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dot12.v
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(* Full safety for DOT *)
(* this version is based on dot10.v *)
(* based on that, it adds intersection types *)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Le.
Require Import Coq.Program.Equality.
Module FSUB.
Definition id := nat.
Inductive ty : Type :=
| TBool : ty
| TBot : ty
| TTop : ty
| TFun : ty -> ty -> ty
| TMem : ty -> ty -> ty
| TSel : id -> ty
| TSelH : id -> ty
| TSelB : id -> ty
| TAll : ty -> ty -> ty
| TBind : ty -> ty
| TAnd : ty -> ty -> ty
.
Inductive tm : Type :=
| ttrue : tm
| tfalse : tm
| tvar : id -> 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
| 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_all: forall k l T1 T2,
closed_rec k l T1 ->
closed_rec (S k) l T2 ->
closed_rec k l (TAll T1 T2)
| cl_bind: forall k l T2,
closed_rec (S k) l T2 ->
closed_rec k l (TBind T2)
| cl_sel: forall k l x,
closed_rec k l (TSel x)
| cl_and: forall k l T1 T2,
closed_rec k l T1 ->
closed_rec k l T2 ->
closed_rec k l (TAnd T1 T2)
| 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)
| TBind T2 => TBind (open_rec (S k) u T2)
| TTop => TTop
| TBot => TBot
| TBool => TBool
| 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)
| TAnd T1 T2 => TAnd (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
| 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)
| TBind T2 => TBind (subst U T2)
| TAnd T1 T2 => TAnd (subst U T1) (subst U T2)
end.
Fixpoint nosubst (T : ty) {struct T} : Prop :=
match T with
| TTop => True
| TBot => True
| TBool => True
| 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
| TBind T2 => nosubst T2
| TAnd T1 T2 => nosubst T1 /\ nosubst T2
end.
Hint Unfold open.
Hint Unfold closed.
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_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_selb1: forall G1 GH TX T2 x,
index x G1 = Some TX ->
stp G1 [] TX (TBind (TMem TBot T2)) -> (* Note GH = [] *)
stp G1 GH (open (TSel x) T2) (open (TSel x) T2) -> (* regularity *)
stp G1 GH (TSel x) (open (TSel x) T2)
| stp_selb2: forall G1 GH TX T1 x,
index x G1 = Some TX ->
stp G1 [] TX (TBind (TMem T1 TTop)) -> (* Note GH = [] *)
stp G1 GH (open (TSel x) T1) (open (TSel x) T1) -> (* regularity *)
stp G1 GH (open (TSel x) 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 (S x) TX ->
stp G1 GH TX (TMem TBot T2) -> (* not using self name for now *)
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 (S x) TX ->
stp G1 GH TX (TMem T1 TTop) -> (* not using self name for now *)
stp G1 GH T1 T1 -> (* regularity of stp2 *)
stp G1 GH T1 (TSelH x)
| stp_selab1: forall G1 GH GU GL TX T2 T2' x,
indexr x GH = Some TX ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem TBot T2)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stp G1 GL TX (TBind (TMem TBot T2)) ->
T2' = (open (TSelH x) T2) ->
stp G1 GH T2' T2' -> (* regularity *)
stp G1 GH (TSelH x) T2'
| stp_selab2: forall G1 GH GU GL TX T1 T1' x,
indexr x GH = Some TX ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem T1 TTop)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stp G1 GL TX (TBind (TMem T1 TTop)) ->
T1' = (open (TSelH x) T1) ->
stp G1 GH T1' T1' -> (* regularity *)
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)
| stp_bindx: forall G1 GH T1 T2 x,
x = length GH ->
closed 1 (length GH) T1 -> (* must not accidentally bind x *)
closed 1 (length GH) T2 ->
stp G1 ((0,open (TSelH x) T2)::GH) (open (TSelH x) T2) (open (TSelH x) T2) -> (* regularity *)
stp G1 ((0,open (TSelH x) T1)::GH) (open (TSelH x) T1) (open (TSelH x) T2) ->
stp G1 GH (TBind T1) (TBind T2)
| stp_and11: forall G GH T1 T2 T,
stp G GH T1 T ->
stp G GH T2 T2 -> (* regularity *)
stp G GH (TAnd T1 T2) T
| stp_and12: forall G GH T1 T2 T,
stp G GH T2 T ->
stp G GH T1 T1 -> (* regularity *)
stp G GH (TAnd T1 T2) T
| stp_and2: forall G GH T1 T2 T,
stp G GH T T1 ->
stp G GH T T2 ->
stp G GH T (TAnd T1 T2)
.
(*
with path_type: tenv -> tenv -> id -> ty -> Prop :=
| pt_var: forall G1 GH TX x,
index x G1 = Some TX ->
path_type G1 GH x TX
| pt_sub: forall G1 GH TX x,
path_type has_type env e T1 ->
stp env [] T1 T2 ->
has_type env e T2
with pathH_type: tenv -> tenv -> id -> ty -> Prop :=
| pth_var: forall G1 GH TX T x,
indexr x GH = Some TX ->
stp G1 GH TX T ->
pathH_type G1 GH x T
*)
Hint Constructors stp.
(* TODO *)
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_var_pack: forall x env T1,
has_type env (tvar x) (open (TSel x) T1) ->
stp env [] (TBind T1) (TBind T1) ->
has_type env (tvar x) (TBind T1)
| t_var_unpack: forall x env T1,
has_type env (tvar x) (TBind T1) ->
stp env [] (open (TSel x) T1) (open (TSel x) T1) ->
has_type env (tvar x) (open (TSel x) T1)
| t_typ: forall env x T1 T1X,
fresh env = x ->
open (TSel x) T1 = T1X ->
stp ((x,TMem T1X T1X)::env) [] T1X T1X ->
stp env [] (TBind (TMem T1 T1)) (TBind (TMem T1 T1)) ->
has_type env (ttyp T1) (TBind (TMem T1 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
| vabs GX _ _ _ => GX
| vtabs GX _ _ _ => GX
end.
Definition MAX := 2.
Inductive stp2: nat -> bool -> venv -> ty -> venv -> ty -> list (id*(venv*ty)) -> nat -> Prop :=
| stp2_topx: forall m G1 G2 GH n1,
stp2 m true G1 TTop G2 TTop GH (S n1)
| stp2_botx: forall m G1 G2 GH n1,
stp2 m true G1 TBot G2 TBot GH (S n1)
| stp2_top: forall m G1 G2 GH T n1,
stp2 m true G1 T G1 T GH n1 -> (* regularity *)
stp2 m true G1 T G2 TTop GH (S n1)
| stp2_bot: forall m G1 G2 GH T n1,
stp2 m true G2 T G2 T GH n1 -> (* regularity *)
stp2 m true G1 TBot G2 T GH (S n1)
| stp2_bool: forall m G1 G2 GH n1,
stp2 m true G1 TBool G2 TBool GH (S n1)
| stp2_fun: forall m G1 G2 T1 T2 T3 T4 GH n1 n2,
stp2 MAX false G2 T3 G1 T1 GH n1 ->
stp2 MAX false G1 T2 G2 T4 GH n2 ->
stp2 m true G1 (TFun T1 T2) G2 (TFun T3 T4) GH (S (n1+n2))
| stp2_mem: forall G1 G2 T1 T2 T3 T4 GH n1 n2,
stp2 0 false G2 T3 G1 T1 GH n1 ->
stp2 0 true G1 T2 G2 T4 GH n2 ->
stp2 0 true G1 (TMem T1 T2) G2 (TMem T3 T4) GH (S (n1+n2))
| stp2_mem2: forall m G1 G2 T1 T2 T3 T4 GH n1 n2,
stp2 (S m) false G2 T3 G1 T1 GH n1 ->
stp2 (S m) false G1 T2 G2 T4 GH n2 ->
stp2 (S m) true G1 (TMem T1 T2) G2 (TMem T3 T4) GH (S (n1+n2))
(* strong version, with precise/invertible bounds *)
| stp2_strong_sel1: forall G1 G2 GX TX x T2 GH GX' TX' n1 n2 nv,
index x G1 = Some (vty GX TX) ->
val_type GX' (vty GX TX) TX' nv -> (* for downgrade *)
stp2 0 false GX' TX' G2 (TMem TBot T2) GH n2 -> (* for downgrade *)
closed 1 0 TX ->
stp2 0 true ((fresh GX, vty GX TX)::GX) (open (TSel (fresh GX)) TX) G2 T2 GH n1 ->
stp2 0 true G1 (TSel x) G2 T2 GH (S (n1+n2))
| stp2_strong_sel2: forall G1 G2 GX TX x T1 GH GX' TX' n1 n2 nv,
index x G2 = Some (vty GX TX) ->
val_type GX' (vty GX TX) TX' nv -> (* for downgrade *)
stp2 0 false GX' TX' G1 (TMem T1 TTop) GH n2 -> (* for downgrade *)
closed 1 0 TX ->
stp2 0 false G1 T1 ((fresh GX, vty GX TX)::GX) (open (TSel (fresh GX)) TX) GH n1 ->
stp2 0 true G1 T1 G2 (TSel x) GH (S (n1+n2))
| stp2_strong_selx: forall G1 G2 v x1 x2 GH n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 0 true G1 (TSel x1) G2 (TSel x2) GH n1
(* existing object, but imprecise type *)
| stp2_sel1: forall m G1 G2 GX TX x T2 GH n1 n2 v nv,
index x G1 = Some v ->
val_type GX v TX nv ->
closed 0 0 TX ->
stp2 (S m) false GX TX G2 (TMem TBot T2) GH n1 ->
stp2 (S m) true G2 T2 G2 T2 GH n2 -> (* regularity *)
stp2 (S m) true G1 (TSel x) G2 T2 GH (S (n1+n2))
| stp2_selb1: forall m G1 G2 GX TX x x' T2 GH n1 n2 v nv,
index x G1 = Some v -> (index x' G2 = Some v \/ closed 0 0 T2) ->
val_type GX v TX nv ->
closed 0 0 TX ->
stp2 (S (S m)) false GX TX G2 (TBind (TMem TBot T2)) [] n1 -> (* Note GH = [] *)
stp2 (S (S m)) true G2 (open (TSel x') T2) G2 (open (TSel x') T2) GH n2 -> (* regularity *)
stp2 (S (S m)) true G1 (TSel x) G2 (open (TSel x') T2) GH (S (n1+n2))
| stp2_sel2: forall m G1 G2 GX TX x T1 GH n1 n2 v nv,
index x G2 = Some v ->
val_type GX v TX nv ->
closed 0 0 TX ->
stp2 (S m) false GX TX G1 (TMem T1 TTop) GH n1 ->
stp2 (S m) true G1 T1 G1 T1 GH n2 -> (* regularity *)
stp2 (S m) true G1 T1 G2 (TSel x) GH (S (n1+n2))
| stp2_selb2: forall m G1 G2 GX TX x x' T1 GH n1 n2 v nv,
index x G2 = Some v -> (index x' G1 = Some v \/ closed 0 0 T1) ->
val_type GX v TX nv ->
closed 0 0 TX ->
stp2 (S (S m)) false GX TX G1 (TBind (TMem T1 TTop)) [] n1 -> (* Note GH = [] *)
stp2 (S (S m)) true G1 (open (TSel x') T1) G1 (open (TSel x') T1) GH n2 -> (* regularity *)
stp2 (S (S m)) true G1 (open (TSel x') T1) G2 (TSel x) GH (S (n1+n2))
| stp2_selx: forall m G1 G2 v x1 x2 GH n1,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stp2 (S m) true G1 (TSel x1) G2 (TSel x2) GH (S n1)
(* hypothetical object *)
| stp2_sela1: forall m G1 G2 GX TX x T2 GH n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
stp2 (S m) false GX TX G2 (TMem TBot T2) GH n1 ->
stp2 (S m) true G2 T2 G2 T2 GH n2 -> (* regularity *)
stp2 (S m) true G1 (TSelH x) G2 T2 GH (S (n1+n2))
| stp2_selab1: forall m G1 G2 GX TX x T2 T2' GH GU GL n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem TBot T2)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stp2 (S m) false GX TX G2 (TBind (TMem TBot T2)) GL n1 ->
T2' = (open (TSelH x) T2) ->
stp2 (S m) true G2 T2' G2 T2' GH n2 -> (* regularity *)
stp2 (S m) true G1 (TSelH x) G2 T2' GH (S (n1+n2))
| stp2_selab2: forall m G1 G2 GX TX x T1 T1' GH GU GL n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem T1 TTop)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stp2 (S m) false GX TX G1 (TBind (TMem T1 TTop)) GL n1 ->
T1' = (open (TSelH x) T1) ->
stp2 (S m) true G1 T1' G1 T1' GH n2 -> (* regularity *)
stp2 (S m) true G1 T1' G2 (TSelH x) GH (S (n1+n2))
| stp2_sela2: forall m G1 G2 GX TX x T1 GH n1 n2,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
stp2 (S m) false GX TX G1 (TMem T1 TTop) GH n1 ->
stp2 (S m) true G1 T1 G1 T1 GH n2 -> (* regularity *)
stp2 (S m) true G1 T1 G2 (TSelH x) GH (S (n1+n2))
| stp2_selax: forall m G1 G2 GX TX x GH n1,
indexr x GH = Some (GX, TX) ->
stp2 (S m) true G1 (TSelH x) G2 (TSelH x) GH (S n1)
| stp2_all: forall m G1 G2 T1 T2 T3 T4 GH n1 n1' n2,
stp2 MAX false G2 T3 G1 T1 GH n1 ->
closed 1 (length GH) T2 -> (* must not accidentally bind x *)
closed 1 (length GH) T4 ->
stp2 MAX false G1 (open (TSelH (length GH)) T2) G1 (open (TSelH (length GH)) T2) ((0,(G1, T1))::GH) n1' -> (* regularity *)
stp2 MAX false G1 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T4) ((0,(G2, T3))::GH) n2 ->
stp2 m true G1 (TAll T1 T2) G2 (TAll T3 T4) GH (S (n1+n1'+n2))
| stp2_bind: forall G1 G2 T1 T2 GH n1 n2,
closed 1 (length GH) T1 -> (* must not accidentally bind x *)
closed 1 (length GH) T2 ->
stp2 1 false G2 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T2) ((0,(G2, open (TSelH (length GH)) T2))::GH) n2 -> (* regularity *)
stp2 1 false G1 (open (TSelH (length GH)) T1) G2 (open (TSelH (length GH)) T2) ((0,(G1, open (TSelH (length GH)) T1))::GH) n1 ->
stp2 0 true G1 (TBind T1) G2 (TBind T2) GH (S (n1+n2))
| stp2_bindb: forall m G1 G2 T1 T2 GH n1 n2,
closed 1 (length GH) T1 -> (* must not accidentally bind x *)
closed 1 (length GH) T2 ->
stp2 (S m) false G2 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T2) ((0,(G2, open (TSelH (length GH)) T2))::GH) n2 -> (* regularity *)
stp2 (S m) false G1 (open (TSelH (length GH)) T1) G2 (open (TSelH (length GH)) T2) ((0,(G1, open (TSelH (length GH)) T1))::GH) n1 ->
stp2 (S m) true G1 (TBind T1) G2 (TBind T2) GH (S (n1+n2))
| stp2_and11: forall m n1 n2 G1 G2 GH T1 T2 T,
stp2 m true G1 T1 G2 T GH n1 ->
stp2 m true G1 T2 G1 T2 GH n2 -> (* regularity *)
stp2 m true G1 (TAnd T1 T2) G2 T GH (S (n1+n2))
| stp2_and12: forall m n1 n2 G1 G2 GH T1 T2 T,
stp2 m true G1 T2 G2 T GH n1 ->
stp2 m true G1 T1 G1 T1 GH n2 -> (* regularity *)
stp2 m true G1 (TAnd T1 T2) G2 T GH (S (n1+n2))
| stp2_and2: forall m n1 n2 G1 G2 GH T1 T2 T,
stp2 m false G1 T G2 T1 GH n1 ->
stp2 m false G1 T G2 T2 GH n2 ->
stp2 m true G1 T G2 (TAnd T1 T2) GH (S (n1+n2))
| stp2_wrapf: forall m G1 G2 T1 T2 GH n1,
stp2 m true G1 T1 G2 T2 GH n1 ->
stp2 m false G1 T1 G2 T2 GH (S n1)
| stp2_transf: forall m G1 G2 G3 T1 T2 T3 GH n1 n2,
stp2 m true G1 T1 G2 T2 GH n1 ->
stp2 m false G2 T2 G3 T3 GH n2 ->
stp2 m false G1 T1 G3 T3 GH (S (n1+n2))
with wf_env : venv -> tenv -> Prop :=
| wfe_nil : wf_env nil nil
| wfe_cons : forall n v t vs ts nv,
val_type ((n,v)::vs) v t nv ->
wf_env vs ts ->
wf_env (cons (n,v) vs) (cons (n,t) ts)
with val_type : venv -> vl -> ty -> nat -> Prop :=
| v_ty: forall env venv tenv T1 T1X TE,
wf_env venv tenv -> (* T1 wf in tenv ? *)
open (TSel (fresh venv)) T1 = T1X ->
(exists n, stp2 0 true ((fresh venv, vty venv T1)::venv) (TMem T1X T1X) env TE [] n)->
val_type env (vty venv T1) TE 1
| v_bool: forall venv b TE,
(exists n, stp2 0 true [] TBool venv TE [] n) ->
val_type venv (vbool b) TE 1
| v_abs: forall env venv tenv f x y T1 T2 TE,
wf_env venv tenv ->
has_type ((x,T1)::(f,TFun T1 T2)::tenv) y T2 ->
fresh venv <= f ->
1 + f <= x ->
(exists n, stp2 0 true venv (TFun T1 T2) env TE [] n)->
val_type env (vabs venv f x y) TE 1
| v_tabs: forall env venv tenv x y T1 T2 TE,
wf_env venv tenv ->
has_type ((x,T1)::tenv) y (open (TSel x) T2) ->
fresh venv = x ->
(exists n, stp2 0 true venv (TAll T1 T2) env TE [] n) ->
val_type env (vtabs venv x T1 y) TE 1
| v_pack: forall venv venv3 x v T T2 T3 n,
index x venv = Some v ->
val_type venv v T n ->
open (TSel x) T2 = T ->
(exists n, stp2 0 true venv (TBind T2) venv3 T3 [] n) ->
val_type venv3 v T3 (S n)
.
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 GH := exists n, stp2 MAX b G1 T1 G2 T2 GH n.
Definition atpd2 b G1 T1 G2 T2 GH := exists n, stp2 1 b G1 T1 G2 T2 GH n.
Definition sstpd2 b G1 T1 G2 T2 GH := exists n, stp2 0 b G1 T1 G2 T2 GH n.
Definition valtpd G v T := exists n, val_type G v T 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: atpd2 _ _ _ _ _ _ |- _ => 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 GH,
stpd2 true G1 TTop G2 TTop GH.
Proof. intros. repeat exists (S 0). eauto. Qed.
Lemma stpd2_botx: forall G1 G2 GH,
stpd2 true G1 TBot G2 TBot GH.
Proof. intros. repeat exists (S 0). eauto. Qed.
Lemma stpd2_top: forall G1 G2 GH T,
stpd2 true G1 T G1 T GH ->
stpd2 true G1 T G2 TTop GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_bot: forall G1 G2 GH T,
stpd2 true G2 T G2 T GH ->
stpd2 true G1 TBot G2 T GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_bool: forall G1 G2 GH,
stpd2 true G1 TBool G2 TBool GH.
Proof. intros. repeat exists (S 0). eauto. Qed.
Lemma stpd2_fun: forall G1 G2 GH T11 T12 T21 T22,
stpd2 false G2 T21 G1 T11 GH ->
stpd2 false G1 T12 G2 T22 GH ->
stpd2 true G1 (TFun T11 T12) G2 (TFun T21 T22) GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_mem: forall G1 G2 GH T11 T12 T21 T22,
stpd2 false G2 T21 G1 T11 GH ->
stpd2 false G1 T12 G2 T22 GH ->
stpd2 true G1 (TMem T11 T12) G2 (TMem T21 T22) GH.
Proof. intros. repeat eu. eauto. unfold stpd2. eexists. eapply stp2_mem2; eauto. Qed.
Lemma stpd2_sel1: forall G1 G2 GX TX x T2 GH v nv,
index x G1 = Some v ->
val_type GX v TX nv ->
closed 0 0 TX ->
stpd2 false GX TX G2 (TMem TBot T2) GH ->
stpd2 true G2 T2 G2 T2 GH ->
stpd2 true G1 (TSel x) G2 T2 GH.
Proof. intros. repeat eu. eexists. eapply stp2_sel1; eauto. Qed.
Lemma stpd2_selb1: forall G1 G2 GX TX x x' T2 GH v nv,
index x G1 = Some v -> (index x' G2 = Some v \/ closed 0 0 T2) ->
val_type GX v TX nv ->
closed 0 0 TX ->
stpd2 false GX TX G2 (TBind (TMem TBot T2)) [] -> (* Note GH = [] *)
stpd2 true G2 (open (TSel x') T2) G2 (open (TSel x') T2) GH ->
stpd2 true G1 (TSel x) G2 (open (TSel x') T2) GH.
Proof. intros. repeat eu. eexists. eapply stp2_selb1; eauto. Qed.
Lemma stpd2_sel2: forall G1 G2 GX TX x T1 GH v nv,
index x G2 = Some v ->
val_type GX v TX nv ->
closed 0 0 TX ->
stpd2 false GX TX G1 (TMem T1 TTop) GH ->
stpd2 true G1 T1 G1 T1 GH ->
stpd2 true G1 T1 G2 (TSel x) GH.
Proof. intros. repeat eu. eexists. eapply stp2_sel2; eauto. Qed.
Lemma stpd2_selb2: forall G1 G2 GX TX x x' T1 GH v nv,
index x G2 = Some v -> (index x' G1 = Some v \/ closed 0 0 T1) ->
val_type GX v TX nv ->
closed 0 0 TX ->
stpd2 false GX TX G1 (TBind (TMem T1 TTop)) [] -> (* Note GH = [] *)
stpd2 true G1 (open (TSel x') T1) G1 (open (TSel x') T1) GH ->
stpd2 true G1 (open (TSel x') T1) G2 (TSel x) GH.
Proof. intros. repeat eu. eexists. eapply stp2_selb2; eauto. Qed.
Lemma stpd2_selx: forall G1 G2 x1 x2 GH v,
index x1 G1 = Some v ->
index x2 G2 = Some v ->
stpd2 true G1 (TSel x1) G2 (TSel x2) GH.
Proof. intros. eauto. exists (S 0). eapply stp2_selx; eauto. Qed.
Lemma stpd2_selab1: forall G1 G2 GX TX x T2 GH GU GL,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem TBot T2)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stpd2 false GX TX G2 (TBind (TMem TBot T2)) GL ->
stpd2 true G2 (open (TSelH x) T2) G2 (open (TSelH x) T2) GH ->
stpd2 true G1 (TSelH x) G2 (open (TSelH x) T2) GH.
Proof. intros. repeat eu. eauto. eexists. eapply stp2_selab1; eauto. Qed.
Lemma stpd2_selab2: forall G1 G2 GX TX x T1 T1' GH GU GL,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
closed 0 0 (TBind (TMem T1 TTop)) ->
length GL = (S x) ->
GH = GU ++ GL ->
stpd2 false GX TX G1 (TBind (TMem T1 TTop)) GL ->
T1' = (open (TSelH x) T1) ->
stpd2 true G1 T1' G1 T1' GH ->
stpd2 true G1 T1' G2 (TSelH x) GH.
Proof. intros. repeat eu. eauto. eexists. eapply stp2_selab2; eauto. Qed.
Lemma stpd2_sela1: forall G1 G2 GX TX x T2 GH,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
stpd2 false GX TX G2 (TMem TBot T2) GH ->
stpd2 true G2 T2 G2 T2 GH ->
stpd2 true G1 (TSelH x) G2 T2 GH.
Proof. intros. repeat eu. eauto. eexists. eapply stp2_sela1; eauto. Qed.
Lemma stpd2_sela2: forall G1 G2 GX TX x T1 GH,
indexr x GH = Some (GX, TX) ->
closed 0 (S x) TX ->
stpd2 false GX TX G1 (TMem T1 TTop) GH ->
stpd2 true G1 T1 G1 T1 GH ->
stpd2 true G1 T1 G2 (TSelH x) GH.
Proof. intros. repeat eu. eauto. eexists. eapply stp2_sela2; eauto. Qed.
Lemma stpd2_selax: forall G1 G2 GX TX x GH,
indexr x GH = Some (GX, TX) ->
stpd2 true G1 (TSelH x) G2 (TSelH x) GH.
Proof. intros. exists (S 0). eauto. eapply stp2_selax; eauto. Qed.
Lemma stpd2_all: forall G1 G2 T1 T2 T3 T4 GH,
stpd2 false G2 T3 G1 T1 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) ((0,(G1, T1))::GH) ->
stpd2 false G1 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T4) ((0,(G2, T3))::GH) ->
stpd2 true G1 (TAll T1 T2) G2 (TAll T3 T4) GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_bind: forall G1 G2 T1 T2 GH,
closed 1 (length GH) T1 ->
closed 1 (length GH) T2 ->
stpd2 false G2 (open (TSelH (length GH)) T2) G2 (open (TSelH (length GH)) T2) ((0,(G2, open (TSelH (length GH)) T2))::GH) ->
stpd2 false G1 (open (TSelH (length GH)) T1) G2 (open (TSelH (length GH)) T2) ((0,(G1, open (TSelH (length GH)) T1))::GH) ->
stpd2 true G1 (TBind T1) G2 (TBind T2) GH.
Proof. intros. repeat eu. eauto. unfold stpd2. eexists. eapply stp2_bindb; eauto. Qed.
Lemma stpd2_and11: forall G1 G2 GH T1 T2 T,
stpd2 true G1 T1 G2 T GH ->
stpd2 true G1 T2 G1 T2 GH ->
stpd2 true G1 (TAnd T1 T2) G2 T GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_and12: forall G1 G2 GH T1 T2 T,
stpd2 true G1 T2 G2 T GH ->
stpd2 true G1 T1 G1 T1 GH ->
stpd2 true G1 (TAnd T1 T2) G2 T GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_and2: forall G1 G2 GH T1 T2 T,
stpd2 false G1 T G2 T1 GH ->
stpd2 false G1 T G2 T2 GH ->
stpd2 true G1 T G2 (TAnd T1 T2) GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_wrapf: forall G1 G2 T1 T2 GH,
stpd2 true G1 T1 G2 T2 GH ->
stpd2 false G1 T1 G2 T2 GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma stpd2_transf: forall G1 G2 G3 T1 T2 T3 GH,
stpd2 true G1 T1 G2 T2 GH ->
stpd2 false G2 T2 G3 T3 GH ->
stpd2 false G1 T1 G3 T3 GH.
Proof. intros. repeat eu. eauto. Qed.
Lemma sstpd2_wrapf: forall G1 G2 T1 T2 GH,
sstpd2 true G1 T1 G2 T2 GH ->
sstpd2 false G1 T1 G2 T2 GH.
Proof. intros. repeat eu. eexists. eapply stp2_wrapf. eauto. Qed.
Lemma sstpd2_transf: forall G1 G2 G3 T1 T2 T3 GH,
sstpd2 true G1 T1 G2 T2 GH ->
sstpd2 false G2 T2 G3 T3 GH ->
sstpd2 false G1 T1 G3 T3 GH.
Proof. intros. repeat eu. eexists. eapply stp2_transf; eauto. Qed.
Lemma atpd2_transf: forall G1 G2 G3 T1 T2 T3 GH,
atpd2 true G1 T1 G2 T2 GH ->
atpd2 false G2 T2 G3 T3 GH ->
atpd2 false G1 T1 G3 T3 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)(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 f x y => Some (Some (vabs env f x y))
| ttabs x T y => Some (Some (vtabs env x T y))
| ttyp T => Some (Some (vty env T))
| 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 (vtabs _ _ _ _)) => Some None
| Some (Some (vabs env2 f x ey)) =>
teval n ((x,vx)::(f,vabs env2 f x ey)::env2) ey
end
end
| ttapp 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 _ _ _ _)) => Some None
| Some (Some (vtabs env2 x T ey)) =>
teval n ((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 (TBind (TMem TBot TTop)) (TFun (TSelB 0) (TSelB 0)).
Example ex1: has_type [] (ttabs 0 (TBind (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. instantiate (1:= (TBind (TMem TBool TBool))).
{ eapply t_sub.
{ eapply t_var. simpl. eauto. crush2. }
{ eapply stp_all; eauto. { eapply stp_bindx; crush2. } compute. eapply cl_fun; eauto.
eapply stp_fun. compute. eapply stp_selax; crush2. crush2.
eapply stp_fun. compute. eapply stp_selab2. crush2.
crush2. instantiate (1:=TBool). crush2.
instantiate (1:=[(0, TBind (TMem TBool TBool))]). simpl. reflexivity.
rewrite app_nil_l. reflexivity.
crush2. crush2. crush2.
simpl. eapply stp_selab1. crush2.
crush2. instantiate (1:=TBool). crush2.
instantiate (1:=[(0, TBind (TMem TBool TBool))]). simpl. reflexivity.
rewrite app_nil_l. reflexivity.
crush2. crush2. crush2.
}
}
{ eapply t_typ; crush2. }
crush2.
Qed.
(* define brand / unbrand client function *)
Definition brandUnbrand :=
TAll (TBind (TMem TBot TTop))
(TFun
(TFun TBool (TSelB 0)) (* brand *)
(TFun
(TFun (TSelB 0) TBool) (* unbrand *)
TBool)).
Example ex3:
has_type []
(ttabs 0 (TBind (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 (TBind (TMem TBool TBool)) (TFun (TFun TBool TBool) (TFun (TFun TBool TBool) TBool))).
Proof.
eapply t_sub. crush2. crush2. eapply stp_all; crush2. compute. eapply stp_fun. eapply stp_fun. crush2.
eapply stp_selab2. crush2. crush2. instantiate(1:=TBool). crush2.
instantiate (1:=[(0, TBind (TMem TBool TBool))]). simpl. reflexivity.
rewrite app_nil_l. reflexivity.