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fsub4.v
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(* Full safety for F-sub *)
(* values well-typed with respect to runtime environment *)
(* inversion lemma structure *)
(* this version adds bottom and lower bounds to fsub0.v *)
(* it also turns types into proper first class objects, *)
(* compared to fsub1.v *)
(* this version adds mutable references to fsub2.v *)
(*
compared to fsub3.v, it does NOT have the limitation:
the underlying type of a mutable reference must be well-formed in empty env
*)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Le.
Module FSUB.
Definition id := nat.
Inductive ty : Type :=
| TBool : ty
| TBot : ty
| TTop : ty
| TFun : ty -> ty -> ty
| TMem : ty -> ty -> ty
| TCell : ty -> ty
| TSel : id -> ty
| TSelH : id -> ty
| TSelB : id -> ty
| TAll : ty -> ty -> ty
.
Inductive tm : Type :=
| ttrue : tm
| tfalse : tm
| tvar : id -> tm
| tnew : tm -> tm
| tget : tm -> tm
| tset : tm -> tm -> tm
| ttyp : ty -> tm
| tapp : tm -> tm -> tm (* f(x) *)
| tabs : id -> id -> tm -> tm (* \f x.y *)
| ttapp : tm -> tm -> tm (* f[X] *)
| ttabs : id -> ty -> tm -> tm (* \f x.y *)
.
Inductive vl : Type :=
| vty : list (id*vl) -> ty -> vl
| vbool : bool -> vl
| vloc : id -> vl
| vabs : list (id*vl) -> id -> id -> tm -> vl
| vtabs : list (id*vl) -> id -> ty -> tm -> vl
.
Definition tenv := list (id*ty).
Definition venv := list (id*vl).
Definition aenv := list (id*(venv*ty)).
Hint Unfold venv.
Hint Unfold tenv.
Fixpoint fresh {X: Type} (l : list (id * X)): nat :=
match l with
| [] => 0
| (n',a)::l' => 1 + n'
end.
Fixpoint index {X : Type} (n : id) (l : list (id * X)) : option X :=
match l with
| [] => None
| (n',a) :: l' =>
if le_lt_dec (fresh l') n' then
if (beq_nat n n') then Some a else index n l'
else None
end.
Fixpoint indexr {X : Type} (n : id) (l : list (id * X)) : option X :=
match l with
| [] => None
| (n',a) :: l' => (* DeBrujin *)
if (beq_nat n (length l')) then Some a else indexr n l'
end.
Fixpoint update {X : Type} (n : nat) (x: X)
(l : list X) { struct l }: list X :=
match l with
| [] => []
| a :: l' => if beq_nat n (length l') then x::l' else a :: update n x l'
end.
(* LOCALLY NAMELESS *)
Inductive closed_rec: nat -> nat -> ty -> Prop :=
| cl_top: forall k l,
closed_rec k l TTop
| cl_bot: forall k l,
closed_rec k l TBot
| cl_bool: forall k l,
closed_rec k l TBool
| cl_fun: forall k l T1 T2,
closed_rec k l T1 ->
closed_rec k l T2 ->
closed_rec k l (TFun T1 T2)
| cl_mem: forall k l T1 T2,
closed_rec k l T1 ->
closed_rec k l T2 ->
closed_rec k l (TMem T1 T2)
| cl_cell: forall k l T1,
closed_rec k l T1 ->
closed_rec k l (TCell T1)
| cl_bind: forall k l T1 T2,
closed_rec k l T1 ->
closed_rec (S k) l T2 ->
closed_rec k l (TAll T1 T2)
| cl_sel: forall k l x,
closed_rec k l (TSel x)
| cl_selh: forall k l x,
l > x ->
closed_rec k l (TSelH x)
| cl_selb: forall k l i,
k > i ->
closed_rec k l (TSelB i)
.
Hint Constructors closed_rec.
Definition closed j l T := closed_rec j l T.
Fixpoint open_rec (k: nat) (u: ty) (T: ty) { struct T }: ty :=
match T with
| TSel x => TSel x (* free var remains free. functional, so we can't check for conflict *)
| TSelH i => TSelH i (*if beq_nat k i then u else TSelH i *)
| TSelB i => if beq_nat k i then u else TSelB i
| TAll T1 T2 => TAll (open_rec k u T1) (open_rec (S k) u T2)
| TTop => TTop
| TBot => TBot
| TBool => TBool
| TCell T1 => TCell (open_rec k u T1)
| TMem T1 T2 => TMem (open_rec k u T1) (open_rec k u T2)
| TFun T1 T2 => TFun (open_rec k u T1) (open_rec k u T2)
end.
Definition open u T := open_rec 0 u T.
(* sanity check *)
Example open_ex1: open (TSel 9) (TAll TBool (TFun (TSelB 1) (TSelB 0))) =
(TAll TBool (TFun (TSel 9) (TSelB 0))).
Proof. compute. eauto. Qed.
Fixpoint subst (U : ty) (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TBot => TBot
| TBool => TBool
| TCell T1 => TCell (subst U T1)
| TMem T1 T2 => TMem (subst U T1) (subst U T2)
| TFun T1 T2 => TFun (subst U T1) (subst U T2)
| TSelB i => TSelB i
| TSel i => TSel i
| TSelH i => if beq_nat i 0 then U else TSelH (i-1)
| TAll T1 T2 => TAll (subst U T1) (subst U T2)
end.
Fixpoint nosubst (T : ty) {struct T} : Prop :=
match T with
| TTop => True
| TBot => True
| TBool => True
| TCell T1 => nosubst T1
| TMem T1 T2 => nosubst T1 /\ nosubst T2
| TFun T1 T2 => nosubst T1 /\ nosubst T2
| TSelB i => True
| TSel i => True
| TSelH i => i <> 0
| TAll T1 T2 => nosubst T1 /\ nosubst T2
end.
Hint Unfold open.
Hint Unfold closed.
(*
the first env is for variables bound in terms
the second env is for variables bound in types
first = TSel, second = TSelH
*)
Inductive stp: tenv -> tenv -> ty -> ty -> Prop :=
| stp_topx: forall G1 GH,
stp G1 GH TTop TTop
| stp_botx: forall G1 GH,
stp G1 GH TBot TBot
| stp_top: forall G1 GH T1,
stp G1 GH T1 T1 -> (* regularity *)
stp G1 GH T1 TTop
| stp_bot: forall G1 GH T2,
stp G1 GH T2 T2 -> (* regularity *)
stp G1 GH TBot T2
| stp_bool: forall G1 GH,
stp G1 GH TBool TBool
| stp_cell: forall G1 GH T1 T2,
stp G1 GH T1 T2 ->
stp G1 GH T2 T1 ->
stp G1 GH (TCell T1) (TCell T2)
| stp_fun: forall G1 GH T1 T2 T3 T4,
stp G1 GH T3 T1 ->
stp G1 GH T2 T4 ->
stp G1 GH (TFun T1 T2) (TFun T3 T4)
| stp_mem: forall G1 GH T1 T2 T3 T4,
stp G1 GH T3 T1 ->
stp G1 GH T2 T4 ->
stp G1 GH (TMem T1 T2) (TMem T3 T4)
| stp_sel1: forall G1 GH TX T2 x,
index x G1 = Some TX ->
closed 0 0 TX ->
stp G1 GH TX (TMem TBot T2) ->
stp G1 GH T2 T2 -> (* regularity of stp2 *)
stp G1 GH (TSel x) T2
| stp_sel2: forall G1 GH TX T1 x,
index x G1 = Some TX ->
closed 0 0 TX ->
stp G1 GH TX (TMem T1 TTop) ->
stp G1 GH T1 T1 -> (* regularity of stp2 *)
stp G1 GH T1 (TSel x)
| stp_selx: forall G1 GH TX x,
index x G1 = Some TX ->
stp G1 GH (TSel x) (TSel x)
| stp_sela1: forall G1 GH TX T2 x,
indexr x GH = Some TX ->
closed 0 x TX ->
stp G1 GH TX (TMem TBot T2) ->
stp G1 GH T2 T2 -> (* regularity of stp2 *)
stp G1 GH (TSelH x) T2
| stp_sela2: forall G1 GH TX T1 x,
indexr x GH = Some TX ->
closed 0 x TX ->
stp G1 GH TX (TMem T1 TTop) ->
stp G1 GH T1 T1 -> (* regularity of stp2 *)
stp G1 GH T1 (TSelH x)
| stp_selax: forall G1 GH TX x,
indexr x GH = Some TX ->
stp G1 GH (TSelH x) (TSelH x)
| stp_all: forall G1 GH T1 T2 T3 T4 x,
stp G1 GH T3 T1 ->
x = length GH ->
closed 1 (length GH) T2 -> (* must not accidentally bind x *)
closed 1 (length GH) T4 ->
stp G1 ((0,T1)::GH) (open (TSelH x) T2) (open (TSelH x) T2) -> (* regularity *)
stp G1 ((0,T3)::GH) (open (TSelH x) T2) (open (TSelH x) T4) ->
stp G1 GH (TAll T1 T2) (TAll T3 T4)
.
Hint Constructors stp.
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_var: forall x env T1,
index x env = Some T1 ->
stp env [] T1 T1 ->
has_type env (tvar x) T1
| t_typ: forall env T1,
stp env [] T1 T1 ->
has_type env (ttyp T1) (TMem T1 T1)
| t_new: forall env x T1,
has_type env x T1 ->
stp env [] T1 T1 ->
has_type env (tnew x) (TCell T1)
| t_get: forall env T1 x,
has_type env x (TCell T1) ->
has_type env (tget x) T1
| t_set: forall env T1 x y,
has_type env x (TCell T1) ->
has_type env y T1 ->
has_type env (tset x y) 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 f x y T1 T2,
has_type ((x,T1)::(f,TFun T1 T2)::env) y T2 ->
stp env [] (TFun T1 T2) (TFun T1 T2) ->
fresh env <= f ->
1+f <= x ->
has_type env (tabs f x y) (TFun T1 T2)
| t_tapp: forall env f x T11 T12,
has_type env f (TAll T11 T12) ->
has_type env x T11 ->
stp env [] T12 T12 ->
has_type env (ttapp f x) T12
(*
NOTE: both the POPLmark paper and Cardelli's paper use this rule:
Does it make a difference? It seems like we can always widen f?
| t_tapp: forall env f T2 T11 T12 ,
has_type env f (TAll T11 T12) ->
stp env T2 T11 ->
has_type env (ttapp f T2) (open T2 T12)
*)
| t_tabs: forall env x y T1 T2,
has_type ((x,T1)::env) y (open (TSel x) T2) ->
stp env [] (TAll T1 T2) (TAll T1 T2) ->
fresh env = x ->
has_type env (ttabs x T1 y) (TAll T1 T2)
| t_sub: forall env e T1 T2,
has_type env e T1 ->
stp env [] T1 T2 ->
has_type env e T2
.
Definition base (v:vl): venv :=
match v with
| vty GX _ => GX
| vbool _ => nil
| vloc _ => nil
| vabs GX _ _ _ => GX
| vtabs GX _ _ _ => GX
end.
Inductive stp2: bool -> bool -> venv -> ty -> venv -> ty -> list(id*(venv*ty)) -> list (id*(venv*ty)) -> nat -> Prop :=
| stp2_topx: forall m G1 G2 STO GH n1,
stp2 m true G1 TTop G2 TTop STO GH (S n1)
| stp2_botx: forall m G1 G2 STO GH n1,
stp2 m true G1 TBot G2 TBot STO GH (S n1)
| stp2_top: forall m G1 G2 STO GH T n1,
stp2 m true G1 T G1 T STO GH n1 -> (* regularity *)
stp2 m true G1 T G2 TTop STO GH (S n1)
| stp2_bot: forall m G1 G2 STO GH T n1,
stp2 m true G2 T G2 T STO GH n1 -> (* regularity *)
stp2 m true G1 TBot G2 T STO GH (S n1)
| stp2_bool: forall m G1 G2 STO GH n1,
stp2 m true G1 TBool G2 TBool STO GH (S n1)
| stp2_fun: forall m G1 G2 T1 T2 T3 T4 STO GH n1 n2,
stp2 false false G2 T3 G1 T1 STO GH n1 ->
stp2 false false G1 T2 G2 T4 STO GH n2 ->
stp2 m true G1 (TFun T1 T2) G2 (TFun T3 T4) STO GH (S (n1+n2))
| stp2_mem: forall G1 G2 T1 T2 T3 T4 STO GH n1 n2,
stp2 true false G2 T3 G1 T1 STO GH n1 ->
stp2 true true G1 T2 G2 T4 STO GH n2 ->
stp2 true true G1 (TMem T1 T2) G2 (TMem T3 T4) STO GH (S (n1+n2))
| stp2_mem2: forall G1 G2 T1 T2 T3 T4 STO GH n1 n2,
stp2 false false G2 T3 G1 T1 STO GH n1 ->
stp2 false false G1 T2 G2 T4 STO GH n2 ->
stp2 false true G1 (TMem T1 T2) G2 (TMem T3 T4) STO GH (S (n1+n2))
| stp2_cell: forall m G1 G2 T1 T2 STO GH n1 n2,
stp2 false false G2 T2 G1 T1 STO GH n1 ->
stp2 false false G1 T1 G2 T2 STO GH n2 ->
stp2 m true G1 (TCell T1) G2 (TCell T2) STO GH (S (n1+n2))
(* strong version, with precise/invertible bounds *)
| stp2_strong_sel1: forall G1 G2 GX TX x T2 STO GH n1,
index x G1 = Some (vty GX TX) ->
val_type STO GX (vty GX TX) (TMem TX TX) -> (* for downgrade *)
closed 0 0 TX ->
stp2 true true GX TX G2 T2 STO GH n1 ->
stp2 true true G1 (TSel x) G2 T2 STO GH (S n1)
| stp2_strong_sel2: forall G1 G2 GX TX x T1 STO GH n1,
index x G2 = Some (vty GX TX) ->
val_type STO GX (vty GX TX) (TMem TX TX) -> (* for downgrade *)
closed 0 0 TX ->
stp2 true false G1 T1 GX TX STO GH n1 ->
stp2 true true G1 T1 G2 (TSel x) STO GH (S n1)
| stp2_strong_selx: forall G1 G2 v x1 x2 STO GH n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 true true G1 (TSel x1) G2 (TSel x2) STO GH (S n1)
(* existing object, but imprecise type *)
| stp2_sel1: forall G1 G2 GX TX x T2 STO GH n1 n2 v,
index x G1 = Some v ->
val_type STO GX v TX ->
closed 0 0 TX ->
stp2 false false GX TX G2 (TMem TBot T2) STO GH n1 ->
stp2 false true G2 T2 G2 T2 STO GH n2 -> (* regularity *)
stp2 false true G1 (TSel x) G2 T2 STO GH (S (n1+n2))
| stp2_sel2: forall G1 G2 GX TX x T1 STO GH n1 n2 v,
index x G2 = Some v ->
val_type STO GX v TX ->
closed 0 0 TX ->
stp2 false false GX TX G1 (TMem T1 TTop) STO GH n1 ->
stp2 false true G1 T1 G1 T1 STO GH n2 -> (* regularity *)
stp2 false true G1 T1 G2 (TSel x) STO GH (S (n1+n2))
| stp2_selx: forall G1 G2 v x1 x2 STO GH n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 false true G1 (TSel x1) G2 (TSel x2) STO GH (S n1)
(* hypothetical object *)
| stp2_sela1: forall G1 G2 GX TX x T2 STO GH n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 x TX ->
stp2 false false GX TX G2 (TMem TBot T2) STO GH n1 ->
stp2 false true G2 T2 G2 T2 STO GH n2 -> (* regularity *)
stp2 false true G1 (TSelH x) G2 T2 STO GH (S (n1+n2))
| stp2_sela2: forall G1 G2 GX TX x T1 STO GH n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 x TX ->
stp2 false false GX TX G1 (TMem T1 TTop) STO GH n1 ->
stp2 false true G1 T1 G1 T1 STO GH n2 -> (* regularity *)
stp2 false true G1 T1 G2 (TSelH x) STO GH (S (n1+n2))
| stp2_selax: forall G1 G2 GX TX x STO GH n1,
indexr x GH = Some (GX, TX) ->
stp2 false true G1 (TSelH x) G2 (TSelH x) STO GH (S n1)
| stp2_all: forall m G1 G2 T1 T2 T3 T4 STO GH n1 n1' n2,
stp2 false false G2 T3 G1 T1 STO GH n1 ->
closed 1 (length GH) T2 -> (* must not accidentally bind x *)
closed 1 (length GH) T4 ->
stp2 false false G1 (open (TSelH (length GH)) T2) G1 (open (TSelH (length GH)) T2) STO ((0,(G1, T1))::GH) n1' -> (* regularity *)
stp2 false false G1 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T4) STO ((0,(G2, T3))::GH) n2 ->
stp2 m true G1 (TAll T1 T2) G2 (TAll T3 T4) STO GH (S (n1+n1'+n2))
| stp2_wrapf: forall m G1 G2 T1 T2 STO GH n1,
stp2 m true G1 T1 G2 T2 STO GH n1 ->
stp2 m false G1 T1 G2 T2 STO GH (S n1)
| stp2_transf: forall m G1 G2 G3 T1 T2 T3 STO GH n1 n2,
stp2 m true G1 T1 G2 T2 STO GH n1 ->
stp2 m false G2 T2 G3 T3 STO GH n2 ->
stp2 m false G1 T1 G3 T3 STO GH (S (n1+n2))
with wf_env : list (id*(venv*ty)) -> venv -> tenv -> Prop :=
| wfe_nil : forall sto, wf_env sto nil nil
| wfe_cons : forall sto n v t vs ts,
val_type sto ((n,v)::vs) v t ->
wf_env sto vs ts ->
wf_env sto (cons (n,v) vs) (cons (n,t) ts)
with val_type : list (id*(venv*ty)) -> venv -> vl -> ty -> Prop :=
| v_ty: forall sto env venv tenv T1 TE n,
wf_env sto venv tenv -> (* T1 wf in tenv ? *)
stp2 true true venv (TMem T1 T1) env TE sto [] n ->
val_type sto env (vty venv T1) TE
| v_bool: forall sto venv b TE n,
stp2 true true [] TBool venv TE sto [] n ->
val_type sto venv (vbool b) TE
| v_loc: forall sto venv venv1 b T1 TE n,
indexr b sto = Some (venv1,T1) ->
stp2 true true venv1 (TCell T1) venv TE sto [] n ->
val_type sto venv (vloc b) TE
| v_abs: forall sto env venv tenv f x y T1 T2 TE n,
wf_env sto venv tenv ->
has_type ((x,T1)::(f,TFun T1 T2)::tenv) y T2 ->
fresh venv <= f ->
1 + f <= x ->
stp2 true true venv (TFun T1 T2) env TE sto [] n ->
val_type sto env (vabs venv f x y) TE
| v_tabs: forall sto env venv tenv x y T1 T2 TE n,
wf_env sto venv tenv ->
has_type ((x,T1)::tenv) y (open (TSel x) T2) ->
fresh venv = x ->
stp2 true true venv (TAll T1 T2) env TE sto [] n ->
val_type sto env (vtabs venv x T1 y) TE
.
Inductive wf_envh : venv -> aenv -> tenv -> Prop :=
| wfeh_nil : forall vvs, wf_envh vvs nil nil
| wfeh_cons : forall n t vs vvs ts,
wf_envh vvs vs ts ->
wf_envh vvs (cons (n,(vvs,t)) vs) (cons (n,t) ts)
.
Inductive valh_type : venv -> aenv -> (venv*ty) -> ty -> Prop :=
| v_tya: forall aenv venv T1,
valh_type venv aenv (venv, T1) T1
.
Definition stpd2 b G1 T1 G2 T2 STO GH := exists n, stp2 false b G1 T1 G2 T2 STO GH n.
Definition sstpd2 b G1 T1 G2 T2 STO GH := exists n, stp2 true b G1 T1 G2 T2 STO GH n.
Ltac ep := match goal with
| [ |- stp2 ?M1 ?M2 ?G1 ?T1 ?G2 ?T2 ?GH ?N ] => assert (exists (x:nat), stp2 M1 M2 G1 T1 G2 T2 GH x) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ _ _ _ |- _ => destruct H as [? H]
| H: sstpd2 _ _ _ _ _ _ _ |- _ => destruct H as [? H]
(* | H: exists n: nat , _ |- _ =>
destruct H as [e P] *)
end.
Hint Constructors stp2.
Hint Unfold stpd2.
Lemma stpd2_topx: forall G1 G2 STO GH,
stpd2 true G1 TTop G2 TTop STO GH.
Proof. intros. exists (S 0). eauto. Qed.
Lemma stpd2_botx: forall G1 G2 STO GH,
stpd2 true G1 TBot G2 TBot STO GH.
Proof. intros. exists (S 0). eauto. Qed.
Lemma stpd2_top: forall G1 G2 STO GH T,
stpd2 true G1 T G1 T STO GH ->
stpd2 true G1 T G2 TTop STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_bot: forall G1 G2 STO GH T,
stpd2 true G2 T G2 T STO GH ->
stpd2 true G1 TBot G2 T STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_bool: forall G1 G2 STO GH,
stpd2 true G1 TBool G2 TBool STO GH.
Proof. intros. exists (S 0). eauto. Qed.
Lemma stpd2_fun: forall G1 G2 STO GH T11 T12 T21 T22,
stpd2 false G2 T21 G1 T11 STO GH ->
stpd2 false G1 T12 G2 T22 STO GH ->
stpd2 true G1 (TFun T11 T12) G2 (TFun T21 T22) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_mem: forall G1 G2 STO GH T11 T12 T21 T22,
stpd2 false G2 T21 G1 T11 STO GH ->
stpd2 false G1 T12 G2 T22 STO GH ->
stpd2 true G1 (TMem T11 T12) G2 (TMem T21 T22) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_cell: forall G1 G2 STO GH T1 T2,
stpd2 false G2 T2 G1 T1 STO GH ->
stpd2 false G1 T1 G2 T2 STO GH ->
stpd2 true G1 (TCell T1) G2 (TCell T2) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_sel1: forall G1 G2 GX TX x T2 STO GH v,
index x G1 = Some v ->
val_type STO GX v TX ->
closed 0 0 TX ->
stpd2 false GX TX G2 (TMem TBot T2) STO GH ->
stpd2 true G2 T2 G2 T2 STO GH ->
stpd2 true G1 (TSel x) G2 T2 STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_sel2: forall G1 G2 GX TX x T1 STO GH v,
index x G2 = Some v ->
val_type STO GX v TX ->
closed 0 0 TX ->
stpd2 false GX TX G1 (TMem T1 TTop) STO GH ->
stpd2 true G1 T1 G1 T1 STO GH ->
stpd2 true G1 T1 G2 (TSel x) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_selx: forall G1 G2 x1 x2 STO GH v,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stpd2 true G1 (TSel x1) G2 (TSel x2) STO GH.
Proof. intros. exists (S 0). eauto. Qed.
Lemma stpd2_sela1: forall G1 G2 GX TX x T2 STO GH,
indexr x GH = Some (GX, TX) ->
closed 0 x TX ->
stpd2 false GX TX G2 (TMem TBot T2) STO GH ->
stpd2 true G2 T2 G2 T2 STO GH ->
stpd2 true G1 (TSelH x) G2 T2 STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_sela2: forall G1 G2 GX TX x T1 STO GH,
indexr x GH = Some (GX, TX) ->
closed 0 x TX ->
stpd2 false GX TX G1 (TMem T1 TTop) STO GH ->
stpd2 true G1 T1 G1 T1 STO GH ->
stpd2 true G1 T1 G2 (TSelH x) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_selax: forall G1 G2 GX TX x STO GH,
indexr x GH = Some (GX, TX) ->
stpd2 true G1 (TSelH x) G2 (TSelH x) STO GH.
Proof. intros. exists (S 0). eauto. Qed.
Lemma stpd2_all: forall G1 G2 T1 T2 T3 T4 STO GH,
stpd2 false G2 T3 G1 T1 STO GH ->
closed 1 (length GH) T2 ->
closed 1 (length GH) T4 ->
stpd2 false G1 (open (TSelH (length GH)) T2) G1 (open (TSelH (length GH)) T2) STO ((0,(G1, T1))::GH) ->
stpd2 false G1 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T4) STO ((0,(G2, T3))::GH) ->
stpd2 true G1 (TAll T1 T2) G2 (TAll T3 T4) STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_wrapf: forall G1 G2 T1 T2 STO GH,
stpd2 true G1 T1 G2 T2 STO GH ->
stpd2 false G1 T1 G2 T2 STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_transf: forall G1 G2 G3 T1 T2 T3 STO GH,
stpd2 true G1 T1 G2 T2 STO GH ->
stpd2 false G2 T2 G3 T3 STO GH ->
stpd2 false G1 T1 G3 T3 STO GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma sstpd2_wrapf: forall G1 G2 T1 T2 STO GH,
sstpd2 true G1 T1 G2 T2 STO GH ->
sstpd2 false G1 T1 G2 T2 STO GH.
Proof. intros. repeat eu. eexists. eapply stp2_wrapf. eauto. Qed.
Lemma sstpd2_transf: forall G1 G2 G3 T1 T2 T3 STO GH,
sstpd2 true G1 T1 G2 T2 STO GH ->
sstpd2 false G2 T2 G3 T3 STO GH ->
sstpd2 false G1 T1 G3 T3 STO GH.
Proof. intros. repeat eu. eexists. eapply stp2_transf; 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)(sto: venv)(env: venv)(t: tm){struct n}: option (option (venv*vl)) :=
match n with
| 0 => None
| S n =>
match t with
| ttrue => Some (Some (sto, vbool true))
| tfalse => Some (Some (sto, vbool false))
| tvar x => Some (match (index x env) with | Some v => Some (sto, v) | None => None end)
| tabs f x y => Some (Some (sto, vabs env f x y))
| ttabs x T y => Some (Some (sto, vtabs env x T y))
| ttyp T => Some (Some (sto, vty env T))
| tnew ex =>
match teval n sto env ex with
| None => None
| Some None => Some None
| Some (Some (sto1, v)) => Some (Some ((0,v)::sto1, vloc (length sto1)))
end
| tget ex =>
match teval n sto env ex with
| None => None
| Some None => Some None
| Some (Some (sto1, vbool _)) => Some None
| Some (Some (sto1, vty _ _)) => Some None
| Some (Some (sto1, vtabs _ _ _ _)) => Some None
| Some (Some (sto1, vabs _ _ _ _)) => Some None
| Some (Some (sto1, vloc i)) =>
Some (match (indexr i sto1) with
| Some v => Some (sto1,v)
| None => None
end)
end
| tset ex ey =>
match teval n sto env ex with
| None => None
| Some None => Some None
| Some (Some (sto1, vbool _)) => Some None
| Some (Some (sto1, vty _ _)) => Some None
| Some (Some (sto1, vtabs _ _ _ _)) => Some None
| Some (Some (sto1, vabs _ _ _ _)) => Some None
| Some (Some (sto1, vloc i)) =>
match teval n sto1 env ey with
| None => None
| Some None => Some None
| Some (Some (sto2, v)) => Some (Some (update i (0,v) sto2, v))
end
end
| tapp ef ex =>
match teval n sto env ex with
| None => None
| Some None => Some None
| Some (Some (sto1, vx)) =>
match teval n sto1 env ef with
| None => None
| Some None => Some None
| Some (Some (sto2, vbool _)) => Some None
| Some (Some (sto2, vty _ _)) => Some None
| Some (Some (sto2, vloc _)) => Some None
| Some (Some (sto2, vtabs _ _ _ _)) => Some None
| Some (Some (sto2, vabs env2 f x ey)) =>
teval n sto2 ((x,vx)::(f,vabs env2 f x ey)::env2) ey
end
end
| ttapp ef ex =>
match teval n sto env ex with
| None => None
| Some None => Some None
| Some (Some (sto1, vx)) =>
match teval n sto1 env ef with
| None => None
| Some None => Some None
| Some (Some (sto2, vbool _)) => Some None
| Some (Some (sto2, vty _ _)) => Some None
| Some (Some (sto2, vloc _)) => Some None
| Some (Some (sto2, vabs _ _ _ _)) => Some None
| Some (Some (sto2, vtabs env2 x T ey)) =>
teval n sto2 ((x,vx)::env2) ey
end
end
end
end.
Hint Constructors ty.
Hint Constructors tm.
Hint Constructors vl.
Hint Constructors closed_rec.
Hint Constructors has_type.
Hint Constructors val_type.
Hint Constructors wf_env.
Hint Constructors stp.
Hint Constructors stp2.
Hint Constructors option.
Hint Constructors list.
Hint Unfold index.
Hint Unfold length.
Hint Unfold closed.
Hint Unfold open.
Hint Resolve ex_intro.
(* ############################################################ *)
(* Examples *)
(* ############################################################ *)
(*
match goal with
| |- has_type _ (tvar _) _ =>
try solve [apply t_vara;
repeat (econstructor; eauto)]
| _ => idtac
end;
*)
Ltac crush_has_tp :=
try solve [eapply stp_selx; compute; eauto; crush_has_tp];
try solve [eapply stp_selax; compute; eauto; crush_has_tp];
try solve [eapply cl_selb; compute; eauto; crush_has_tp];
try solve [(econstructor; compute; eauto; crush_has_tp)].
Ltac crush2 :=
try solve [(eapply stp_selx; compute; eauto; crush2)];
try solve [(eapply stp_selax; compute; eauto; crush2)];
try solve [(eapply stp_sel1; compute; eauto; crush2)];
try solve [(eapply stp_sela1; compute; eauto; crush2)];
try solve [(eapply cl_selb; compute; eauto; crush2)];
try solve [(econstructor; compute; eauto; crush2)];
try solve [(eapply t_sub; eapply t_var; compute; eauto; crush2)].
(* define polymorphic identity function *)
Definition polyId := TAll (TMem TBot TTop) (TFun (TSelB 0) (TSelB 0)).
Example ex1: has_type [] (ttabs 0 (TMem TBot TTop) (tabs 1 2 (tvar 2))) polyId.
Proof.
crush2.
Qed.
(* instantiate it to bool *)
Example ex2: has_type [(0,polyId)] (ttapp (tvar 0) (ttyp TBool)) (TFun TBool TBool).
Proof.
eapply t_tapp. eapply t_sub. eapply t_var. simpl. eauto.
eapply stp_all. eauto. eauto. crush_has_tp. crush_has_tp. crush_has_tp.
compute. crush_has_tp.
eapply stp_all. instantiate (1:= (TMem TBool TBool)).
eapply stp_mem. crush2. crush2. crush2. crush2. crush2.
eapply stp_fun. crush2. crush2. crush2.
eapply t_sub. eapply t_typ. crush2. crush2.
eapply stp_fun; crush2.
Qed.
Example rex1: has_type [] (tnew (ttabs 0 (TMem TBot TTop) (tabs 1 2 (tvar 2)))) (TCell polyId).
Proof.
crush2.
Qed.
Example rex2: has_type [(0, TCell polyId)] (ttapp (tget (tvar 0)) (ttyp TBool)) (TFun TBool TBool).
Proof.
unfold polyId. eapply t_tapp.
eapply t_sub. eapply t_get. eapply t_var. compute. eauto. crush2.
instantiate (1:=(TMem TBool TBool)). crush2. crush2. crush2.
Qed.
(* define brand / unbrand client function *)
Definition brandUnbrand :=
TAll (TMem TBot TTop)
(TFun
(TFun TBool (TSelB 0)) (* brand *)
(TFun
(TFun (TSelB 0) TBool) (* unbrand *)
TBool)).
Example ex3:
has_type []
(ttabs 0 (TMem TBot TTop)
(tabs 1 2
(tabs 3 4
(tapp (tvar 4) (tapp (tvar 2) ttrue)))))
brandUnbrand.
Proof.
crush2.
Qed.
(* instantiating it at bool is admissible *)
Example ex4:
has_type [(1,TFun TBool TBool);(0,brandUnbrand)]
(tvar 0) (TAll (TMem TBool TBool) (TFun (TFun TBool TBool) (TFun (TFun TBool TBool) TBool))).
Proof.
eapply t_sub. crush2. crush2.
Qed.
Hint Resolve ex4.
(* apply it to identity functions *)
Example ex5:
has_type [(1,TFun TBool TBool);(0,brandUnbrand)]
(tapp (tapp (ttapp (tvar 0) (ttyp TBool)) (tvar 1)) (tvar 1)) TBool.
Proof.
crush2.
Qed.
(* ############################################################ *)
(* Proofs *)
(* ############################################################ *)
Lemma wf_fresh : forall sto vs ts,
wf_env sto vs ts ->
(fresh vs = fresh ts).
Proof.
intros. induction H. auto.
compute. eauto.
Qed.
Hint Immediate wf_fresh.
Lemma wfh_length : forall vvs vs ts,
wf_envh vvs vs ts ->
(length vs = length ts).
Proof.
intros. induction H. auto.
compute. eauto.
Qed.
Hint Immediate wf_fresh.
Lemma index_max : forall X vs n (T: X),
index n vs = Some T ->
n < fresh vs.
Proof.
intros X vs. induction vs.
- Case "nil". intros. inversion H.
- Case "cons".
intros. inversion H. destruct a.
case_eq (le_lt_dec (fresh vs) i); intros ? E1.
+ SCase "ok".
rewrite E1 in H1.
case_eq (beq_nat n i); intros E2.
* SSCase "hit".
eapply beq_nat_true in E2. subst n. compute. eauto.
* SSCase "miss".
rewrite E2 in H1.
assert (n < fresh vs). eapply IHvs. apply H1.
compute. omega.
+ SCase "bad".
rewrite E1 in H1. inversion H1.
Qed.
Lemma indexr_max : forall X vs n (T: X),
indexr n vs = Some T ->
n < length vs.
Proof.
intros X vs. induction vs.
- Case "nil". intros. inversion H.
- Case "cons".
intros. inversion H. destruct a.
case_eq (beq_nat n (length vs)); intros E2.
+ SSCase "hit".
eapply beq_nat_true in E2. subst n. compute. eauto.
+ SSCase "miss".
rewrite E2 in H1.
assert (n < length vs). eapply IHvs. apply H1.
compute. eauto.
Qed.
Lemma le_xx : forall a b,
a <= b ->
exists E, le_lt_dec a b = left E.
Proof. intros.
case_eq (le_lt_dec a b). intros. eauto.
intros. omega.
Qed.
Lemma le_yy : forall a b,
a > b ->
exists E, le_lt_dec a b = right E.
Proof. intros.
case_eq (le_lt_dec a b). intros. omega.
intros. eauto.
Qed.
Lemma index_extend : forall X vs n n' x (T: X),
index n vs = Some T ->
fresh vs <= n' ->
index n ((n',x)::vs) = Some T.
Proof.
intros.
assert (n < fresh vs). eapply index_max. eauto.
assert (n <> n'). omega.
assert (beq_nat n n' = false) as E. eapply beq_nat_false_iff; eauto.
assert (fresh vs <= n') as E2. omega.
elim (le_xx (fresh vs) n' E2). intros ? EX.
unfold index. unfold index in H. rewrite H. rewrite E. rewrite EX. reflexivity.
Qed.
Lemma indexr_extend : forall X vs n n' x (T: X),
indexr n vs = Some T ->
indexr n ((n',x)::vs) = Some T.
Proof.
intros.
assert (n < length vs). eapply indexr_max. eauto.
assert (beq_nat n (length vs) = false) as E. eapply beq_nat_false_iff. omega.
unfold indexr. unfold indexr in H. rewrite H. rewrite E. reflexivity.
Qed.
(* splicing -- for stp_extend. *)
Fixpoint splice n (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TBot => TBot
| TBool => TBool
| TCell T1 => TCell (splice n T1)
| TMem T1 T2 => TMem (splice n T1) (splice n T2)
| TFun T1 T2 => TFun (splice n T1) (splice n T2)
| TSelB i => TSelB i
| TSel i => TSel i
| TSelH i => if le_lt_dec n i then TSelH (i+1) else TSelH i
| TAll T1 T2 => TAll (splice n T1) (splice n T2)
end.
Definition splicett n (V: (id*ty)) :=
match V with
| (x,T) => (x,(splice n T))
end.
Definition spliceat n (V: (id*(venv*ty))) :=
match V with
| (x,(G,T)) => (x,(G,splice n T))
end.