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dot.v
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dot.v
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(*
DOT
T ::= Bot | Top | T1 /\ T2 | T1 \/ T2 |
{ def m(x: S): U^x } | { type A: S..U } | x.A | { z => T^z }
t ::= x | { y => d^y... } | t.m(t)
d ::= { def m(x: S): U^x = t^x } | { type A = T }
*)
(* in small-step *)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Lt.
(*# Syntax #*)
Definition id := nat. (* identifiers for variables: x,y,z *)
Definition lb := nat. (* labels for records: L, m *)
Inductive vr : Type :=
| TVar : bool(*true for concrete context, false for abstract context *) ->
id(*absolute position in context, from origin, invariant under context extension*) -> vr
| TVarB : id(*bound variable, de Bruijn, locally nameless style -- see open *) -> vr
.
Inductive ty : Type :=
| TBot : ty (* bottom type *)
| TTop : ty (* top type *)
| TFun : lb -> ty -> ty -> ty (* dependent function / method member type:
{ def m(x: S): U^x },
where x is locally bound in U *)
| TTyp : lb -> ty -> ty -> ty (* type member type: { type L: S..U } *)
| TSel : vr -> lb -> ty (* type selection: x.L *)
| TBind : ty -> ty (* Recursive binder: { z => T^z },
where z is locally bound in T *)
| TAnd : ty -> ty -> ty (* Intersection Type: T1 /\ T2 *)
| TOr : ty -> ty -> ty (* Union Type: T1 \/ T2 *)
.
Inductive tm : Type :=
| tvar : bool(*like TVar: true for concrete, false for hypothetical *) -> id -> tm (* variable: x *)
(* N.B.: no varB -- terms just use absolute identifers directly *)
| tobj : dms(*self is next slot in abstract context -- see subst_tm*) -> tm (* new object instance: { z => d... } *)
| tapp : tm -> lb -> tm -> tm (* method invocation: t.m(t) *)
with dm : Type := (* initialization / member definition --
the labels, e.g. m & A, are determined from the position in member list, dms *)
| dfun : option ty -> option ty -> tm -> dm (* method: { def m(x[: S])[: U] = t }, where the types [: S] and [: U] are optional *)
(* Church vs Curry: we show that all options work, by making parameter and return types optional,
when defining a method. *)
| dty : ty -> dm (* type: { type L = T } *)
(* we use our own list-like structure for easy recursion, e.g. in subst_tm *)
with dms : Type := (* list of member defs *)
| dnil : dms
| dcons : dm -> dms -> dms
.
Fixpoint dms_to_list (ds: dms) : list dm :=
match ds with
| dnil => []
| dcons d ds => d :: dms_to_list ds
end.
Inductive vl : Type :=
| vobj : dms -> vl
.
Definition venv := list vl. (*rho G*)
Definition tenv := list ty. (*Gamma GH*)
Hint Unfold venv.
Hint Unfold tenv.
(*# Variable Binding #*)
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 i j k -- well-bound in
an abstract environment GH of size >= i
a concrete environment G of size >= j
under >= k binders/de Bruijn levels
*)
Inductive vr_closed: nat(*abstract, TVar false i*) -> nat(*concrete, TVar true j*) -> nat(*bound, TVarB k*) -> vr -> Prop :=
| cl_var0: forall i j k x,
i > x ->
vr_closed i j k (TVar false x)
| cl_var1: forall i j k x,
j > x ->
vr_closed i j k (TVar true x)
| cl_varB: forall i j k x,
k > x ->
vr_closed i j k (TVarB x).
Inductive closed: nat(*abstract, TVar false i*) -> nat(*concrete, TVar true j*) -> nat(*bound, TVarB k*) -> 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_fun: forall i j k l T1 T2,
closed i j k T1 ->
closed i j (S k) T2 ->
closed i j k (TFun l T1 T2)
| cl_typ: forall i j k l T1 T2,
closed i j k T1 ->
closed i j k T2 ->
closed i j k (TTyp l T1 T2)
| cl_sel: forall i j k p1 l,
vr_closed i j k p1 ->
closed i j k (TSel p1 l)
| cl_bind: forall i j k T1,
closed i j (S k) T1 ->
closed i j k (TBind T1)
| cl_and: forall i j k T1 T2,
closed i j k T1 ->
closed i j k T2 ->
closed i j k (TAnd T1 T2)
| cl_or: forall i j k T1 T2,
closed i j k T1 ->
closed i j k T2 ->
closed i j k (TOr T1 T2)
.
(* substitute a locally bound variable at de Brujin level k with variable u in type T *)
Definition vr_open (k: nat) (u: vr) (p: vr) : vr :=
match p 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
end.
Fixpoint open (k: nat) (u: vr) (T: ty) { struct T }: ty :=
match T with
| TTop => TTop
| TBot => TBot
| TSel p1 l => TSel (vr_open k u p1) l
| TFun l T1 T2 => TFun l (open k u T1) (open (S k) u T2)
| TTyp l T1 T2 => TTyp l (open k u T1) (open k u T2)
| TBind T1 => TBind (open (S k) u T1)
| TAnd T1 T2 => TAnd (open k u T1) (open k u T2)
| TOr T1 T2 => TOr (open k u T1) (open k u T2)
end.
(* substitute the first abstract variable (id 0) with variable u in type T --
all other abstract variables are shifted (id decremented) to fit the shrinked abstract context
*)
Definition vr_subst (u : vr) (X : vr): vr :=
match X with
| TVarB i => TVarB i
| TVar true i => TVar true i
(* subst the _first_ aka _oldest_ abstract variables,
the other abstract variables are shifted to resolve in the shrinked context *)
| TVar false i => if beq_nat i 0 then u else TVar false (i-1)
end.
Fixpoint subst (u : vr) (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TBot => TBot
| TTyp l T1 T2 => TTyp l (subst u T1) (subst u T2)
| TSel p1 l => TSel (vr_subst u p1) l
| TFun l T1 T2 => TFun l (subst u T1) (subst u T2)
| TBind T2 => TBind (subst u T2)
| TAnd T1 T2 => TAnd (subst u T1) (subst u T2)
| TOr T1 T2 => TOr (subst u T1) (subst u T2)
end.
(* substitute the first hypothetical variable with term u in term t --
like subst, shifts other hypothetical variables *)
Fixpoint subst_tm (u:nat) (t : tm) {struct t} : tm :=
match t with
| tvar true i => tvar true i
| tvar false i => if beq_nat i 0 then (tvar true u) else tvar false (i-1)
| tobj ds => tobj (subst_dms u ds)
| tapp t1 l t2 => tapp (subst_tm u t1) l (subst_tm u t2)
end
with subst_dm (u:nat) (d: dm) {struct d} : dm :=
match d with
| dty T => dty (subst (TVar true u) T)
| dfun T1 T2 t => dfun (option_map (subst (TVar true u)) T1) (option_map (subst (TVar true u)) T2) (subst_tm u t)
end
with subst_dms (u:nat) (ds: dms) {struct ds} : dms :=
match ds with
| dnil => dnil
| dcons d ds1 => dcons (subst_dm u d) (subst_dms u ds1)
end.
(* Shortcut for the common case of replacing abstract with concrete. *)
Definition substt x T := (subst (TVar true x) T).
Hint Immediate substt.
(*# Operational Semantics #*)
(* Reduction semantics *)
Inductive step : venv -> tm -> venv -> tm -> Prop :=
(* Computation Rules *)
| ST_Obj : forall G1 D,
step G1 (tobj D) (vobj (subst_dms (length G1) D)::G1) (tvar true (length G1))
| ST_AppAbs : forall G1 f l x ds T1 T2 t12,
index f G1 = Some (vobj ds) ->
index l (dms_to_list ds) = Some (dfun T1 T2 t12) ->
step G1 (tapp (tvar true f) l (tvar true x)) G1 (subst_tm x t12)
(* Congruence Rules *)
| ST_App1 : forall G1 G1' t1 t1' l t2,
step G1 t1 G1' t1' ->
step G1 (tapp t1 l t2) G1' (tapp t1' l t2)
| ST_App2 : forall G1 G1' f t2 l t2',
step G1 t2 G1' t2' ->
step G1 (tapp (tvar true f) l t2) G1' (tapp (tvar true f) l t2')
.
(*# Static Semantics #*)
Definition eq_some {X} (OT:option X) (T:X) := OT=None \/ OT=Some T.
(* : -- typing *)
Inductive has_type : tenv -> venv -> tm -> ty -> nat -> Prop :=
| T_Vary : forall GH G1 x ds ds' T T' n1,
index x G1 = Some (vobj ds) ->
dms_has_type [T'] G1 ds' T' n1 ->
subst_dms x ds' = ds ->
substt x T' = T ->
closed 0 (length G1) 0 T ->
has_type GH G1 (tvar true x) T (S n1)
| T_Varz : forall G1 GH x T n1,
index x GH = Some T ->
closed (length GH) (length G1) 0 T ->
has_type GH G1 (tvar false x) T (S n1)
| T_VarPack : forall GH G1 b x T1 T1' n1,
has_type GH G1 (tvar b x) T1' n1 ->
T1' = (open 0 (TVar b x) T1) ->
closed (length GH) (length G1) 1 T1 ->
has_type GH G1 (tvar b x) (TBind T1) (S n1)
| T_VarUnpack : forall GH G1 b x T1 T1' n1,
has_type GH G1 (tvar b x) (TBind T1) n1 ->
T1' = (open 0 (TVar b x) T1) ->
closed (length GH) (length G1) 0 T1' ->
has_type GH G1 (tvar b x) T1' (S n1)
| T_Obj : forall GH G1 ds T T' n1,
dms_has_type (T'::GH) G1 ds T' n1 ->
T' = open 0 (TVar false (length GH)) T ->
closed (length GH) (length G1) 1 T ->
has_type GH G1 (tobj ds) (TBind T) (S n1)
| T_App : forall l T1 T2 GH G1 t1 t2 n1 n2,
has_type GH G1 t1 (TFun l T1 T2) n1 ->
has_type GH G1 t2 T1 n2 ->
closed (length GH) (length G1) 0 T2 ->
has_type GH G1 (tapp t1 l t2) T2 (S (n1+n2))
| T_AppVar : forall l T1 T2 T2' GH G1 t1 b2 x2 n1 n2,
has_type GH G1 t1 (TFun l T1 T2) n1 ->
has_type GH G1 (tvar b2 x2) T1 n2 ->
T2' = (open 0 (TVar b2 x2) T2) ->
closed (length GH) (length G1) 0 T2' ->
has_type GH G1 (tapp t1 l (tvar b2 x2)) T2' (S (n1+n2))
| T_Sub : forall GH G1 t T1 T2 n1 n2,
has_type GH G1 t T1 n1 ->
stp GH G1 T1 T2 n2 ->
has_type GH G1 t T2 (S (n1 + n2))
(* : -- member initialization *)
with dms_has_type: tenv -> venv -> dms -> ty -> nat -> Prop :=
| D_Nil : forall GH G1 n1,
dms_has_type GH G1 dnil TTop (S n1)
| D_Typ : forall GH G1 l T11 ds TS T n1,
dms_has_type GH G1 ds TS n1 ->
closed (length GH) (length G1) 0 T11 ->
l = length (dms_to_list ds) ->
T = TAnd (TTyp l T11 T11) TS ->
dms_has_type GH G1 (dcons (dty T11) ds) T (S n1)
| D_Fun : forall GH G1 l OT11 T11 OT12 T12 T12' t12 ds TS T n1 n2,
dms_has_type GH G1 ds TS n1 ->
has_type (T11::GH) G1 t12 T12' n2 ->
T12' = (open 0 (TVar false (length GH)) T12) ->
closed (length GH) (length G1) 0 T11 ->
closed (length GH) (length G1) 1 T12 ->
l = length (dms_to_list ds) ->
T = TAnd (TFun l T11 T12) TS ->
eq_some OT11 T11 ->
eq_some OT12 T12 ->
dms_has_type GH G1 (dcons (dfun OT11 OT12 t12) ds) T (S (n1+n2))
(* <: -- subtyping *)
with stp: tenv -> venv -> ty -> ty -> nat -> Prop :=
| stp_bot: forall GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp GH G1 TBot T (S n1)
| stp_top: forall GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp GH G1 T TTop (S n1)
| stp_fun: forall GH G1 l T1 T2 T3 T4 T2' T4' n1 n2,
T2' = (open 0 (TVar false (length GH)) T2) ->
T4' = (open 0 (TVar false (length GH)) T4) ->
closed (length GH) (length G1) 1 T2 ->
closed (length GH) (length G1) 1 T4 ->
stp GH G1 T3 T1 n1 ->
stp (T3::GH) G1 T2' T4' n2 ->
stp GH G1 (TFun l T1 T2) (TFun l T3 T4) (S (n1+n2))
| stp_typ: forall GH G1 l T1 T2 T3 T4 n1 n2,
stp GH G1 T3 T1 n2 ->
stp GH G1 T2 T4 n1 ->
stp GH G1 (TTyp l T1 T2) (TTyp l T3 T4) (S (n1+n2))
| stp_strong_sel1: forall GH G1 l T2 ds TX x n1,
index x G1 = Some (vobj ds) ->
index l (dms_to_list ds) = Some (dty TX) ->
stp [] G1 TX T2 n1 ->
stp GH G1 (TSel (TVar true x) l) T2 (S n1)
| stp_strong_sel2: forall GH G1 l T1 ds TX x n1,
index x G1 = Some (vobj ds) ->
index l (dms_to_list ds) = Some (dty TX) ->
stp [] G1 T1 TX n1 ->
stp GH G1 T1 (TSel (TVar true x) l) (S n1)
| stp_sel1: forall GH G1 l T2 x n1,
htp GH G1 x (TTyp l TBot T2) n1 ->
stp GH G1 (TSel (TVar false x) l) T2 (S n1)
| stp_sel2: forall GH G1 l T1 x n1,
htp GH G1 x (TTyp l T1 TTop) n1 ->
stp GH G1 T1 (TSel (TVar false x) l) (S n1)
| stp_selx: forall GH G1 l p1 n1,
vr_closed (length GH) (length G1) 0 p1 ->
stp GH G1 (TSel p1 l) (TSel p1 l) (S n1)
| stp_bind1: forall GH G1 T1 T1' T2 n1,
stp (T1'::GH) G1 T1' T2 n1 ->
T1' = (open 0 (TVar false (length GH)) T1) ->
closed (length GH) (length G1) 1 T1 ->
closed (length GH) (length G1) 0 T2 ->
stp GH G1 (TBind T1) T2 (S n1)
| stp_bindx: forall GH G1 T1 T1' T2 T2' n1,
stp (T1'::GH) G1 T1' T2' n1 ->
T1' = (open 0 (TVar false (length GH)) T1) ->
T2' = (open 0 (TVar false (length GH)) T2) ->
closed (length GH) (length G1) 1 T1 ->
closed (length GH) (length G1) 1 T2 ->
stp GH G1 (TBind T1) (TBind T2) (S n1)
| stp_and11: forall GH G1 T1 T2 T n1,
stp GH G1 T1 T n1 ->
closed (length GH) (length G1) 0 T2 ->
stp GH G1 (TAnd T1 T2) T (S n1)
| stp_and12: forall GH G1 T1 T2 T n1,
stp GH G1 T2 T n1 ->
closed (length GH) (length G1) 0 T1 ->
stp GH G1 (TAnd T1 T2) T (S n1)
| stp_and2: forall GH G1 T1 T2 T n1 n2,
stp GH G1 T T1 n1 ->
stp GH G1 T T2 n2 ->
stp GH G1 T (TAnd T1 T2) (S (n1+n2))
| stp_or21: forall GH G1 T1 T2 T n1,
stp GH G1 T T1 n1 ->
closed (length GH) (length G1) 0 T2 ->
stp GH G1 T (TOr T1 T2) (S n1)
| stp_or22: forall GH G1 T1 T2 T n1,
stp GH G1 T T2 n1 ->
closed (length GH) (length G1) 0 T1 ->
stp GH G1 T (TOr T1 T2) (S n1)
| stp_or1: forall GH G1 T1 T2 T n1 n2,
stp GH G1 T1 T n1 ->
stp GH G1 T2 T n2 ->
stp GH G1 (TOr T1 T2) T (S (n1+n2))
| stp_trans: forall GH G1 T1 T2 T3 n1 n2,
stp GH G1 T1 T2 n1 ->
stp GH G1 T2 T3 n2 ->
stp GH G1 T1 T3 (S (n1+n2))
(* :! -- typing for type selection in subtyping *)
with htp: tenv -> venv -> id -> ty -> nat -> Prop :=
| htp_var: forall GH G1 x TX n1,
index x GH = Some TX ->
closed (S x) (length G1) 0 TX ->
htp GH G1 x TX (S n1)
| htp_unpack: forall GH G1 x TX n1,
htp GH G1 x (TBind TX) n1 ->
closed (S x) (length G1) 1 TX ->
htp GH G1 x (open 0 (TVar false x) TX) (S n1)
| htp_sub: forall GH GU GL G1 x T1 T2 n1 n2,
(* use restricted GH. note: this is slightly different
from the big-step version b/c here we do not distinguish
if variables are bound in terms vs types. it would be easy
to do exactly the same thing by adding this distinction. *)
htp GH G1 x T1 n1 ->
stp GL G1 T1 T2 n2 ->
length GL = S x ->
GH = GU ++ GL ->
htp GH G1 x T2 (S (n1+n2)).
Definition has_typed GH G1 x T1 := exists n, has_type GH G1 x T1 n.
Definition stpd GH G1 T1 T2 := exists n, stp GH G1 T1 T2 n.
Definition htpd GH G1 x T1 := exists n, htp GH G1 x T1 n.
Hint Constructors stp.
Ltac ep := match goal with
| [ |- stp ?GH ?G1 ?T1 ?T2 ?N ] => assert (exists (n:nat), stp GH G1 T1 T2 n) as EEX
end.
Ltac eu := match goal with
| H: has_typed _ _ _ _ |- _ => destruct H as [? H]
| H: stpd _ _ _ _ |- _ => destruct H as [? H]
| H: htpd _ _ _ _ |- _ => destruct H as [? H]
end.
Lemma stpd_bot: forall GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd GH G1 TBot T.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd_top: forall GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd GH G1 T TTop.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd_fun: forall GH G1 l T1 T2 T3 T4 T2' T4',
T2' = (open 0 (TVar false (length GH)) T2) ->
T4' = (open 0 (TVar false (length GH)) T4) ->
closed (length GH) (length G1) 1 T2 ->
closed (length GH) (length G1) 1 T4 ->
stpd GH G1 T3 T1 ->
stpd (T3::GH) G1 T2' T4' ->
stpd GH G1 (TFun l T1 T2) (TFun l T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd_typ: forall GH G1 l T1 T2 T3 T4,
stpd GH G1 T3 T1 ->
stpd GH G1 T2 T4 ->
stpd GH G1 (TTyp l T1 T2) (TTyp l T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd_trans: forall GH G1 T1 T2 T3,
stpd GH G1 T1 T2 ->
stpd GH G1 T2 T3 ->
stpd GH G1 T1 T3.
Proof. intros. repeat eu. eexists. eauto. Qed.
Hint Constructors ty.
Hint Constructors vl.
Hint Constructors stp.
Hint Constructors htp.
Hint Constructors has_type.
Hint Unfold has_typed.
Hint Unfold stpd.
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.
(*# Regularity #*)
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 plus_lt_contra: forall a b,
a + b < b -> False.
Proof.
intros a b H. induction a.
- simpl in H. apply lt_irrefl in H. assumption.
- simpl in H. apply IHa. omega.
Qed.
Lemma index_extend_mult: forall {X} G0 G2 x0 (T:X),
index x0 G0 = Some T ->
index x0 (G2++G0) = Some T.
Proof.
intros X G0 G2. induction G2; intros.
- simpl. assumption.
- simpl.
case_eq (beq_nat x0 (length (G2 ++ G0))); intros E.
+ eapply beq_nat_true_iff in E.
apply index_max in H. subst.
rewrite app_length in H. apply plus_lt_contra in H. inversion H.
+ apply IHG2. assumption.
Qed.
Lemma vr_closed_extend : forall p X (a:X) i k G,
vr_closed i (length G) k p ->
vr_closed i (length (a::G)) k p.
Proof.
intros. inversion H; subst; econstructor; eauto. simpl. omega.
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 using vr_closed_extend.
Qed.
Lemma all_extend: forall ni,
(forall GH v1 G1 T1 T2 n,
stp GH G1 T1 T2 n -> n < ni ->
stp GH (v1::G1) T1 T2 n) /\
(forall v1 x GH G1 T2 n,
htp GH G1 x T2 n -> n < ni ->
htp GH (v1::G1) x T2 n) /\
(forall GH G1 t T v n,
has_type GH G1 t T n -> n < ni ->
has_type GH (v::G1) t T n) /\
(forall GH G1 ds T v n,
dms_has_type GH G1 ds T n -> n < ni ->
dms_has_type GH (v::G1) ds T n).
Proof.
intros n. induction n. repeat split; intros; omega.
repeat split; intros; inversion H.
(* stp *)
- econstructor. eapply closed_extend. eauto.
- econstructor. eapply closed_extend. eauto.
- econstructor. eauto. eauto.
eapply closed_extend. eauto. eapply closed_extend. eauto.
eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply index_extend. eauto. eauto. eapply IHn. eauto. omega.
- econstructor. eapply index_extend. eauto. eauto. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega.
- econstructor. eapply vr_closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto. eapply closed_extend. eauto.
- eapply stp_bindx. eapply IHn. eauto. omega. eauto. eauto. eapply closed_extend. eauto. eapply closed_extend. eauto.
- eapply stp_and11. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp_and12. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp_and2. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- eapply stp_or21. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp_or22. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp_or1. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- eapply stp_trans. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
(* htp *)
- econstructor. eauto. eapply closed_extend. eauto.
- eapply htp_unpack. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply htp_sub. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eauto. eauto.
(* has_type *)
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega. eauto. eauto. eapply closed_extend. eauto.
- econstructor. eauto. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto.
- econstructor. subst. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply T_AppVar. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
(* dms_has_type *)
- econstructor.
- econstructor. eapply IHn. eauto. omega. eapply closed_extend. eauto. eauto. eauto.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eauto.
eapply closed_extend. eauto. eapply closed_extend. eauto. eauto. eauto. eauto. eauto.
Qed.
Lemma vr_closed_upgrade_gh: forall i i1 j k p1,
vr_closed i j k p1 -> i <= i1 -> vr_closed i1 j k p1.
Proof.
intros. inversion H; subst; econstructor; eauto. omega.
Qed.
Lemma closed_upgrade_gh: forall i i1 j k T1,
closed i j k T1 -> i <= i1 -> closed i1 j k T1.
Proof.
intros. generalize dependent i1. induction H; intros; econstructor; eauto using vr_closed_upgrade_gh.
Qed.
Lemma vr_closed_extend_mult : forall p i j j' k,
vr_closed i j k p -> j <= j' ->
vr_closed i j' k p.
Proof.
intros. inversion H; subst; econstructor; eauto. omega.
Qed.
Lemma closed_extend_mult : forall T i j j' k,
closed i j k T -> j <= j' ->
closed i j' k T.
Proof.
intros. generalize dependent j'. induction H; intros; econstructor; eauto using vr_closed_extend_mult.
Qed.
Lemma vr_closed_upgrade: forall i j k k1 p1,
vr_closed i j k p1 -> k <= k1 -> vr_closed i j k1 p1.
Proof.
intros. inversion H; subst; econstructor; eauto. omega.
Qed.
Lemma closed_upgrade: forall i j k k1 T1,
closed i j k T1 -> k <= k1 -> closed i j k1 T1.
Proof.
intros. generalize dependent k1. induction H; intros; econstructor; eauto using vr_closed_upgrade.
eapply IHclosed2. omega.
eapply IHclosed. omega.
Qed.
Lemma vr_closed_open: forall j k n b V p, vr_closed k n (j+1) p -> vr_closed k n j (TVar b V) -> vr_closed k n j (vr_open j (TVar b V) p).
Proof.
intros. inversion H; try econstructor; eauto; subst.
- Case "TVarB". simpl.
case_eq (beq_nat j x); intros E. eauto.
econstructor. eapply beq_nat_false_iff in E. omega.
Qed.
Lemma closed_open: forall j k n b V T, closed k n (j+1) T -> vr_closed k n j (TVar b V) -> closed k n j (open j (TVar b V) T).
Proof.
intros. generalize dependent j. induction T; intros; inversion H;
try econstructor; try eapply IHT1; try eapply IHT2; eauto; try eapply IHT; eauto using vr_closed_open; subst;
eapply vr_closed_upgrade; eauto.
Qed.
Lemma all_closed: forall ni,
(forall GH G1 T1 T2 n,
stp GH G1 T1 T2 n -> n < ni ->
closed (length GH) (length G1) 0 T1) /\
(forall GH G1 T1 T2 n,
stp GH G1 T1 T2 n -> n < ni ->
closed (length GH) (length G1) 0 T2) /\
(forall x GH G1 T2 n,
htp GH G1 x T2 n -> n < ni ->
x < length GH) /\
(forall x GH G1 T2 n,
htp GH G1 x T2 n -> n < ni ->
closed (length GH) (length G1) 0 T2) /\
(forall GH G1 t T n,
has_type GH G1 t T n -> n < ni ->
closed (length GH) (length G1) 0 T) /\
(forall GH G1 ds T n,
dms_has_type GH G1 ds T n -> n < ni ->
closed (length GH) (length G1) 0 T).
Proof.
intros n. induction n. repeat split; intros; omega.
repeat split; intros; inversion H; destruct IHn as [IHS1 [IHS2 [IHH1 [IHH2 [IHT IHD]]]]].
(* stp left *)
- econstructor.
- eauto.
- econstructor. eapply IHS2. eauto. omega. eauto.
- econstructor. eapply IHS2. eauto. omega. eapply IHS1. eauto. omega.
- econstructor. econstructor. eapply index_max. eauto.
- eapply closed_upgrade_gh. eapply IHS1. eauto. omega. simpl. omega.
- econstructor. econstructor. eapply IHH1. eauto. omega.
- eapply closed_upgrade_gh. eapply IHH2 in H1. inversion H1. eauto. omega. simpl. omega.
- econstructor. eauto.
- econstructor. eauto.
- econstructor. eauto.
- econstructor. eapply IHS1. eauto. omega. eauto.
- econstructor. eauto. eapply IHS1. eauto. omega.
- eapply IHS1. eauto. omega.
- eapply IHS1. eauto. omega.
- eapply IHS1. eauto. omega.
- econstructor. eapply IHS1. eauto. omega. eapply IHS1. eauto. omega.
- eapply IHS1. eauto. omega.
(* stp right *)
- eauto.
- econstructor.
- econstructor. eapply IHS1. eauto. omega. eauto.
- econstructor. eapply IHS1. eauto. omega. eapply IHS2. eauto. omega.
- eapply closed_upgrade_gh. eapply IHS2. eauto. omega. simpl. omega.
- econstructor. econstructor. eapply index_max. eauto.
- eapply closed_upgrade_gh. eapply IHH2 in H1. inversion H1. eauto. omega. simpl. omega.
- econstructor. econstructor. eapply IHH1. eauto. omega.
- econstructor. eauto.
- eauto.
- econstructor. eauto.
- eapply IHS2. eauto. omega.
- eapply IHS2. eauto. omega.
- econstructor. eapply IHS2. eauto. omega. eapply IHS2. eauto. omega.
- econstructor. eapply IHS2. eauto. omega. eauto.
- econstructor. eauto. eapply IHS2. eauto. omega.
- eapply IHS2. eauto. omega.
- eapply IHS2. eauto. omega.
(* htp left *)
- eapply index_max. eauto.
- eapply IHH1. eauto. omega.
- eapply IHH1. eauto. omega.
(* htp right *)
- eapply closed_upgrade_gh. eauto. subst. eapply index_max in H1. omega.
- eapply IHH1 in H1. eapply closed_open. simpl. eapply closed_upgrade_gh. eauto. omega. econstructor. eauto. omega.
- eapply closed_upgrade_gh. eapply IHS2. eauto. omega. rewrite H4. rewrite app_length. omega.
(* has_type *)
- subst. eapply closed_upgrade_gh. eauto. omega.
- eauto.
- econstructor. eapply closed_upgrade_gh. eauto. omega.
- eapply IHT in H1. inversion H1; subst. eauto. omega.
- econstructor. eauto.
- eapply IHT in H1. inversion H1. eauto. omega.
- eapply IHT in H1. inversion H1. eauto. omega.
- eapply IHS2. eauto. omega.
(* dms_has_type *)
- econstructor.
- subst. econstructor. econstructor. eauto. eauto. eapply IHD. eauto. omega.
- subst. econstructor. econstructor. eauto. eauto. eapply IHD. eauto. omega.
Qed.
Lemma htp_extend : forall v1 x GH G1 T2 n,
htp GH G1 x T2 n ->
htp GH (v1::G1) x T2 n.
Proof. intros. eapply all_extend. eauto. eauto. Qed.
Lemma stp_extend : forall GH v1 G1 T1 T2 n,
stp GH G1 T1 T2 n ->
stp GH (v1::G1) T1 T2 n.
Proof. intros. eapply all_extend. eauto. eauto. Qed.
Lemma stp_extend_mult : forall GH G1 G' T1 T2 n,
stp GH G1 T1 T2 n ->
stp GH (G'++G1) T1 T2 n.
Proof. intros. induction G'. simpl. eauto. simpl. eapply stp_extend. eauto. Qed.
Lemma has_type_extend: forall GH G1 t T v n1,
has_type GH G1 t T n1 ->
has_type GH (v::G1) t T n1.
Proof. intros. eapply all_extend. eauto. eauto. Qed.
Lemma dms_has_type_extend: forall GH G1 t T v n1,
dms_has_type GH G1 t T n1 ->
dms_has_type GH (v::G1) t T n1.
Proof. intros. eapply all_extend. eauto. eauto. Qed.
Lemma has_type_extend_mult: forall GH G1 t T G' n1,
has_type GH G1 t T n1 ->
has_type GH (G'++G1) t T n1.
Proof. intros. induction G'. simpl. eauto. simpl. eapply has_type_extend. eauto. Qed.
Lemma htp_closed: forall x GH G1 T2 n,
htp GH G1 x T2 n ->
closed (length GH) (length G1) 0 T2.
Proof. intros. eapply all_closed with (x:=x). eauto. eauto. Qed.
Lemma htp_closed1: forall x GH G1 T2 n,
htp GH G1 x T2 n ->
x < length GH.
Proof. intros. eapply all_closed with (x:=x). eauto. eauto. Qed.
Lemma has_type_closed: forall GH G1 t T n1,
has_type GH G1 t T n1 ->
closed (length GH) (length G1) 0 T.
Proof. intros. eapply all_closed with (t:=t). eauto. eauto. Qed.
Lemma dms_has_type_closed: forall GH G1 t T n1,
dms_has_type GH G1 t T n1 ->
closed (length GH) (length G1) 0 T.
Proof. intros. eapply all_closed with (ds:=t). eauto. eauto. Qed.
Lemma has_type_closed_z: forall GH G1 z T n1,
has_type GH G1 (tvar false z) T n1 ->
z < length GH.
Proof.
intros. remember (tvar false z) as t. generalize dependent z.
induction H; intros; inversion Heqt; subst; eauto using index_max.
Qed.
Lemma has_type_closed1: forall GH G1 x T n1,
has_type GH G1 (tvar true x) T n1 ->
x < length G1.
Proof.
intros. remember (tvar true x) as t. generalize dependent x.
induction H; intros; inversion Heqt; subst; eauto using index_max.
Qed.
Lemma has_type_closed_b: forall G1 b x T n1,
has_type [] G1 (tvar b x) T n1 ->
b = true /\ x < length G1.
Proof.
intros.
remember [] as GH.
remember (tvar b x) as t.
generalize dependent x. generalize dependent b. generalize HeqGH. clear HeqGH.
induction H; intros; inversion Heqt; subst; eauto using index_max.
- simpl in H. inversion H.
Qed.
Lemma stp_closed1 : forall GH G1 T1 T2 n1,
stp GH G1 T1 T2 n1 ->
closed (length GH) (length G1) 0 T1.
Proof. intros. edestruct all_closed. eapply H0. eauto. eauto. Qed.
Lemma stp_closed2 : forall GH G1 T1 T2 n1,
stp GH G1 T1 T2 n1 ->
closed (length GH) (length G1) 0 T2.
Proof. intros. edestruct all_closed. destruct H1. eapply H1. eauto. eauto. Qed.
Lemma stpd_closed1 : forall GH G1 T1 T2,
stpd GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T1.
Proof. intros. eu. eapply stp_closed1. eauto. Qed.
Lemma stpd_closed2 : forall GH G1 T1 T2,
stpd GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T2.
Proof. intros. eu. eapply stp_closed2. eauto. Qed.
Lemma beq_nat_true_eq: forall A, beq_nat A A = true.
Proof. intros. eapply beq_nat_true_iff. eauto. Qed.
Fixpoint tsize (T: ty) { struct T }: nat :=
match T with
| TTop => 1
| TBot => 1
| TSel T1 l => 1
| TFun l T1 T2 => S (tsize T1 + tsize T2)
| TTyp l T1 T2 => S (tsize T1 + tsize T2)
| TBind T1 => S (tsize T1)
| TAnd T1 T2 => S (tsize T1 + tsize T2)
| TOr T1 T2 => S (tsize T1 + tsize T2)
end.
Lemma open_preserves_size: forall T b x j,
tsize T = tsize (open j (TVar b x) T).
Proof.
intros T. induction T; intros; simpl; eauto.
Qed.
Lemma stpd_refl_aux: forall n GH G1 T1,
closed (length GH) (length G1) 0 T1 ->
tsize T1 < n ->
stpd GH G1 T1 T1.
Proof.
intros n. induction n; intros; try omega.
inversion H; subst; simpl in H0.
- Case "bot". exists 1. eauto.
- Case "top". exists 1. eauto.
- Case "fun". eapply stpd_fun; eauto.
eapply IHn; eauto; omega.
eapply IHn; eauto.
simpl. apply closed_open. simpl. eapply closed_upgrade_gh. eauto. omega.
econstructor. omega.
rewrite <- open_preserves_size. omega.
- Case "typ". eapply stpd_typ; try eapply IHn; eauto; try omega.
- Case "sel". exists 1. eapply stp_selx. eauto.
- Case "bind".
remember (open 0 (TVar false (length GH)) T0) as T0'.
destruct (IHn (T0'::GH) G1 T0').
subst. eapply closed_open. eapply closed_upgrade_gh. eauto.
simpl. omega. simpl. econstructor. omega.
subst. rewrite <- open_preserves_size. omega.
eexists. eapply stp_bindx; eauto.
- Case "and".
destruct (IHn GH G1 T0 H1). omega.
destruct (IHn GH G1 T2 H2). omega.
eexists. eapply stp_and2. eapply stp_and11. eauto. eauto. eapply stp_and12. eauto. eauto.
- Case "or".
destruct (IHn GH G1 T0 H1). omega.
destruct (IHn GH G1 T2 H2). omega.
eexists. eapply stp_or1. eapply stp_or21. eauto. eauto. eapply stp_or22. eauto. eauto.
Qed.
Lemma stpd_refl: forall GH G1 T1,
closed (length GH) (length G1) 0 T1 ->
stpd GH G1 T1 T1.
Proof.
intros. apply stpd_refl_aux with (n:=S (tsize T1)); eauto.
Qed.
Lemma stpd_reg1 : forall GH G1 T1 T2,
stpd GH G1 T1 T2 ->
stpd GH G1 T1 T1.
Proof. intros. eapply stpd_refl. eapply stpd_closed1. eauto. Qed.
Lemma stpd_reg2 : forall GH G1 T1 T2,
stpd GH G1 T1 T2 ->
stpd GH G1 T2 T2.
Proof. intros. eapply stpd_refl. eapply stpd_closed2. eauto. Qed.
(*# Infrastructure Lemmas #*)
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: TSel _ _ = _ |- _ => inversion H1
| H1: TTyp _ _ _ = _ |- _ => inversion H1
| H1: TVar _ _ = _ |- _ => inversion H1
| H1: TFun _ _ _ = _ |- _ => inversion H1
| H1: TBind _ = _ |- _ => inversion H1
| H1: TAnd _ _ = _ |- _ => inversion H1
| H1: TOr _ _ = _ |- _ => inversion H1
| _ => idtac
end.
Ltac invstp_var := match goal with
| H1: stp _ true _ _ TBot (TVar _ _) _ |- _ => inversion H1
| H1: stp _ true _ _ TTop (TVar _ _) _ |- _ => inversion H1
| H1: stp _ true _ _ (TFun _ _ _) (TVar _ _) _ |- _ => inversion H1
| H1: stp _ true _ _ (TTyp _ _ _) (TVar _ _) _ |- _ => inversion H1
| H1: stp _ true _ _ (TAnd _ _) (TVar _ _) _ |- _ => inversion H1
| H1: stp _ true _ _ (TOr _ _) (TVar _ _) _ |- _ => inversion H1
| _ => idtac
end.
Lemma map_eq_some: forall {X} {Y} (f: X -> Y) OT (T: X),
eq_some OT T ->
eq_some (option_map f OT) (f T).
Proof.
intros. destruct H as [EN | ES]; subst; unfold eq_some; simpl; auto.
Qed.
Lemma subst_eq_some: forall OT T U,
eq_some OT T ->
eq_some (option_map (subst U) OT) (subst U T).
Proof.
intros. apply map_eq_some; auto.
Qed.
Lemma vr_closed_no_open: forall p b x k l j,
vr_closed l k j p ->
p = vr_open j (TVar b x) p.
Proof.
intros. inversion H; eauto; subst.
simpl.
assert (j <> x0) as A. omega.
apply beq_nat_false_iff in A.
rewrite A. auto.
Qed.
Lemma closed_no_open: forall T b x k l j,
closed l k j T ->
T = open j (TVar b x) T.
Proof.
intros. induction H; intros; eauto;
try solve [compute; compute in IHclosed; rewrite <-IHclosed; auto];
try solve [compute; compute in IHclosed1; compute in IHclosed2; rewrite <-IHclosed1; rewrite <-IHclosed2; auto].
simpl. f_equal. erewrite <- vr_closed_no_open; eauto.
Qed.
Lemma vr_closed_no_subst: forall p j k TX,
vr_closed 0 j k p ->
vr_subst TX p = p.
Proof.
intros. inversion H; simpl; eauto; subst. omega.
Qed.
Lemma closed_no_subst: forall T j k TX,
closed 0 j k T ->
subst TX T = T.
Proof.