vs) v t ->
wf_env vs ts ->
wf_env (cons v vs) (cons t ts)
with val_type0 : venv -> vl -> ty -> Prop :=
| v_bool: forall b,
val_type0 [] (vbool b) TBool
| v_abs: forall venv tenv y T1 T2,
wf_env venv tenv ->
has_type (T1::(TFun T1 T2)::tenv) y T2 ->
val_type0 venv (vabs venv y) (TFun T1 T2)
| v_var: forall venv x v,
index x venv = Some v ->
val_type0 venv v (TVar x)
| v_ty: forall venv T1,
val_type0 venv (vty venv T1) (TMem T1 T1)
with val_type : venv -> vl -> ty -> Prop :=
| v_sub: forall G1 T1 G2 T2 v,
val_type0 G1 v T1 ->
(exists n, stp2 true G1 T1 G2 T2 n) ->
val_type G2 v T2
.
Definition stpd2 m G1 T1 G2 T2 := exists n, stp2 m G1 T1 G2 T2 n.
Hint Constructors stp2.
Ltac ep := match goal with
| [ |- stp2 ?M ?G1 ?T1 ?G2 ?T2 ?N ] => assert (exists (n:nat), stp2 M G1 T1 G2 T2 n) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ _ |- _ => destruct H as [? H]
end.
Lemma stpd2_bot: forall G1 G2 T,
closed (length G2) 0 T ->
stpd2 true G1 TBot G2 T.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_top: forall G1 G2 T,
closed (length G1) 0 T ->
stpd2 true G1 T G2 TTop.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_bool: forall G1 G2,
stpd2 true G1 TBool G2 TBool.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_fun: forall G1 G2 T1 T2 T3 T4,
stpd2 false G2 T3 G1 T1 ->
stpd2 false G1 T2 G2 T4 ->
stpd2 true G1 (TFun T1 T2) G2 (TFun T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_varx: forall G1 G2 x1 x2 v,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stpd2 true G1 (TVar x1) G2 (TVar x2).
Proof. intros. repeat eu. exists 1. eauto. Qed.
Lemma stpd2_var1: forall G1 GX TX G2 x T2 v,
index x G1 = Some v ->
val_type0 GX v TX -> (* slack for: val_type G2 v T2 *)
stpd2 true GX TX G2 T2 ->
stpd2 true G1 (TVar x) G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_mem: forall G1 G2 T1 T2 T3 T4,
stpd2 false G2 T3 G1 T1 ->
stpd2 true G1 T2 G2 T4 ->
stpd2 true G1 (TMem T1 T2) G2 (TMem T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_sel: forall G1 G2 T1 T2,
stpd2 false G2 T2 G1 T1 ->
stpd2 true G1 T1 G2 T2 ->
stpd2 true G1 (TSel T1) G2 (TSel T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red: forall G1 G2 T1 T2,
stpd2 true G1 T1 G2 (TMem TBot T2) ->
stpd2 true G1 (TSel T1) G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red2: forall G1 G2 GX TX T1 T2 GY T0,
real T0 ->
stpd2 true GY T0 G2 T2 ->
stpd2 false G2 T2 GX (TMem TX TTop) ->
stpd2 true GY T0 GX (TMem TX TTop) ->
stpd2 false G1 T1 GX TX ->
stpd2 true G1 T1 G2 (TSel T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_wrapf: forall G1 G2 T1 T2,
stpd2 true G1 T1 G2 T2 ->
stpd2 false G1 T1 G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_transf: forall G1 G2 G3 T1 T2 T3,
stpd2 true G1 T1 G2 T2 ->
stpd2 false G2 T2 G3 T3 ->
stpd2 false G1 T1 G3 T3.
Proof. intros. repeat eu. eexists. eauto. Qed.
(*
None means timeout
Some None means stuck
Some (Some v)) means result v
Could use do-notation to clean up syntax.
*)
Fixpoint teval(n: nat)(env: venv)(t: tm){struct n}: option (option vl) :=
match n with
| 0 => None
| S n =>
match t with
| ttrue => Some (Some (vbool true))
| tfalse => Some (Some (vbool false))
| tvar x => Some (index x env)
| tabs y => Some (Some (vabs env y))
| tapp ef ex =>
match teval n env ex with
| None => None
| Some None => Some None
| Some (Some vx) =>
match teval n env ef with
| None => None
| Some None => Some None
| Some (Some (vbool _)) => Some None
| Some (Some (vty _ _)) => Some None
| Some (Some (vabs env2 ey)) =>
teval n (vx::(vabs env2 ey)::env2) ey
end
end
end
end.
Hint Constructors ty.
Hint Constructors tm.
Hint Constructors vl.
Hint Constructors has_type.
Hint Constructors stp2.
Hint Constructors val_type.
Hint Constructors wf_env.
Hint Unfold stpd2.
Hint Constructors option.
Hint Constructors list.
Hint Unfold index.
Hint Unfold length.
Hint Resolve ex_intro.
Ltac ev := repeat match goal with
| H: exists _, _ |- _ => destruct H
| H: _ /\ _ |- _ => destruct H
end.
Lemma wf_length : forall vs ts,
wf_env vs ts ->
(length vs = length ts).
Proof.
intros. induction H. auto.
assert ((length (v::vs)) = 1 + length vs). constructor.
assert ((length (t::ts)) = 1 + length ts). constructor.
rewrite IHwf_env in H1. auto.
Qed.
Hint Immediate wf_length.
Lemma index_max : forall X vs n (T: X),
index n vs = Some T ->
n < length vs.
Proof.
intros X vs. induction vs.
Case "nil". intros. inversion H.
Case "cons".
intros. inversion H.
case_eq (beq_nat n (length vs)); intros E.
SCase "hit".
rewrite E in H1. inversion H1. subst.
eapply beq_nat_true in E.
unfold length. unfold length in E. rewrite E. eauto.
SCase "miss".
rewrite E in H1.
assert (n < length vs). eapply IHvs. apply H1.
compute. eauto.
Qed.
Lemma index_exists : forall X vs n,
n < length vs ->
exists (T:X), index n vs = Some T.
Proof.
intros X vs. induction vs.
Case "nil". intros. inversion H.
Case "cons".
intros. inversion H.
SCase "hit".
assert (beq_nat n (length vs) = true) as E. eapply beq_nat_true_iff. eauto.
simpl. subst n. rewrite E. eauto.
SCase "miss".
assert (beq_nat n (length vs) = false) as E. eapply beq_nat_false_iff. omega.
simpl. rewrite E. eapply IHvs. eauto.
Qed.
Lemma index_extend : forall X vs n a (T: X),
index n vs = Some T ->
index n (a::vs) = Some T.
Proof.
intros.
assert (n < length vs). eapply index_max. eauto.
assert (n <> length vs). omega.
assert (beq_nat n (length vs) = false) as E. eapply beq_nat_false_iff; eauto.
unfold index. unfold index in H. rewrite H. rewrite E. reflexivity.
Qed.
Lemma closed_extend : forall X (a:X) k G T,
closed (length G) k T ->
closed (length (a::G)) k T.
Proof.
intros. induction T; inversion H; econstructor; eauto.
simpl. omega.
Qed.
Lemma stp2_extend : forall m v1 G1 G2 T1 T2 n,
stp2 m G1 T1 G2 T2 n ->
stp2 m (v1::G1) T1 G2 T2 n /\
stp2 m G1 T1 (v1::G2) T2 n /\
stp2 m (v1::G1) T1 (v1::G2) T2 n.
Proof.
intros. induction H; try solve [repeat split; econstructor; try eauto;
try eapply IHstp2; eauto;
try eapply index_extend; eauto; try eapply closed_extend; eauto;
try eapply IHstp2_1; try eapply IHstp2_2;
try eapply IHstp2_3; try eapply IHstp2_4].
(* trans case left *)
repeat split; eapply stp2_transf; try eapply IHstp2_1; eauto; try eapply IHstp2_2; eauto.
Qed.
Lemma stpd2_extend : forall m v1 G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
stpd2 m (v1::G1) T1 G2 T2 /\
stpd2 m G1 T1 (v1::G2) T2 /\
stpd2 m (v1::G1) T1 (v1::G2) T2.
Proof.
intros. repeat eu. repeat split; eexists; eapply stp2_extend; eauto.
Qed.
Lemma stp2_extend1 : forall m v1 G1 G2 T1 T2 n, stp2 m G1 T1 G2 T2 n -> stp2 m (v1::G1) T1 G2 T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stp2_extend2 : forall m v1 G1 G2 T1 T2 n, stp2 m G1 T1 G2 T2 n -> stp2 m G1 T1 (v1::G2) T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stpd2_extend1 : forall m v1 G1 G2 T1 T2, stpd2 m G1 T1 G2 T2 -> stpd2 m (v1::G1) T1 G2 T2.
Proof. intros. eapply stpd2_extend. eauto. Qed.
Lemma stpd2_extend2 : forall m v1 G1 G2 T1 T2, stpd2 m G1 T1 G2 T2 -> stpd2 m G1 T1 (v1::G2) T2.
Proof. intros. eapply stpd2_extend. eauto. Qed.
Lemma stp_closed : forall G1 T1 T2,
stp G1 T1 T2 ->
closed (length G1) 0 T1 /\
closed (length G1) 0 T2.
Proof.
intros. induction H; repeat split; try econstructor; try eapply IHstp1; try eapply IHstp2; eauto; try eapply IHstp; eauto; try eapply index_max; eauto.
destruct IHstp. inversion H1. eauto.
destruct IHstp2. inversion H3. eauto.
Qed.
Lemma stpd2_closed : forall m G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
closed (length G1) 0 T1 /\
closed (length G2) 0 T2.
Proof.
intros. eu. induction H; repeat split; try econstructor; try eapply IHstp2_1; try eapply IHstp2_2; eauto; try eapply IHstp2; eauto; try eapply index_max; eauto.
destruct IHstp2. inversion H1. eauto.
eapply IHstp2_4.
Qed.
Lemma stpd2_closed1 : forall m G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
closed (length G1) 0 T1.
Proof. intros. eapply (stpd2_closed m G1 G2); eauto. Qed.
Lemma stpd2_closed2 : forall m G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
closed (length G2) 0 T2.
Proof. intros. eapply (stpd2_closed m G1 G2); eauto. Qed.
Lemma valtp_extend : forall vs v v1 T,
val_type vs v T ->
val_type (v1::vs) v T.
Proof.
intros. induction H; econstructor; eauto; try eapply stpd2_extend; eauto; try eapply index_extend; eauto.
Qed.
Lemma index_safe_ex: forall H1 G1 TF i,
wf_env H1 G1 ->
index i G1 = Some TF ->
exists v, index i H1 = Some v /\ val_type H1 v TF.
Proof. intros. induction H.
Case "nil". inversion H0.
Case "cons". inversion H0.
case_eq (beq_nat i (length ts)).
SCase "hit".
intros E.
rewrite E in H3. inversion H3. subst t.
assert (beq_nat i (length vs) = true). eauto.
assert (index i (v :: vs) = Some v). eauto. unfold index. rewrite H2. eauto.
eauto.
SCase "miss".
intros E.
assert (beq_nat i (length vs) = false). eauto.
rewrite E in H3.
assert (exists v0, index i vs = Some v0 /\ val_type vs v0 TF) as HI. eapply IHwf_env. eauto.
inversion HI as [v0 HI1]. inversion HI1.
eexists. econstructor. eapply index_extend; eauto. eapply valtp_extend; eauto.
Qed.
Inductive res_type: venv -> option vl -> ty -> Prop :=
| not_stuck: forall venv v T,
val_type venv v T ->
res_type venv (Some v) T.
Hint Constructors res_type.
Hint Resolve not_stuck.
Lemma stpd2_refl: forall G1 T1,
closed (length G1) T1 ->
stpd2 true G1 T1 G1 T1.
Proof.
intros. induction T1; inversion H.
- Case "bool". eapply stpd2_bool; eauto.
- Case "fun". eapply stpd2_fun; try eapply stpd2_wrapf; eauto.
- Case "var".
assert (exists v, index i G1 = Some v) as E. eapply index_exists; eauto.
destruct E.
eapply stpd2_varx; eauto.
- Case "mem". eapply stpd2_mem; try eapply stpd2_wrapf; eauto.
- Case "sel". eapply stpd2_sel; try eapply stpd2_wrapf; eauto.
- Case "bot". exists 1. eauto.
- Case "top". exists 1. eauto.
Qed.
Lemma stpd2_reg1 : forall m G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
stpd2 true G1 T1 G1 T1.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m G1 G2). eauto.
Qed.
Lemma stpd2_reg2 : forall m G1 G2 T1 T2,
stpd2 m G1 T1 G2 T2 ->
stpd2 true G2 T2 G2 T2.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m G1 G2). eauto.
Qed.
Lemma invert_mem1: forall n, forall venv vf T1 T2 GX TX n1,
val_type0 GX vf TX -> stp2 true GX TX venv (TMem T1 T2) n1 -> n1 < n ->
exists env T3 n2 n3,
vf = (vty env T3) /\
(n2 + n3) < n /\
stp2 false venv T1 env T3 n2 /\
stp2 true env T3 venv T2 n3.
Proof.
intros n. induction n; intros. solve by inversion.
inversion H; subst.
- Case "bool". solve by inversion.
- Case "fun". solve by inversion.
- Case "var". subst. inversion H0; subst.
assert (vf = v) as A. rewrite H2 in H4. inversion H4. eauto.
rewrite A. assert (n0 < n) as B. omega.
specialize (IHn venv0 v T1 T2 GX0 TX n0 H5 H6 B).
ev. repeat eexists; eauto.
(* repeat eapply IHn; eauto. omega. *)
- Case "mem". inversion H0. subst. repeat eexists; eauto. omega.
Qed.
Lemma invert_mem: forall venv vf T1 T2,
val_type venv vf (TMem T1 T2) ->
exists env T3,
vf = (vty env T3) /\
stpd2 false venv T1 env T3 /\
stpd2 true env T3 venv T2.
Proof.
intros. inversion H. ev. assert (x < S x) as E. omega.
specialize (invert_mem1 (S x) venv0 vf T1 T2 G1 T0 x H0 H1 E). intros IH.
ev. repeat eexists; eauto.
Qed.
Lemma stpd2_trans_axiom_aux: forall n, forall G1 G2 G3 T1 T2 T3 n1,
stp2 false G1 T1 G2 T2 n1 -> n1 < n ->
stpd2 false G2 T2 G3 T3 ->
stpd2 false G1 T1 G3 T3.
Proof.
intros n. induction n; intros; try omega; repeat eu; subst; inversion H; subst.
- Case "wrapf". eapply stpd2_transf. eexists. eauto. eexists. eauto.
- Case "transf". eapply stpd2_transf. eexists. eauto. eapply IHn. eauto. omega. eexists. eauto.
Qed.
Lemma stp2_trans_axiom: forall m G1 G2 G3 T1 T2 T3,
stpd2 m G1 T1 G2 T2 ->
stpd2 false G2 T2 G3 T3 ->
stpd2 false G1 T1 G3 T3.
Proof.
intros. destruct m; eu; eu; eapply stpd2_trans_axiom_aux; eauto.
Qed.
Lemma stp2_trans: forall n, forall m G1 G2 G3 T1 T2 T3 n1 n2,
stp2 m G1 T1 G2 T2 n1 ->
stp2 true G2 T2 G3 T3 n2 ->
n1 < n ->
stpd2 true G1 T1 G3 T3.
Proof.
intros n. induction n; intros. solve by inversion.
inversion H.
- Case "bot". eapply stpd2_bot; eauto. eapply stpd2_closed2; eauto.
- Case "top". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "bool". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. econstructor.
+ SCase "bool". eapply stpd2_bool; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "fun". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "fun". inversion H14. subst. eapply stpd2_fun; eapply stp2_trans_axiom; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "varx". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "varx". inversion H14. subst. rewrite H3 in H10. inversion H10. subst. eapply stpd2_varx; eauto.
+ SCase "var1". inversion H15. subst. rewrite H3 in H10. inversion H10. subst. eapply stpd2_var1; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "var1".
eapply stpd2_var1; eauto. eapply IHn; eauto. omega.
- Case "mem". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "mem". inversion H14; subst. eapply stpd2_mem. eapply stp2_trans_axiom; eauto. eapply IHn; eauto; try omega.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "sel". inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "sel". inversion H14; subst. eapply stpd2_sel. eapply stp2_trans_axiom; eauto. eapply IHn; eauto; try omega.
+ SCase "red". inversion H13; subst. eapply stpd2_red. eapply IHn. eauto. eauto. omega.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "red".
eapply stpd2_red. eauto. eapply IHn; eauto. repeat econstructor. eauto. omega.
- Case "red2".
inversion H0; subst; try solve by inversion.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "sel". inversion H17; subst.
eapply stpd2_red2. eauto.
eapply IHn. eapply H3. eauto. omega. eapply stp2_trans_axiom; eauto. eauto. eauto.
+ SCase "red". clear H. inversion H16; subst.
(* pull this out as stp2_trans_cross lemma? *)
assert (stp2 true GY T5 G2 T4 n0) as A0. eauto.
assert (stp2 true GY T5 GX (TMem TX TTop) n5) as A1. eauto.
assert (stpd2 true GY T5 G3 (TMem TBot T3)) as A2. eapply IHn. eapply A0. eauto. omega. eu.
destruct H2; inversion A1; inversion A2; subst. (* could pull out into a lemma: invert_real *)
* SSCase "var".
rewrite H2 in H15. inversion H15. subst v0.
assert (exists GX1 TX1 m1 m2, v = (vty GX1 TX1) /\ m1 + m2 < S n1 /\
stp2 false GX TX GX1 TX1 m1 /\ stp2 true GX1 TX1 GX TTop m2) as B1.
eapply invert_mem1. eapply H7. eauto. eauto.
assert (exists GX2 TX2 m3 m4, v = (vty GX2 TX2) /\ m3 + m4 < S n2 /\
stp2 false G3 TBot GX2 TX2 m3 /\ stp2 true GX2 TX2 G3 T3 m4) as B2.
eapply invert_mem1; eauto.
destruct B1 as [GZ1 [TZ1 [m1 [m2 [V1 [M1 [B1 _]]]]]]].
destruct B2 as [GZ2 [TZ2 [m3 [m4 [V2 [M2 [_ B2 ]]]]]]].
subst v. inversion V2. subst.
assert (stpd2 true GX TX G3 T3) as S3. eapply IHn. eapply B1. eapply B2. omega. eu.
assert (stpd2 true G1 T1 G3 T3) as S1. eapply IHn. eauto. eapply S3. omega.
eapply S1.
* SSCase "mem".
assert (stp2 false GX TX GY T n2) as B1. eauto.
assert (stp2 true GY T G3 T3 n7) as B2. eauto.
assert (stpd2 true GX TX G3 T3) as S3. eapply IHn. eapply B1. eapply B2. omega. eu.
assert (stpd2 true G1 T1 G3 T3) as S1. eapply IHn. eauto. eapply S3. omega.
eapply S1.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
- Case "wrapf".
subst. eapply IHn. eapply H2. eapply H0. omega.
- Case "transf".
assert (stpd2 true G4 T4 G3 T3). eapply IHn. eapply H3. eapply H0. omega.
eu. subst. eapply IHn. eapply H2. eauto. omega.
Grab Existential Variables.
apply 0.
Qed.
(* idea: can we use the same strong_sel as before, and just translate the new rules? *)
Lemma stpd2_trans: forall m G1 G2 G3 T1 T2 T3,
stpd2 m G1 T1 G2 T2 ->
stpd2 true G2 T2 G3 T3 ->
stpd2 true G1 T1 G3 T3.
Proof.
intros. repeat eu. eapply stp2_trans; eauto.
Qed.
Lemma invert_var1: forall n, forall venv v x GX TX n1,
val_type0 GX v TX -> stp2 true GX TX venv (TVar x) n1 -> n1 < n ->
index x venv = Some v.
Proof.
intros n. induction n; intros. solve by inversion.
inversion H; subst.
- Case "bool". solve by inversion.
- Case "fun". solve by inversion.
- Case "var". subst. inversion H0; subst.
+ SCase "varx".
assert (v = v0) as A. rewrite H2 in H5. inversion H5. eauto.
rewrite A. eauto.
+ SCase "var1".
assert (v = v0) as A. rewrite H2 in H4. inversion H4. eauto.
rewrite A. eapply IHn; eauto. omega.
- Case "mem". solve by inversion.
Qed.
Lemma invert_var: forall venv v x,
val_type venv v (TVar x) ->
index x venv = Some v.
Proof.
intros. inversion H. ev. eapply invert_var1; eauto.
Qed.
Lemma stp_to_stpd2: forall G1 T1 T2,
stp G1 T1 T2 ->
forall GX, wf_env GX G1 ->
stpd2 true GX T1 GX T2.
Proof.
intros G1 T1 T2 H. induction H; intros.
- Case "bool". eapply stpd2_bool; eauto.
- Case "fun". eapply stpd2_fun; eapply stpd2_wrapf; eauto.
- Case "varx".
assert (exists v, index x GX = Some v) as E. eapply index_exists. rewrite (wf_length GX G1). eauto. eauto.
destruct E.
eapply stpd2_varx; eauto.
- Case "vary".
assert (exists v, index x GX = Some v /\ val_type GX v (TVar y)) as E. eapply index_safe_ex; eauto.
destruct E. destruct H2. inversion H3. subst.
eapply stpd2_varx. eapply invert_var. eauto. eauto.
- Case "var1".
assert (exists v, index x GX = Some v /\ val_type GX v T1) as E. eapply index_safe_ex; eauto.
destruct E. destruct H2. destruct H3.
eapply stpd2_var1. eauto. eauto. eauto.
- Case "mem". eapply stpd2_mem; try eapply stpd2_wrapf; eauto.
- Case "sel". eapply stpd2_sel; eauto; try eapply stpd2_wrapf; eauto.
- Case "red". eapply stpd2_red. eauto.
- Case "red2". eapply stpd2_red2. eauto.
eapply IHstp1. eauto. eapply stpd2_wrapf.
eapply IHstp2. eauto.
eapply stpd2_trans. eapply IHstp1. eauto. eapply IHstp2. eauto.
assert (closed (length G1) (TMem T1 TTop)) as C. eapply stp_closed; eauto.
eapply stpd2_wrapf. eapply stpd2_refl. rewrite (wf_length GX G1). inversion C. eauto. eauto.
Qed.
Lemma valtp_widen: forall vf H1 H2 T1 T2,
val_type H1 vf T1 ->
stpd2 true H1 T1 H2 T2 ->
val_type H2 vf T2.
Proof.
intros. inversion H.
- econstructor; eauto. eapply stpd2_trans; eauto.
Qed.
Lemma restp_widen: forall vf H1 H2 T1 T2,
res_type H1 vf T1 ->
stpd2 true H1 T1 H2 T2 ->
res_type H2 vf T2.
Proof.
intros. inversion H. eapply not_stuck. eapply valtp_widen; eauto.
Qed.
Lemma invert_abs1: forall n, forall venv vf T1 T2 GX TX n1,
val_type0 GX vf TX -> stp2 true GX TX venv (TFun T1 T2) n1 -> n1 < n ->
exists env tenv y T3 T4,
vf = (vabs env y) /\
wf_env env tenv /\
has_type (T3::(TFun T3 T4)::tenv) y T4 /\
stpd2 true venv T1 env T3 /\
stpd2 true env T4 venv T2.
Proof.
intros n. induction n; intros. solve by inversion.
inversion H; subst.
- Case "bool". solve by inversion.
- Case "fun". subst. inversion H0. subst.
eexists. eexists. eexists. eexists. eexists. repeat split; eauto.
eapply stpd2_trans. eauto. eapply stpd2_reg2. eauto.
eapply stpd2_trans. eauto. eapply stpd2_reg2. eauto.
- Case "var". subst. inversion H0. subst.
assert (vf = v) as A. rewrite H2 in H4. inversion H4. eauto.
rewrite A. eapply IHn; eauto. omega.
- Case "mem". solve by inversion.
Qed.
Lemma invert_abs: forall venv vf T1 T2,
val_type venv vf (TFun T1 T2) ->
exists env tenv y T3 T4,
vf = (vabs env y) /\
wf_env env tenv /\
has_type (T3::(TFun T3 T4)::tenv) y T4 /\
stpd2 true venv T1 env T3 /\
stpd2 true env T4 venv T2.
Proof.
intros. inversion H. ev. eapply invert_abs1; eauto.
Qed.
(* if not a timeout, then result not stuck and well-typed *)
Theorem full_safety : forall n e tenv venv res T,
teval n venv e = Some res -> has_type tenv e T -> wf_env venv tenv ->
res_type venv res T.
Proof.
intros n. induction n.
(* 0 *) intros. inversion H.
(* S n *) intros. destruct e; inversion H.
- Case "True".
remember (ttrue) as e. induction H0; inversion Heqe; subst.
+ eapply not_stuck. eapply v_sub; eauto. eapply v_bool; eauto.
+ eapply restp_widen. eapply IHhas_type; eauto. eapply stp_to_stpd2; eauto.
- Case "False".
remember (tfalse) as e. induction H0; inversion Heqe; subst.
+ eapply not_stuck. eapply v_sub; eauto. eapply v_bool; eauto.
+ eapply restp_widen. eapply IHhas_type; eauto. eapply stp_to_stpd2; eauto.
- Case "Var".
remember (tvar i) as e. induction H0; inversion Heqe; subst.
+ destruct (index_safe_ex venv0 env T1 i) as [v [I V]]; eauto. (* Var *)
rewrite I. eapply not_stuck. destruct V. eapply v_sub. eapply v_var. eauto. eapply stpd2_varx. eauto. eauto.
+ eapply restp_widen. eapply IHhas_type; eauto. eapply stp_to_stpd2; eauto.
- Case "App".
remember (tapp e1 e2) as e. induction H0; inversion Heqe; subst.
+
remember (teval n venv0 e1) as tf.
remember (teval n venv0 e2) as tx.
destruct tx as [rx|]; try solve by inversion.
assert (res_type venv0 rx T1) as HRX. SCase "HRX". subst. eapply IHn; eauto.
inversion HRX as [? vx].
destruct tf as [rf|]; subst rx; try solve by inversion.
assert (res_type venv0 rf (TFun T1 T2)) as HRF. SCase "HRF". subst. eapply IHn; eauto.
inversion HRF as [? vf].
destruct (invert_abs venv0 vf T1 T2) as
[env1 [tenv [y0 [T3 [T4 [EF [WF [HTY [STX STY]]]]]]]]]. eauto.
(* now we know it's a closure, and we have has_type evidence *)
assert (res_type (vx::vf::env1) res T4) as HRY.
SCase "HRY".
subst. eapply IHn. eauto. eauto.
(* wf_env f x *) econstructor. eapply valtp_widen; eauto. eapply stpd2_extend2; eauto. eapply stpd2_extend2; eauto.
(* wf_env f *) econstructor. eapply v_sub. eapply v_abs; eauto. eapply stpd2_extend2. eapply stpd2_fun. eapply stpd2_wrapf. eapply stpd2_reg2; eauto. eapply stpd2_wrapf. eapply stpd2_reg1; eauto.
eauto.
inversion HRY as [? vy].
eapply not_stuck. eapply valtp_widen; eauto. eapply stpd2_extend1. eapply stpd2_extend1. eauto.
+ eapply restp_widen. eapply IHhas_type; eauto. eapply stp_to_stpd2; eauto.
- Case "Abs".
remember (tabs e) as xe. induction H0; inversion Heqxe; subst.
+ eapply not_stuck. eapply v_sub. eapply v_abs; eauto. eapply stpd2_refl. rewrite (wf_length venv0 env). eauto. eauto.
+ eapply restp_widen. eapply IHhas_type; eauto. eapply stp_to_stpd2; eauto.
Grab Existential Variables.
apply 0. apply 0.
Qed.
End STLC.