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singletons.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.Lt.
Module STLC.
Definition id := nat.
Inductive ty : Type :=
| TBot : ty
| TTop : ty
| TBool : ty
| TFun : ty -> ty -> ty
| TVar : bool -> 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 -> nat -> ty -> Prop :=
| cl_bot: forall i j k,
closed i j k TBot
| cl_top: forall i j k,
closed i j k TTop
| cl_bool: forall i j k,
closed i j k TBool
| cl_fun: forall i j k T1 T2,
closed i j k T1 ->
closed i j k T2 ->
closed i j k (TFun T1 T2)
| cl_var0: forall i j k x,
i > x ->
closed i j k (TVar false x)
| cl_var1: forall i j k x,
j > x ->
closed i j k (TVar true x)
| cl_varB: forall i j k x,
k > x ->
closed i j k (TVarB x)
| cl_mem: forall i j k T1 T2,
closed i j k T1 ->
closed i j k T2 ->
closed i j k (TMem T1 T2)
| cl_sel: forall i j k T1,
closed i j k T1 ->
closed i j k (TSel T1)
| cl_bind: forall i j k T1,
closed i j (S k) T1 ->
closed i j k (TBind T1)
.
Fixpoint open (k: nat) (u: ty) (T: ty) { struct T }: ty :=
match T with
| TVar b x => TVar b 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
| 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.
Fixpoint subst (k: nat) (U : ty) (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TBot => TBot
| TBool => TBool
| TMem T1 T2 => TMem (subst k U T1) (subst k U T2)
| TSel T1 => TSel (subst k U T1)
| TVarB i => TVarB i
| TVar true i => TVar true i
| TVar false i =>
match nat_compare i k with
| Lt => TVar false i
| Eq => U
| Gt => TVar false (i-1)
end
| TFun T1 T2 => TFun (subst k U T1) (subst k U T2)
| TBind T2 => TBind (subst k U T2)
end.
(*
Fixpoint nosubst (T : ty) {struct T} : Prop :=
match T with
| TTop => True
| TBot => True
| TBool => True
| TMem m T1 T2 => nosubst T1 /\ nosubst T2
| TSel (varB i) m => True
| TSel (varF i) m => True
| TSel (varH i) m => i <> 0
| TAll m T1 T2 => nosubst T1 /\ nosubst T2
| TBind T2 => nosubst T2
| TAnd T1 T2 => nosubst T1 /\ nosubst T2
| TOr T1 T2 => nosubst T1 /\ nosubst T2
end.
*)
Inductive real: ty -> Prop :=
| real_var: forall b x,
real (TVar b 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 false x) (TVar false x)
| stp_vary: forall G1 x y,
index x G1 = Some (TVar false y) ->
y < length G1 ->
stp G1 (TVar false y) (TVar false x)
| stp_var1: forall G1 x T1,
index x G1 = Some T1 ->
closed (length G1) 0 0 T1 ->
stp G1 (TVar false 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 false 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 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 -> tenv -> venv -> ty -> ty -> nat -> Prop :=
| stp2_bot: forall m GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp2 m true GH G1 TBot T (S n1)
| stp2_top: forall m GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp2 m true GH G1 T TTop (S n1)
| stp2_bool: forall m GH G1 n1,
stp2 m true GH G1 TBool TBool (S n1)
| stp2_fun: forall m GH G1 T1 T2 T3 T4 n1 n2,
stp2 m false GH G1 T3 T1 n1 ->
stp2 m false GH G1 T2 T4 n2 ->
stp2 m true GH G1 (TFun T1 T2) (TFun T3 T4) (S (n1+n2))
| stp2_varx: forall m GH G1 x n1,
x < length G1 ->
stp2 m true GH G1 (TVar true x) (TVar true x) (S n1)
| stp2_var1: forall m GH G1 TX x T2 v n1,
index x G1 = Some v ->
val_type0 G1 v TX -> (* slack for: val_type G2 v T2 *)
stp2 m true GH G1 TX T2 n1 ->
stp2 m true GH G1 (TVar true x) T2 (S n1)
| stp2_var1b: forall m GH G1 TX x T2 v n1 n2,
index x G1 = Some v ->
val_type0 G1 v TX -> (* slack for: val_type G2 v T2 *)
stp2 0 true ((TVar true x)::GH) G1 TX (open 0 (TVar false (length GH)) T2) n1 ->
stp2 0 true GH G1 TX (open 0 (TVar true x) T2) n2 ->
stp2 (S m) true GH G1 (TVar true x) (TBind T2) (S (n1+n2))
| stp2_varax: forall m GH G1 x n1,
x < length GH ->
stp2 m true GH G1 (TVar false x) (TVar false x) (S n1)
| stp2_varay: forall m GH G1 TX x1 x2 n1,
index x1 GH = Some TX ->
stp2 m false GH G1 TX (TVar false x2) n1 ->
stp2 m true GH G1 (TVar false x2) (TVar false x1) (S n1)
| stp2_vary: forall m GH G1 x1 x2 n1,
index x1 GH = Some (TVar true x2) ->
x2 < length G1 ->
stp2 m true GH G1 (TVar true x2) (TVar false x1) (S n1)
| stp2_vara1: forall m GH G1 TX x T2 n1,
index x GH = Some TX ->
stp2 m true GH G1 TX T2 n1 ->
stp2 m true GH G1 (TVar false x) T2 (S n1)
| stp2_mem: forall m GH G1 T1 T2 T3 T4 n1 n2,
stp2 m false GH G1 T3 T1 n2 ->
stp2 m true GH G1 T2 T4 n1 ->
stp2 m true GH G1 (TMem T1 T2) (TMem T3 T4) (S (n1+n2))
| stp2_sel: forall m GH G1 T1 T2 n1 n2,
stp2 m false GH G1 T2 T1 n2 ->
stp2 m true GH G1 T1 T2 n1 ->
stp2 m true GH G1 (TSel T1) (TSel T2) (S (n1+n2))
| stp2_red: forall m GH G1 T1 T2 n1,
stp2 m true GH G1 T1 (TMem TBot T2) n1 ->
stp2 m true GH G1 (TSel T1) T2 (S n1)
| stp2_red2: forall m GH G1 TX T1 T2 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 G1 T0 T2 n0 ->
stp2 m false GH G1 T2 (TMem TX TTop) n1 ->
stp2 m true GH G1 T0 (TMem TX TTop) n3 ->
stp2 m false GH G1 T1 TX n2 ->
stp2 m true GH G1 T1 (TSel T2) (S (n0+n1+n2+n3))
| stp2_bind1: forall m GH G1 T1 T2 n1,
stp2 m true ((open 0 (TVar false (length GH)) T1)::GH) G1 (TVar false (length GH)) T2 n1 ->
stp2 (S m) true GH G1 (TBind T1) T2 (S n1)
| stp2_bind2: forall m GH G1 TX x T2 n1 n2,
index x GH = Some TX ->
stp2 0 true ((TVar false x)::GH) G1 TX (open 0 (TVar false (length GH)) T2) n1 ->
stp2 0 true GH G1 TX (open 0 (TVar false x) T2) n2 ->
stp2 (S m) true GH G1 (TVar false x) (TBind T2) (S (n1+n2))
| stp2_wrapf: forall m GH G1 T1 T2 n1,
stp2 m true GH G1 T1 T2 n1 ->
stp2 m false GH G1 T1 T2 (S n1)
| stp2_transf: forall m GH G1 T1 T2 T3 n1 n2,
stp2 m true GH G1 T1 T2 n1 ->
stp2 m false GH G1 T2 T3 n2 ->
stp2 m false GH G1 T1 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 T2 v,
val_type0 G1 v T1 ->
(exists n, stp2 0 true [] G1 T1 T2 n) ->
val_type G1 v T2
.
Definition stpd2 m b GH G1 T1 T2 := exists n, stp2 m b GH G1 T1 T2 n.
Hint Constructors stp2.
Ltac ep := match goal with
| [ |- stp2 ?M ?B ?GH ?G1 ?T1 ?T2 ?N ] => assert (exists (n:nat), stp2 M B GH G1 T1 T2 n) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ _ _ |- _ => destruct H as [? H]
end.
Lemma stpd2_bot: forall m GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd2 m true GH G1 TBot T.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_top: forall m GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd2 m true GH G1 T TTop.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_bool: forall m GH G1,
stpd2 m true GH G1 TBool TBool.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_fun: forall m GH G1 T1 T2 T3 T4,
stpd2 m false GH G1 T3 T1 ->
stpd2 m false GH G1 T2 T4 ->
stpd2 m true GH G1 (TFun T1 T2) (TFun T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_varx: forall m GH G1 x,
x < length G1 ->
stpd2 m true GH G1 (TVar true x) (TVar true x).
Proof. intros. repeat eu. exists 1. eauto. Qed.
Lemma stpd2_var1: forall m GH G1 TX x T2 v,
index x G1 = Some v ->
val_type0 G1 v TX -> (* slack for: val_type G2 v T2 *)
stpd2 m true GH G1 TX T2 ->
stpd2 m true GH G1 (TVar true x) 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 true x2) T2) ->
stpd2 m true GH G1 (TVar true x) G2 (TBind T2).
Proof. intros. repeat eu. eexists. eauto. Qed.*)
Lemma stpd2_mem: forall m GH G1 T1 T2 T3 T4,
stpd2 m false GH G1 T3 T1 ->
stpd2 m true GH G1 T2 T4 ->
stpd2 m true GH G1 (TMem T1 T2) (TMem T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_sel: forall m GH G1 T1 T2,
stpd2 m false GH G1 T2 T1 ->
stpd2 m true GH G1 T1 T2 ->
stpd2 m true GH G1 (TSel T1) (TSel T2).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red: forall m GH G1 T1 T2,
stpd2 m true GH G1 T1 (TMem TBot T2) ->
stpd2 m true GH G1 (TSel T1) T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_red2: forall m GH G1 TX T1 T2 T0,
real T0 ->
stpd2 m true GH G1 T0 T2 ->
stpd2 m false GH G1 T2 (TMem TX TTop) ->
stpd2 m true GH G1 T0 (TMem TX TTop) ->
stpd2 m false GH G1 T1 TX ->
stpd2 m true GH G1 T1 (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 T1 T2,
closed (length GH) (length G2) 0 T2 ->
stpd2 m true ((G1,(open 0 (TVar false (length GH)) T1))::GH)
G1 (TVar false (length GH)) 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 T1 T2,
stpd2 m true GH G1 T1 T2 ->
stpd2 m false GH G1 T1 T2.
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_transf: forall m GH G1 T1 T2 T3,
stpd2 m true GH G1 T1 T2 ->
stpd2 m false GH G1 T2 T3 ->
stpd2 m false GH G1 T1 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) i k G,
closed i (length G) k T ->
closed i (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.
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 0 T1 /\
closed (length G1) 0 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 T1 T2,
stpd2 m b GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T1 /\
closed (length GH) (length G1) 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 T1 T2,
stpd2 m b GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T1.
Proof. intros. eapply (stpd2_closed m b GH G1 T1 T2); eauto. Qed.
Lemma stpd2_closed2 : forall m b GH G1 T1 T2,
stpd2 m b GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T2.
Proof. intros. eapply (stpd2_closed m b GH G1); eauto. Qed.
Lemma valtp_extend : forall vs v v1 T,
val_type vs v T ->
val_type (v1::vs) v T.
Proof.
admit.
(*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 GH) (length G1) 0 T1 ->
stpd2 m true GH G1 T1 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 "var0". exists 1. eauto.
- Case "var1".
assert (exists v, index i G1 = Some v) as E. eapply index_exists; eauto.
destruct E.
eapply stpd2_varx; eauto.
- Case "varb". inversion H4.
- Case "mem". eapply stpd2_mem; try eapply stpd2_wrapf; eauto.
- Case "sel". eapply stpd2_sel; try eapply stpd2_wrapf; eauto.
- Case "bind".
admit.
(* don't have index & val_type0 evidence *)
(*exists 3. eapply stp2_bind1. eauto.
eapply stp2_bind2.
simpl. eauto.
eapply stp2_vara1. simpl. rewrite <-beq_nat_refl. eauto. *)
(* TODO:
stp2 m true ((G1, open 0 (TVar false (length GH)) T1) :: GH) G1
(open 0 (TVar false (length GH)) T1) G1
(open 0 (TVar false (length GH)) T1) 0
*)
Qed.
Lemma stpd2_reg1 : forall m b GH G1 T1 T2,
stpd2 m b GH G1 T1 T2 ->
stpd2 m true GH G1 T1 T1.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m b GH G1 T1 T2). eauto.
Qed.
Lemma stpd2_reg2 : forall m b GH G1 T1 T2,
stpd2 m b GH G1 T1 T2 ->
stpd2 m true GH G1 T2 T2.
Proof.
intros. eapply stpd2_refl. eapply (stpd2_closed m b GH G1). 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 m venv GH vf T1 T2 GX TX n1,
val_type0 GX vf TX -> stp2 m true GH GX TX (TMem T1 T2) n1 -> n1 < n ->
exists T3 n2 n3,
vf = (vty [] T3) /\
(n2 + n3) < n /\
stp2 m false GH venv T1 T3 n2 /\
stp2 m true GH venv T3 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.
*)
- admit.
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 T1 T2 T3 n1,
stp2 m false GH G1 T1 T2 n1 -> n1 < n ->
stpd2 m false GH G1 T2 T3 ->
stpd2 m false GH G1 T1 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 T1 T2 T3,
stpd2 m b GH G1 T1 T2 ->
stpd2 m false GH G1 T2 T3 ->
stpd2 m false GH G1 T1 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 _ _ TBot (TVar _ _) _ |- _ => inversion H1
| H1: stp2 _ true _ _ TTop (TVar _ _) _ |- _ => inversion H1
| 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_substitute_aux: forall ni nj, forall d m G1 G2 T1 T2 GH n1,
stp2 (S d) m G1 T1 G2 T2 GH n1 -> n1 < nj ->
forall GH0 GH0' GX TX TX' l T1' T2' V,
GH = (GH0 ++ [(0,(GX, TX))]) ->
val_type GX V (subst TX' TX) ni ->
(* When we're replacing binds from a pack/unpack sequence, the
type in GH may refer to itself (contain TSelH 0).
It should be safe for TX to refer to itself. *)
closed 0 1 TX ->
closed 0 0 (TSel TX' l) ->
compat GX TX TX' (Some V) G1 T1 T1' ->
compat GX TX TX' (Some V) G2 T2 T2' ->
Forall2 (compat2 GX TX TX' (Some V)) GH0 GH0' ->
compat GX TX TX' (Some V) GX TX (subst TX' TX) ->
exists n1', stp2 (S d) m G1 T1' G2 T2' GH0' n1'.
*)
Lemma closed_no_subst: forall T i j k TX,
closed i j k T ->
subst i TX T = T.
Proof.
admit.
(* intros T. induction T; intros; inversion H; simpl; eauto;
try rewrite (IHT (S j) TX); eauto;
try rewrite (IHT2 (S j) TX); eauto;
try rewrite (IHT j TX); eauto;
try rewrite (IHT1 j TX); eauto;
try rewrite (IHT2 j TX); eauto.
eapply closed_upgrade. eauto. eauto.
subst. omega.
subst. eapply closed_upgrade. eassumption. omega.
subst. eapply closed_upgrade. eassumption. omega. *)
Qed.
Lemma stp2_subst: forall m b GH G1 T1 T2 TX n1,
stp2 m b (TX::GH) G1 T1 T2 n1 ->
stpd2 m b GH G1 (subst (length GH) TX T1) (subst (length GH) TX T2).
Proof.
admit.
Qed.
Lemma stp2_narrow: forall m b GH GX1 TX1 TX2 G1 T1 T2 n1 n2,
stp2 0 true (TX1::GH) GX1 TX1 TX2 n1 ->
stp2 m b (TX2::GH) G1 T1 T2 n2 ->
stpd2 m b (TX1::GH) G1 T1 T2.
Proof.
admit.
Qed.
Lemma stp2_downgrade: forall m b GH G1 T1 T2 n1,
stp2 m b GH G1 T1 T2 n1 ->
stp2 (S m) b GH G1 T1 T2 n1.
Proof.
admit.
Qed.
Lemma stp2_trans: forall l, forall n, forall m b GH G1 T1 T2 T3 n1 n2,
stp2 m b GH G1 T1 T2 n1 ->
stp2 m true GH G1 T2 T3 n2 ->
n1 < n -> m < l ->
stpd2 m true GH G1 T1 T3.
Proof.
intros l. induction l. intros. solve by inversion.
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.
+ 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; 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.
- Case "varx". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto.
+ SCase "varx". eauto.
+ SCase "var1". eauto.
+ SCase "var1b". eauto.
+ SCase "vary". 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 "var1b". inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top; eauto. eapply stpd2_closed1; eauto.
+ SCase "red2". eapply stpd2_red2. eauto. eauto. eauto. eauto. eapply stp2_trans_axiom; eauto.
+ SCase "bind1".
inversion H15. subst.
assert (stpd2 m0 true (TVar true x::GH) G1
(TVar false (length GH)) T3) as NRW.
eapply stp2_narrow. eapply stp2_var1. eauto. eauto. eauto. eauto.
(* result may be false but we can call untrans on it using outer induction on m! *)
assert (closed (length GH) (length G1) 0 T3). eapply stpd2_closed2; eauto.
assert (closed (length GH) (length G1) 0 (TVar true x)). eapply stpd2_closed1; eauto.
destruct NRW as [? NRW]. eapply stp2_subst in NRW.
simpl in NRW. assert (nat_compare (length GH) (length GH) = Eq). eapply nat_compare_eq_iff. eauto. rewrite H9 in NRW.
rewrite (closed_no_subst _ (length GH) (length G1) 0) in NRW.
destruct NRW as [? NRW].
eexists. eapply stp2_downgrade. eauto. eauto.
- Case "varax". subst. inversion H0; subst; invty; eauto.
- Case "varay". subst. inversion H0; subst; invty.
+ SCase "top". eapply stpd2_top. eapply stpd2_closed1; eauto.
+ SCase "varax". eauto.