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nano4-6.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
.
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
.
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.
Inductive closed: nat -> ty -> Prop :=
| cl_bool: forall k,
closed k TBool
| cl_fun: forall k T1 T2,
closed k T1 ->
closed k T2 ->
closed k (TFun T1 T2)
| cl_selb: forall k x,
k > x ->
closed k (TVar x)
.
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 ->
stp G1 (TVar x) T1
.
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) (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: venv -> ty -> venv -> ty -> nat -> Prop :=
| stp2_bool: forall G1 G2 n1,
stp2 G1 TBool G2 TBool (S n1)
| stp2_fun: forall G1 G2 T1 T2 T3 T4 n1 n2,
stp2 G2 T3 G1 T1 n1 ->
stp2 G1 T2 G2 T4 n2 ->
stp2 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 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 GX TX G2 T2 n1 ->
stp2 G1 (TVar x) G2 T2 (S n1)
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)
with val_type : venv -> vl -> ty -> Prop :=
| v_sub: forall G1 T1 G2 T2 v,
val_type0 G1 v T1 ->
(exists n, stp2 G1 T1 G2 T2 n) ->
val_type G2 v T2
.
Definition stpd2 G1 T1 G2 T2 := exists n, stp2 G1 T1 G2 T2 n.
Hint Constructors stp2.
Ltac ep := match goal with
| [ |- stp2 ?G1 ?T1 ?G2 ?T2 ?N ] => assert (exists (n:nat), stp2 G1 T1 G2 T2 n) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ |- _ => destruct H as [? H]
end.
Lemma stpd2_bool: forall G1 G2,
stpd2 G1 TBool G2 TBool.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_fun: forall G1 G2 T1 T2 T3 T4,
stpd2 G2 T3 G1 T1 ->
stpd2 G1 T2 G2 T4 ->
stpd2 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 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 GX TX G2 T2 ->
stpd2 G1 (TVar x) G2 T2.
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 (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 stp2_extend : forall v1 G1 G2 T1 T2 n,
stp2 G1 T1 G2 T2 n ->
stp2 (v1::G1) T1 G2 T2 n /\
stp2 G1 T1 (v1::G2) T2 n /\
stp2 (v1::G1) T1 (v1::G2) T2 n.
Proof.
intros. induction H; repeat split; econstructor; try eapply IHstp2_1; try eapply IHstp2_2; try eapply index_extend; eauto; try eapply IHstp2; eauto.
Qed.
Lemma stpd2_extend : forall v1 G1 G2 T1 T2,
stpd2 G1 T1 G2 T2 ->
stpd2 (v1::G1) T1 G2 T2 /\
stpd2 G1 T1 (v1::G2) T2 /\
stpd2 (v1::G1) T1 (v1::G2) T2.
Proof.
intros. repeat eu. repeat split; eexists; eapply stp2_extend; eauto.
Qed.
Lemma stp2_extend1 : forall v1 G1 G2 T1 T2 n, stp2 G1 T1 G2 T2 n -> stp2 (v1::G1) T1 G2 T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stp2_extend2 : forall v1 G1 G2 T1 T2 n, stp2 G1 T1 G2 T2 n -> stp2 G1 T1 (v1::G2) T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stpd2_extend1 : forall v1 G1 G2 T1 T2, stpd2 G1 T1 G2 T2 -> stpd2 (v1::G1) T1 G2 T2.
Proof. intros. eapply stpd2_extend. eauto. Qed.
Lemma stpd2_extend2 : forall v1 G1 G2 T1 T2, stpd2 G1 T1 G2 T2 -> stpd2 G1 T1 (v1::G2) T2.
Proof. intros. eapply stpd2_extend. eauto. Qed.
Lemma stpd2_closed : forall G1 G2 T1 T2,
stpd2 G1 T1 G2 T2 ->
closed (length G1) T1 /\
closed (length G2) 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.
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 G1 T1 G1 T1.
Proof.
intros. induction T1; inversion H.
- Case "bool". eapply stpd2_bool; eauto.
- Case "fun". eapply stpd2_fun; eauto.
- Case "var".
assert (exists v, index i G1 = Some v) as E. eapply index_exists; eauto.
destruct E.
eapply stpd2_varx; eauto.
Qed.
Lemma stpd2_reg1 : forall G1 G2 T1 T2,
stpd2 G1 T1 G2 T2 ->
stpd2 G1 T1 G1 T1.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed G1 G2). eauto.
Qed.
Lemma stpd2_reg2 : forall G1 G2 T1 T2,
stpd2 G1 T1 G2 T2 ->
stpd2 G2 T2 G2 T2.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed G1 G2). eauto.
Qed.
Lemma stp2_trans: forall n, forall G1 G2 G3 T1 T2 T3 n1 n2,
stp2 G1 T1 G2 T2 n1 ->
stp2 G2 T2 G3 T3 n2 ->
(n1+n2) < n ->
stpd2 G1 T1 G3 T3.
Proof.
intros n. induction n; intros. solve by inversion.
inversion H.
- Case "bool". inversion H0; subst; try solve by inversion.
+ SCase "bool". eapply stpd2_bool; eauto.
- Case "fun". inversion H0; subst; try solve by inversion.
+ SCase "fun". inversion H12. subst. eapply stpd2_fun; eauto; try eapply IHn; eauto; try omega.
- Case "varx". inversion H0; subst; try solve by inversion.
+ SCase "varx". inversion H12. subst. rewrite H3 in H9. inversion H9. subst. eapply stpd2_varx; eauto.
+ SCase "var1". inversion H13. subst. rewrite H3 in H9. inversion H9. subst. eapply stpd2_var1; eauto.
- Case "var1".
eapply stpd2_var1; eauto. eapply IHn; eauto. omega.
Qed.
Lemma stpd2_trans: forall G1 G2 G3 T1 T2 T3,
stpd2 G1 T1 G2 T2 ->
stpd2 G2 T2 G3 T3 ->
stpd2 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 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 H6. inversion H6. eauto.
rewrite A. eauto.
+ SCase "var1".
assert (v = v0) as A. rewrite H2 in H4. inversion H4. eauto.
rewrite A. eapply IHn; eauto. omega.
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 GX T1 GX T2.
Proof.
intros G1 T1 T2 H. induction H; intros.
- Case "bool". eapply stpd2_bool; eauto.
- Case "fun". eapply stpd2_fun; 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 H1. destruct H2.
eapply stpd2_var1. eauto. eauto. eauto.
Qed.
Lemma valtp_widen: forall vf H1 H2 T1 T2,
val_type H1 vf T1 ->
stpd2 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 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 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 venv T1 env T3 /\
stpd2 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. repeat eexists; 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.
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 venv T1 env T3 /\
stpd2 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_reg2; eauto. 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.