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singletons1.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 *)
(* small-step style !! *)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Le.
Module STLC.
Definition id := nat.
Inductive ty : Type :=
| TBot : ty
| TTop : ty
| TBool : ty
| TFun : ty -> ty -> ty
| TVar : id -> ty
| TVarB : id -> ty
| TMem : ty -> ty -> ty (* intro *)
| TSel : ty -> ty (* elim *)
| TBind : ty -> 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.
Definition aenv := list (venv*ty).
Hint Unfold venv.
Hint Unfold tenv.
Hint Unfold aenv.
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_bot: forall j k,
closed j k TBot
| cl_top: forall j k,
closed j k TTop
| 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_bind: forall j k T1,
closed j (S k) T1 ->
closed j k (TBind T1)
.
Fixpoint open (k: nat) (u: ty) (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 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)
| TBind T1 => TBind (open (S k) u T1)
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)
(* bind! *)
.
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: nat -> bool -> aenv -> venv -> ty -> venv -> ty -> nat -> Prop :=
| stp2_bot: forall m GH G1 G2 T n1,
closed (length G2) 0 T ->
stp2 m true GH G1 TBot G2 T (S n1)
| stp2_top: forall m GH G1 G2 T n1,
closed (length G1) 0 T ->
stp2 m true GH G1 T G2 TTop (S n1)
| stp2_bool: forall m GH G1 G2 n1,
stp2 m true GH G1 TBool G2 TBool (S n1)
| stp2_fun: forall m GH G1 G2 T1 T2 T3 T4 n1 n2,
stp2 m false GH G2 T3 G1 T1 n1 ->
stp2 m false GH G1 T2 G2 T4 n2 ->
stp2 m true GH G1 (TFun T1 T2) G2 (TFun T3 T4) (S (n1+n2))
| stp2_varx: forall m GH G1 G2 x1 x2 v n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 m true GH G1 (TVar x1) G2 (TVar x2) (S n1)
| stp2_var1: forall m GH 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 m true GH GX TX G2 T2 n1 ->
stp2 m true GH G1 (TVar x) G2 T2 (S n1)
| stp2_var1b: forall m GH G1 GX TX G2 x x2 T2 v n1,
index x G1 = Some v ->
index x2 G2 = Some v ->
val_type0 GX v TX -> (* slack for: val_type G2 v T2 *)
stp2 m true GH GX TX G2 (open 0 (TVar x2) T2) n1 ->
stp2 m true GH G1 (TVar x) G2 (TBind T2) (S n1)
| stp2_mem: forall m GH G1 G2 T1 T2 T3 T4 n1 n2,
stp2 m false GH G2 T3 G1 T1 n2 ->
stp2 m true GH G1 T2 G2 T4 n1 ->
stp2 m true GH G1 (TMem T1 T2) G2 (TMem T3 T4) (S (n1+n2))
| stp2_sel: forall m GH G1 G2 T1 T2 n1 n2,
stp2 m false GH G2 T2 G1 T1 n2 ->
stp2 m true GH G1 T1 G2 T2 n1 ->
stp2 m true GH G1 (TSel T1) G2 (TSel T2) (S (n1+n2))
| stp2_red: forall m GH G1 G2 T1 T2 n1,
stp2 m true GH G1 T1 G2 (TMem TBot T2) n1 ->
stp2 m true GH G1 (TSel T1) G2 T2 (S n1)
| stp2_red2: forall m GH 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 m true GH GY T0 G2 T2 n0 ->
stp2 m false GH G2 T2 GX (TMem TX TTop) n1 ->
stp2 m true GH GY T0 GX (TMem TX TTop) n3 ->
stp2 m false GH G1 T1 GX TX n2 ->
stp2 m true GH G1 T1 G2 (TSel T2) (S (n0+n1+n2+n3))
(*
| stp2_bindI: 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 (open 0 (TVar x) T2) n1 ->
stp2 true G1 (TVar x) G2 (TBind T2) (S n1)
*)
(*
| stp2_bindE: forall G1 G2 T2 x n1,
stp2 true G1 (TVar x) G2 (TBind T2) n1 ->
stp2 true G1 (TVar x) G2 (open 0 (TVar x) T2) (S n1)
*)
| stp2_bind2: forall m GH G1 G2 T1 T2 v n1,
(* issue here: we want x.this < { z => T2 }
and z:x.this |- *)
closed (length G1) 0 T1 ->
val_type0 G1 v T1 ->
stp2 m true GH G1 T1 (v::G2) (TVar (length G2)) n1 ->
stp2 m true GH G1 T1 (v::G2) (open 0 (TVar (length G2)) T2) n1 ->
stp2 m true GH G1 T1 G2 (TBind T2) (S n1)
| stp2_bind1: forall m GH G1 G2 T1 T2 v n1,
closed (length G2) 0 T2 ->
val_type0 G1 v (open 0 (TVar (length G1)) T1) ->
stp2 m true GH (v::G1) (TVar (length G1)) G2 T2 n1 ->
stp2 m true GH G1 (TBind T1) G2 T2 (S n1)
(*
x < T^x
----------------
x < { z => T^z }
T < { z => T^z } T
----------------
x < T^x
*)
| stp2_wrapf: forall m GH G1 G2 T1 T2 n1,
stp2 m true GH G1 T1 G2 T2 n1 ->
stp2 m false GH G1 T1 G2 T2 (S n1)
| stp2_transf: forall m GH G1 G2 G3 T1 T2 T3 n1 n2,
stp2 m true GH G1 T1 G2 T2 n1 ->
stp2 m false GH G2 T2 G3 T3 n2 ->
stp2 m false GH 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 0 true [] G1 T1 G2 T2 n) ->
val_type G2 v T2
.
Definition stpd2 m b GH G1 T1 G2 T2 := exists n, stp2 m b GH G1 T1 G2 T2 n.
Hint Constructors stp2.
Ltac ep := match goal with
| [ |- stp2 ?M ?B ?GH ?G1 ?T1 ?G2 ?T2 ?N ] => assert (exists (n:nat), stp2 M B GH G1 T1 G2 T2 n) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ _ _ _ |- _ => destruct H as [? H]
end.
Lemma stpd2_bot: forall m GH G1 G2 T,
closed (length G2) 0 T ->
stpd2 m true GH G1 TBot G2 T.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_top: forall m GH G1 G2 T,
closed (length G1) 0 T ->
stpd2 m true GH G1 T G2 TTop.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_bool: forall m GH G1 G2,
stpd2 m true GH G1 TBool G2 TBool.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_fun: forall m GH G1 G2 T1 T2 T3 T4,
stpd2 m false GH G2 T3 G1 T1 ->
stpd2 m false GH G1 T2 G2 T4 ->
stpd2 m true GH G1 (TFun T1 T2) G2 (TFun T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_varx: forall m GH G1 G2 x1 x2 v,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stpd2 m true GH G1 (TVar x1) G2 (TVar x2).
Proof. intros. repeat eu. exists 1. eauto. Qed.
Lemma stpd2_var1: forall m GH 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 m true GH GX TX G2 T2 ->
stpd2 m true GH G1 (TVar x) G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_var1b: forall m GH G1 GX TX G2 x x2 T2 v,
index x G1 = Some v ->
index x2 G2 = Some v ->
val_type0 GX v TX -> (* slack for: val_type G2 v T2 *)
stpd2 m true GH GX TX G2 (open 0 (TVar x2) T2) ->
stpd2 m true GH G1 (TVar x) G2 (TBind T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_mem: forall m GH G1 G2 T1 T2 T3 T4,
stpd2 m false GH G2 T3 G1 T1 ->
stpd2 m true GH G1 T2 G2 T4 ->
stpd2 m true GH G1 (TMem T1 T2) G2 (TMem T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_sel: forall m GH G1 G2 T1 T2,
stpd2 m false GH G2 T2 G1 T1 ->
stpd2 m true GH G1 T1 G2 T2 ->
stpd2 m true GH G1 (TSel T1) G2 (TSel T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red: forall m GH G1 G2 T1 T2,
stpd2 m true GH G1 T1 G2 (TMem TBot T2) ->
stpd2 m true GH G1 (TSel T1) G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red2: forall m GH G1 G2 GX TX T1 T2 GY T0,
real T0 ->
stpd2 m true GH GY T0 G2 T2 ->
stpd2 m false GH G2 T2 GX (TMem TX TTop) ->
stpd2 m true GH GY T0 GX (TMem TX TTop) ->
stpd2 m false GH G1 T1 GX TX ->
stpd2 m true GH G1 T1 G2 (TSel T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
(*
Lemma stpd2_bindI: forall G1 G2 T2 x,
stpd2 true G1 (TVar x) G2 (open 0 (TVar x) T2) ->
stpd2 true G1 (TVar x) G2 (TBind T2).
Proof. intros. repeat eu. eexists. eauto. Qed. *
Lemma stpd2_bindE: forall G1 G2 T2 x,
stpd2 true G1 (TVar x) G2 (TBind T2) ->
stpd2 true G1 (TVar x) G2 (open 0 (TVar x) T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
*)
Lemma stpd2_bind1: forall m GH G1 G2 T1 T2 v,
closed (length G2) 0 T2 ->
val_type0 G1 v (open 0 (TVar (length G1)) T1) ->
stpd2 m true GH (v::G1) (TVar (length G1)) G2 T2 ->
stpd2 m true GH G1 (TBind T1) G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_wrapf: forall m GH G1 G2 T1 T2,
stpd2 m true GH G1 T1 G2 T2 ->
stpd2 m false GH G1 T1 G2 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_transf: forall m GH G1 G2 G3 T1 T2 T3,
stpd2 m true GH G1 T1 G2 T2 ->
stpd2 m false GH G2 T2 G3 T3 ->
stpd2 m false GH 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 T X (a:X) k G,
closed (length G) k T ->
closed (length (a::G)) k T.
Proof.
intros T. induction T; intros; inversion H; econstructor; eauto.
simpl. omega.
Qed.
Lemma stp2_extend : forall m b GH v1 G1 G2 T1 T2 n,
stp2 m b GH G1 T1 G2 T2 n ->
stp2 m b GH (v1::G1) T1 G2 T2 n /\
stp2 m b GH G1 T1 (v1::G2) T2 n /\
stp2 m b GH (v1::G1) T1 (v1::G2) T2 n.
Proof.
intros. induction H; try solve [repeat split; econstructor; try eauto;
try eapply index_extend; eauto; try eapply closed_extend; eauto;
try eapply IHstp2; eauto;
try eapply IHstp2_1; try eapply IHstp2_2;
try eapply IHstp2_3; try eapply IHstp2_4].
repeat split; eapply stp2_var1b. eapply index_extend; eauto. eauto. eauto.
eauto. eauto. eapply index_extend; eauto. eauto. eapply IHstp2. eapply index_extend; eauto.
eapply index_extend; eauto. eauto.
eapply IHstp2.
admit. (* bind1 *)
admit.
repeat split; eapply stp2_transf; try eapply IHstp2_1; eauto; try eapply IHstp2_2; eauto.
Qed.
Lemma stpd2_extend : forall m b GH v1 G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
stpd2 m b GH (v1::G1) T1 G2 T2 /\
stpd2 m b GH G1 T1 (v1::G2) T2 /\
stpd2 m b GH (v1::G1) T1 (v1::G2) T2.
Proof.
intros. repeat eu. repeat split; eexists; eapply stp2_extend; eauto.
Qed.
Lemma stp2_extend1 : forall m b GH v1 G1 G2 T1 T2 n, stp2 m b GH G1 T1 G2 T2 n -> stp2 m b GH (v1::G1) T1 G2 T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stp2_extend2 : forall m b GH v1 G1 G2 T1 T2 n, stp2 m b GH G1 T1 G2 T2 n -> stp2 m b GH G1 T1 (v1::G2) T2 n.
Proof. intros. eapply stp2_extend. eauto. Qed.
Lemma stpd2_extend1 : forall m b GH v1 G1 G2 T1 T2, stpd2 m b GH G1 T1 G2 T2 -> stpd2 m b GH (v1::G1) T1 G2 T2.
Proof. intros. eapply stpd2_extend. eauto. Qed.
Lemma stpd2_extend2 : forall m b GH v1 G1 G2 T1 T2, stpd2 m b GH G1 T1 G2 T2 -> stpd2 m b GH 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 b GH G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
closed (length G1) 0 T1 /\
closed (length G2) 0 T2.
Proof.
admit. (*
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 b GH G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
closed (length G1) 0 T1.
Proof. intros. eapply (stpd2_closed m b GH G1 G2); eauto. Qed.
Lemma stpd2_closed2 : forall m b GH G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
closed (length G2) 0 T2.
Proof. intros. eapply (stpd2_closed m b GH 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 m GH G1 T1,
closed (length G1) 0 T1 ->
stpd2 m true GH G1 T1 G1 T1.
Proof.
intros. induction T1; inversion H.
- Case "bot". exists 1. eauto.
- Case "top". exists 1. eauto.
- 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 "varb". inversion H3.
- Case "mem". eapply stpd2_mem; try eapply stpd2_wrapf; eauto.
- Case "sel". eapply stpd2_sel; try eapply stpd2_wrapf; eauto.
- Case "bind". admit. (* problem: need val_type *)
Qed.
Lemma stpd2_reg1 : forall m b GH G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
stpd2 m true GH G1 T1 G1 T1.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m b GH G1 G2). eauto.
Qed.
Lemma stpd2_reg2 : forall m b GH G1 G2 T1 T2,
stpd2 m b GH G1 T1 G2 T2 ->
stpd2 m true GH G2 T2 G2 T2.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m b GH G1 G2). eauto.
Qed.
(*
Lemma invert_bind1: forall n, forall venv vf T1 GX TX n1,
val_type0 GX vf TX -> stp2 true GX TX venv (TBind T1) n1 -> n1 < n ->
exists x n2,
index x venv = Some vf ->
n2 < n1 ->
stp2 true GX TX venv (open 0 (TVar x) T1) n2.
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 "normal".
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 GX0 TX n0 H5 H6 B).
ev. repeat eexists; eauto.
(* repeat eapply IHn; eauto. omega. *)
+ SCase "bindE". eauto.
+ eauto. *)
- Case "mem". solve by inversion.
Qed.
*)
Lemma invert_mem1: forall n, forall venv m GH vf T1 T2 GX TX n1,
val_type0 GX vf TX -> stp2 m true GH GX TX venv (TMem T1 T2) n1 -> n1 < n ->
exists env T3 n2 n3,
vf = (vty env T3) /\
(n2 + n3) < n /\
stp2 m false GH venv T1 env T3 n2 /\
stp2 m true GH 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.
+ SCase "normal".
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. *)
+ SCase "bindE".
destruct T0; simpl in H6; try destruct i; inversion H6.
*)
- 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 m GH G1 G2 G3 T1 T2 T3 n1,
stp2 m false GH G1 T1 G2 T2 n1 -> n1 < n ->
stpd2 m false GH G2 T2 G3 T3 ->
stpd2 m false GH G1 T1 G3 T3.
Proof.
intros n. induction n; intros; try omega; repeat eu; subst; inversion H; clear 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 b GH G1 G2 G3 T1 T2 T3,
stpd2 m b GH G1 T1 G2 T2 ->
stpd2 m false GH G2 T2 G3 T3 ->
stpd2 m false GH G1 T1 G3 T3.
Proof.
intros. destruct b; eu; eu; eapply stpd2_trans_axiom_aux; eauto.
Qed.
Ltac index_subst := match goal with
| H1: index ?x ?G = ?V1 , H2: index ?x ?G = ?V2 |- _ => rewrite H1 in H2; inversion H2; subst
| _ => idtac
end.
Ltac invty := match goal with
| H1: TBot = _ |- _ => inversion H1
| H1: TBool = _ |- _ => inversion H1
| H1: TSel _ = _ |- _ => inversion H1
| H1: TMem _ _ = _ |- _ => inversion H1
| H1: TVar _ = _ |- _ => inversion H1
| H1: TFun _ _ = _ |- _ => inversion H1
| H1: TBind _ = _ |- _ => inversion H1
| _ => idtac
end.
Ltac invstp_var := match goal with
| H1: stp2 _ true _ _ TBool _ (TVar _) _ |- _ => inversion H1
| H1: stp2 _ true _ _ (TFun _ _) _ (TVar _) _ |- _ => inversion H1
| H1: stp2 _ true _ _ (TMem _ _) _ (TVar _) _ |- _ => inversion H1
| H1: val_type0 _ _ _ |- _ => inversion H1
| _ => idtac
end.
Lemma stp2_trans: forall n, forall m b GH G1 G2 G3 T1 T2 T3 n1 n2,
stp2 m b GH G1 T1 G2 T2 n1 ->
stp2 m true GH G2 T2 G3 T3 n2 ->
n1 < n ->
stpd2 m true GH 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; invty.
+ SCase "top". eapply stpd2_top; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind2". invstp_var.
- Case "bool". inversion H0; subst; invty.
+ 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.
+ SCase "bind2". invstp_var.
- Case "fun". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "fun". invty. subst. eapply stpd2_fun; eapply stp2_trans_axiom; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind2". invstp_var.
- Case "varx". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "varx". invty. subst. index_subst. eapply stpd2_varx; eauto.
+ SCase "var1". invty. subst. index_subst. eapply stpd2_var1; eauto.
+ SCase "var1b". invty. subst. index_subst. eapply stpd2_var1b; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind2". invstp_var.
- Case "var1".
eapply stpd2_var1; eauto. eapply IHn; eauto. omega.
- Case "var1b". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind".
admit.
+ SCase "bind2".
admit.
- Case "mem". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "mem". 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.
+ SCase "bind2". invstp_var.
- Case "sel". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "sel". subst. eapply stpd2_sel. eapply stp2_trans_axiom; eauto. eapply IHn; eauto; try omega.
+ SCase "red". subst. eapply stpd2_red. eapply IHn. eauto. eauto. omega.
+ SCase "red2". subst. eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind2". invstp_var.
- Case "red".
eapply stpd2_red. eauto. eapply IHn; eauto. repeat econstructor. eauto. omega.
- Case "red2".
inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "sel". subst.
eapply stpd2_red2. eauto.
eapply IHn. eapply H3. eauto. omega. eapply stp2_trans_axiom; eauto. eauto. eauto.
+ SCase "red". clear H. subst.
(* pull this out as stp2_trans_cross lemma? *)
assert (stp2 m true GH GY T5 G2 T4 n0) as A0. eauto.
assert (stp2 m true GH GY T5 GX (TMem TX TTop) n5) as A1. eauto.
assert (stpd2 m true GH 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".
index_subst.
assert (exists GX1 TX1 m1 m2, v0 = (vty GX1 TX1) /\ m1 + m2 < S n1 /\
stp2 m false GH GX TX GX1 TX1 m1 /\ stp2 m true GH GX1 TX1 GX TTop m2) as B1.
eapply invert_mem1. eapply H7. eauto. eauto.
assert (exists GX2 TX2 m3 m4, v0 = (vty GX2 TX2) /\ m3 + m4 < S n2 /\
stp2 m false GH G3 TBot GX2 TX2 m3 /\ stp2 m true GH GX2 TX2 G3 T3 m4) as B2.
eapply invert_mem1. eapply H18. eauto. eauto.
destruct B1 as [GZ1 [TZ1 [m1 [m2 [V1 [M1 [B1 _]]]]]]].
destruct B2 as [GZ2 [TZ2 [m3 [m4 [V2 [M2 [_ B2 ]]]]]]].
subst v0. inversion V2. subst.
assert (stpd2 m true GH GX TX G3 T3) as S3. eapply IHn. eapply B1. eapply B2. omega. eu.
assert (stpd2 m true GH G1 T1 G3 T3) as S1. eapply IHn. eauto. eapply S3. omega.
eapply S1.
* SSCase "mem".
assert (stp2 m false GH GX TX GY T n2) as B1. eauto.
assert (stp2 m true GH GY T G3 T3 n7) as B2. eauto.
assert (stpd2 m true GH GX TX G3 T3) as S3. eapply IHn. eapply B1. eapply B2. omega. eu.
assert (stpd2 m true GH 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.
+ SCase "bind2". invstp_var.
- Case "bind2".
inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto.
+ SCase "red2". eapply stpd2_red2; eauto.
+ SCase "bind2". invstp_var.
+ SCase "bind".
(* cross, between v and v0 *)
assert (stp2 m true GH G1 T1 (v :: G2) (TVar (length G2)) n0) as ST. assumption.
(* to be narrowed *)
assert (stp2 m true GH (v0 :: G2) (TVar (length G2)) G3 T3 n3) as A1. assumption.
(* narrowed *)
assert (stp2 m true GH (v :: G2) (TVar (length G2)) G3 T3 n3) as A2. admit.
(* result *)
eapply IHn. eauto. eauto. omega.
- Case "bind1". subst.
eapply stpd2_bind1. eapply stpd2_closed2; eauto. eauto. eapply IHn; eauto. omega.
- Case "wrapf".
subst. eapply IHn. eapply H2. eapply H0. omega.
- Case "transf".
assert (stpd2 m true GH G4 T4 G3 T3). eapply IHn. eapply H3. eapply H0. omega.
eu. subst. eapply IHn. eapply H2. eauto. omega.
Grab Existential Variables.
apply 0. apply 0. apply []. apply 0.
Qed.
(* idea: can we use the same strong_sel as before, and just translate the new rules? *)
Lemma stpd2_trans: forall m b GH G1 G2 G3 T1 T2 T3,
stpd2 m b GH G1 T1 G2 T2 ->
stpd2 m true GH G2 T2 G3 T3 ->
stpd2 m true GH 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.