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dot-smallstep2.v
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dot-smallstep2.v
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(* smallstep proof *)
(* compared to 1, adds packing/unpacking in term typing *)
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
| TMem : ty -> ty -> ty (* intro *)
| TVar : bool -> id -> ty
| TVarB : id -> ty
| TSel : ty -> ty (* elim *)
| TBind : ty -> ty
| TAnd : ty -> ty -> ty
.
Inductive tm : Type :=
| tvar : bool -> id -> tm
| tbool : bool -> tm
| tobj : dm -> tm (* todo: multiple members *)
| tapp : tm -> tm -> tm
with dm : Type :=
| dfun : ty -> ty -> tm -> dm
| dty : ty -> dm
.
Inductive vl : Type :=
| vbool : bool -> vl
| vobj : dm -> vl
.
Definition venv := list vl.
Definition tenv := list ty.
Hint Unfold venv.
Hint Unfold tenv.
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.
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_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_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_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)
| 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)
.
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)
| TAnd T1 T2 => TAnd (open k u T1) (open k u T2)
end.
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)
| TSel T1 => TSel (subst U T1)
| TVarB i => TVarB i
| TVar true i => TVar true i
| TVar false i => if beq_nat i 0 then U else TVar false (i-1)
| TFun T1 T2 => TFun (subst U T1) (subst U T2)
| TBind T2 => TBind (subst U T2)
| TAnd T1 T2 => TAnd (subst U T1) (subst U T2)
end.
Inductive has_type : tenv -> venv -> tm -> ty -> nat -> Prop :=
| T_Varx : forall m GH G1 x T n1,
vtp m G1 x T n1 ->
has_type GH G1 (tvar true x) T (S n1)
| T_Vary : 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)
(* todo: recursive objects with multiple members *)
| T_Mem : forall GH G1 T11 n1,
closed (length GH) (length G1) 0 T11 ->
has_type GH G1 (tobj (dty T11)) (TMem T11 T11) (S n1)
| T_Abs : forall GH G1 T11 T12 t12 n1,
has_type (T11::GH) G1 t12 T12 n1 ->
closed (length GH) (length G1) 0 T11 ->
closed (length GH) (length G1) 0 T12 ->
has_type GH G1 (tobj (dfun T11 T12 t12)) (TFun T11 T12) (S n1)
| T_App : forall T1 T2 GH G1 t1 t2 n1 n2,
has_type GH G1 t1 (TFun T1 T2) n1 ->
has_type GH G1 t2 T1 n2 ->
has_type GH G1 (tapp t1 t2) T2 (S (n1+n2))
(* todo: add app-var and use dependent functions *)
| T_Sub : forall GH G1 t T1 T2 n1 n2,
has_type GH G1 t T1 n1 ->
stp2 GH G1 T1 T2 n2 ->
has_type GH G1 t T2 (S (n1 + n2))
with stp2: tenv -> venv -> ty -> ty -> nat -> Prop :=
| stp2_bot: forall GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp2 GH G1 TBot T (S n1)
| stp2_top: forall GH G1 T n1,
closed (length GH) (length G1) 0 T ->
stp2 GH G1 T TTop (S n1)
| stp2_bool: forall GH G1 n1,
stp2 GH G1 TBool TBool (S n1)
(* todo: enabled dependent functions *)
| stp2_fun: forall GH G1 T1 T2 T3 T4 n1 n2,
stp2 GH G1 T3 T1 n1 ->
stp2 GH G1 T2 T4 n2 ->
stp2 GH G1 (TFun T1 T2) (TFun T3 T4) (S (n1+n2))
| stp2_mem: forall GH G1 T1 T2 T3 T4 n1 n2,
stp2 GH G1 T3 T1 n2 ->
stp2 GH G1 T2 T4 n1 ->
stp2 GH G1 (TMem T1 T2) (TMem T3 T4) (S (n1+n2))
| stp2_varx: forall GH G1 x n1,
x < length G1 ->
stp2 GH G1 (TVar true x) (TVar true x) (S n1)
| stp2_varax: forall GH G1 x n1,
x < length GH ->
stp2 GH G1 (TVar false x) (TVar false x) (S n1)
| stp2_strong_sel1: forall GH G1 T2 TX x n1,
index x G1 = Some (vobj (dty TX)) ->
stp2 [] G1 TX T2 n1 ->
stp2 GH G1 (TSel (TVar true x)) T2 (S n1)
| stp2_strong_sel2: forall GH G1 T1 TX x n1,
index x G1 = Some (vobj (dty TX)) ->
stp2 [] G1 T1 TX n1 ->
stp2 GH G1 T1 (TSel (TVar true x)) (S n1)
| stp2_sel1: forall GH G1 T2 x n1,
htp GH G1 x (TMem TBot T2) n1 ->
stp2 GH G1 (TSel (TVar false x)) T2 (S n1)
| stp2_sel2: forall GH G1 T1 x n1,
htp GH G1 x (TMem T1 TTop) n1 ->
stp2 GH G1 T1 (TSel (TVar false x)) (S n1)
| stp2_selx: forall GH G1 T1 n1,
closed (length GH) (length G1) 0 T1 ->
stp2 GH G1 (TSel T1) (TSel T1) (S n1)
| stp2_bind1: forall GH G1 T1 T1' T2 n1,
htp (T1'::GH) G1 (length GH) T2 n1 ->
T1' = (open 0 (TVar false (length GH)) T1) ->
closed (length GH) (length G1) 1 T1 ->
closed (length GH) (length G1) 0 T2 ->
stp2 GH G1 (TBind T1) T2 (S n1)
| stp2_bindx: forall GH G1 T1 T1' T2 T2' n1,
htp (T1'::GH) G1 (length GH) 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 ->
stp2 GH G1 (TBind T1) (TBind T2) (S n1)
| stp2_and11: forall GH G1 T1 T2 T n1,
stp2 GH G1 T1 T n1 ->
closed (length GH) (length G1) 0 T2 ->
stp2 GH G1 (TAnd T1 T2) T (S n1)
| stp2_and12: forall GH G1 T1 T2 T n1,
stp2 GH G1 T2 T n1 ->
closed (length GH) (length G1) 0 T1 ->
stp2 GH G1 (TAnd T1 T2) T (S n1)
| stp2_and2: forall GH G1 T1 T2 T n1 n2,
stp2 GH G1 T T1 n1 ->
stp2 GH G1 T T2 n2 ->
stp2 GH G1 T (TAnd T1 T2) (S (n1+n2))
| stp2_transf: forall GH G1 T1 T2 T3 n1 n2,
stp2 GH G1 T1 T2 n1 ->
stp2 GH G1 T2 T3 n2 ->
stp2 GH G1 T1 T3 (S (n1+n2))
with htp: tenv -> venv -> nat -> ty -> nat -> Prop :=
| htp_var: forall GH G1 x TX n1,
index x GH = Some TX ->
closed (length GH) (length G1) 0 TX ->
htp GH G1 x TX (S n1)
| htp_bind: forall GH G1 x TX n1,
htp GH G1 x (TBind TX) n1 ->
closed 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 ->
stp2 GL G1 T1 T2 n2 ->
length GL = S x ->
GH = GU ++ GL ->
htp GH G1 x T2 (S (n1+n2))
with vtp : nat -> venv -> nat -> ty -> nat -> Prop :=
| vtp_top: forall m G1 x n1,
x < length G1 ->
vtp m G1 x TTop (S n1)
| vtp_bool: forall m G1 x b n1,
index x G1 = Some (vbool b) ->
vtp m G1 x (TBool) (S (n1))
| vtp_mem: forall m G1 x TX T1 T2 n1 n2,
index x G1 = Some (vobj (dty TX)) ->
stp2 [] G1 T1 TX n1 ->
stp2 [] G1 TX T2 n2 ->
vtp m G1 x (TMem T1 T2) (S (n1+n2))
| vtp_fun: forall m G1 x T1 T2 T3 T4 t n1 n2 n3,
index x G1 = Some (vobj (dfun T1 T2 t)) ->
has_type [T1] G1 t T2 n3 ->
stp2 [] G1 T3 T1 n1 ->
stp2 [] G1 T2 T4 n2 ->
vtp m G1 x (TFun T3 T4) (S (n1+n2+n3))
| vtp_bind: forall m G1 x T2 n1,
vtp m G1 x (open 0 (TVar true x) T2) n1 ->
closed 0 (length G1) 1 T2 ->
vtp (S m) G1 x (TBind T2) (S (n1))
| vtp_sel: forall m G1 x y TX n1,
index y G1 = Some (vobj (dty TX)) ->
vtp m G1 x TX n1 ->
vtp m G1 x (TSel (TVar true y)) (S (n1))
| vtp_and: forall m m1 m2 G1 x T1 T2 n1 n2,
vtp m1 G1 x T1 n1 ->
vtp m2 G1 x T2 n2 ->
m1 <= m -> m2 <= m ->
vtp m G1 x (TAnd T1 T2) (S (n1+n2))
.
Definition has_typed GH G1 x T1 := exists n, has_type GH G1 x T1 n.
Definition stpd2 GH G1 T1 T2 := exists n, stp2 GH G1 T1 T2 n.
Definition htpd GH G1 x T1 := exists n, htp GH G1 x T1 n.
Definition vtpd m G1 x T1 := exists n, vtp m G1 x T1 n.
Definition vtpdd m G1 x T1 := exists m1 n, vtp m1 G1 x T1 n /\ m1 <= m.
Hint Constructors stp2.
Hint Constructors vtp.
Ltac ep := match goal with
| [ |- stp2 ?GH ?G1 ?T1 ?T2 ?N ] => assert (exists (n:nat), stp2 GH G1 T1 T2 n) as EEX
end.
Ltac eu := match goal with
| H: has_typed _ _ _ _ |- _ => destruct H as [? H]
| H: stpd2 _ _ _ _ |- _ => destruct H as [? H]
| H: htpd _ _ _ _ |- _ => destruct H as [? H]
| H: vtpd _ _ _ _ |- _ => destruct H as [? H]
| H: vtpdd _ _ _ _ |- _ => destruct H as [? [? [H ?]]]
end.
Lemma stpd2_bot: forall GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd2 GH G1 TBot T.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_top: forall GH G1 T,
closed (length GH) (length G1) 0 T ->
stpd2 GH G1 T TTop.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_bool: forall GH G1,
stpd2 GH G1 TBool TBool.
Proof. intros. exists 1. eauto. Qed.
Lemma stpd2_fun: forall GH G1 T1 T2 T3 T4,
stpd2 GH G1 T3 T1 ->
stpd2 GH G1 T2 T4 ->
stpd2 GH G1 (TFun T1 T2) (TFun T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_mem: forall GH G1 T1 T2 T3 T4,
stpd2 GH G1 T3 T1 ->
stpd2 GH G1 T2 T4 ->
stpd2 GH G1 (TMem T1 T2) (TMem T3 T4).
Proof. intros. repeat eu. eexists. eauto. Qed.
Lemma stpd2_transf: forall GH G1 T1 T2 T3,
stpd2 GH G1 T1 T2 ->
stpd2 GH G1 T2 T3 ->
stpd2 GH G1 T1 T3.
Proof. intros. repeat eu. eexists. eauto. Qed.
Hint Constructors ty.
Hint Constructors vl.
Hint Constructors stp2.
Hint Constructors vtp.
Hint Constructors htp.
Hint Constructors has_type.
Hint Unfold has_typed.
Hint Unfold stpd2.
Hint Unfold vtpd.
Hint Unfold vtpdd.
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 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 all_extend: forall ni,
(forall GH v1 G1 T1 T2 n,
stp2 GH G1 T1 T2 n -> n < ni ->
stp2 GH (v1::G1) T1 T2 n) /\
(forall m v1 x G1 T2 n,
vtp m G1 x T2 n -> n < ni ->
vtp m (v1::G1) x 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).
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.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. simpl. eauto.
- econstructor. eauto.
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega.
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega.
- econstructor. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eauto. eapply closed_extend. eauto. eapply closed_extend. eauto.
- eapply stp2_bindx. eapply IHn. eauto. omega. eauto. eauto. eapply closed_extend. eauto. eapply closed_extend. eauto.
- eapply stp2_and11. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp2_and12. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- eapply stp2_and2. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- eapply stp2_transf. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
(* vtp *)
- econstructor. simpl. eauto.
- econstructor. eapply index_extend. eauto.
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega. eapply closed_extend. eauto.
- econstructor. eapply index_extend. eauto. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega. eauto. eauto.
(* htp *)
- econstructor. eauto. eapply closed_extend. eauto.
- eapply htp_bind. 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 IHn. eauto. omega.
- 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 closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eapply closed_extend. eauto. eapply closed_extend. eauto.
- econstructor. eapply IHn. eauto. omega. eapply IHn. eauto. omega.
- econstructor. eapply IHn. eauto. omega. eapply IHn. 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. 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. omega.
eapply IHclosed. omega.
Qed.
Lemma closed_open: forall j k n b V T, closed k n (j+1) T -> 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; eauto; try eapply IHT2; eauto; try eapply IHT; eauto.
- Case "TVarB". simpl.
case_eq (beq_nat j i); intros E. eauto.
econstructor. eapply beq_nat_false_iff in E. omega.
- eapply closed_upgrade; eauto.
Qed.
Lemma all_closed: forall ni,
(forall GH G1 T1 T2 n,
stp2 GH G1 T1 T2 n -> n < ni ->
closed (length GH) (length G1) 0 T1) /\
(forall GH G1 T1 T2 n,
stp2 GH G1 T1 T2 n -> n < ni ->
closed (length GH) (length G1) 0 T2) /\
(forall m x G1 T2 n,
vtp m G1 x T2 n -> n < ni ->
x < length G1) /\
(forall m x G1 T2 n,
vtp m G1 x T2 n -> n < ni ->
closed 0 (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).
Proof.
intros n. induction n. repeat split; intros; omega.
repeat split; intros; inversion H; destruct IHn as [IHS1 [IHS2 [IHV1 [IHV2 [IHH1 [IHH2 IHT]]]]]].
(* stp left *)
- econstructor.
- eauto.
- econstructor.
- econstructor. eapply IHS2. eauto. omega. eapply IHS1. eauto. omega.
- econstructor. eapply IHS2. eauto. omega. eapply IHS1. eauto. omega.
- econstructor. simpl. eauto.
- econstructor. eauto.
- econstructor. econstructor. eapply index_max. eauto.
- eapply closed_upgrade_gh. eapply IHS1. eapply H2. 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.
(* stp right *)
- eauto.
- econstructor.
- econstructor.
- econstructor. eapply IHS1. eauto. omega. eapply IHS2. eauto. omega.
- econstructor. eapply IHS1. eauto. omega. eapply IHS2. eauto. omega.
- econstructor. simpl. eauto.
- econstructor. eauto.
- eapply closed_upgrade_gh. eapply IHS2. eapply H2. 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.
- eapply IHS2. eauto. omega.
(* vtp left *)
- eauto.
- eapply index_max. eauto.
- eapply index_max. eauto.
- eapply index_max. eauto.
- eapply IHV1. eauto. omega.
- eapply IHV1. eauto. omega.
- eapply IHV1. eauto. omega.
(* vtp right *)
- econstructor.
- econstructor.
- change 0 with (length ([]:tenv)) at 1. econstructor. eapply IHS1. eauto. omega. eapply IHS2. eauto. omega.
- change 0 with (length ([]:tenv)) at 1. econstructor. eapply IHS1. eauto. omega. eapply IHS2. eauto. omega.
- econstructor. eauto. (* eapply IHV2 in H1. eauto. omega. *)
- econstructor. econstructor. eapply index_max. eauto.
- econstructor. eapply IHV2. eauto. omega. eapply IHV2. eauto. omega.
(* htp left *)
- eapply index_max. eauto.
- eapply IHH1. eauto. omega.
- eapply IHH1. eauto. omega.
(* htp right *)
- eauto.
- 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 *)
- eapply closed_upgrade_gh. eapply IHV2. eauto. omega. omega.
- eauto.
- econstructor. eapply closed_upgrade_gh. eauto. omega.
- eapply IHT in H1. inversion H1; subst. eauto. omega.
- econstructor. eauto. eauto.
- econstructor. eauto. eauto.
- eapply IHT in H1. inversion H1. eauto. omega.
- eapply IHS2. eauto. omega.
Qed.
Lemma vtp_extend : forall m v1 x G1 T2 n,
vtp m G1 x T2 n ->
vtp m (v1::G1) x T2 n.
Proof. intros. eapply all_extend. eauto. eauto. 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 stp2_extend : forall GH v1 G1 T1 T2 n,
stp2 GH G1 T1 T2 n ->
stp2 GH (v1::G1) T1 T2 n.
Proof. intros. eapply all_extend. eauto. eauto. Qed.
Lemma stp2_extend_mult : forall GH G1 G' T1 T2 n,
stp2 GH G1 T1 T2 n ->
stp2 GH (G'++G1) T1 T2 n.
Proof. intros. induction G'. simpl. eauto. simpl. eapply stp2_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 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 vtp_closed: forall m G1 x T2 n1,
vtp m G1 x T2 n1 ->
closed 0 (length G1) 0 T2.
Proof. intros. eapply all_closed. eauto. eauto. Qed.
Lemma vtp_closed1: forall m G1 x T2 n1,
vtp m G1 x T2 n1 ->
x < length G1.
Proof. intros. eapply all_closed. 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. 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_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.
- split; eauto. eapply vtp_closed1; eauto.
- simpl in H. inversion H.
Qed.
Lemma stp2_closed1 : forall GH G1 T1 T2 n1,
stp2 GH G1 T1 T2 n1 ->
closed (length GH) (length G1) 0 T1.
Proof. intros. edestruct all_closed. eapply H0. eauto. eauto. Qed.
Lemma stp2_closed2 : forall GH G1 T1 T2 n1,
stp2 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 stpd2_closed1 : forall GH G1 T1 T2,
stpd2 GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T1.
Proof. intros. eu. eapply stp2_closed1. eauto. Qed.
Lemma stpd2_closed2 : forall GH G1 T1 T2,
stpd2 GH G1 T1 T2 ->
closed (length GH) (length G1) 0 T2.
Proof. intros. eu. eapply stp2_closed2. eauto. Qed.
Lemma beq_nat_true_eq: forall A, beq_nat A A = true.
Proof. intros. eapply beq_nat_true_iff. eauto. Qed.
Lemma stpd2_refl: forall GH G1 T1,
closed (length GH) (length G1) 0 T1 ->
stpd2 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 "mem". eapply stpd2_mem; 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.
eexists. eapply stp2_varx; eauto.
- Case "varb". inversion H4.
- Case "sel". exists 1. eapply stp2_selx. eauto.
- Case "bind".
eexists. eapply stp2_bindx. eapply htp_var. simpl. rewrite beq_nat_true_eq. eauto.
instantiate (1:=open 0 (TVar false (length GH)) T1).
eapply closed_open. simpl. eapply closed_upgrade_gh. eauto. omega. econstructor. simpl. omega.
eauto. eauto. eauto. eauto.
- Case "and".
destruct IHT1_1. eauto. destruct IHT1_2. eauto.
eexists. eapply stp2_and2. eapply stp2_and11. eauto. eauto. eapply stp2_and12. eauto. eauto.
Grab Existential Variables.
apply 0. apply 0.
Qed.
Lemma stpd2_reg1 : forall GH G1 T1 T2,
stpd2 GH G1 T1 T2 ->
stpd2 GH G1 T1 T1.
Proof. intros. eapply stpd2_refl. eapply stpd2_closed1. eauto. Qed.
Lemma stpd2_reg2 : forall GH G1 T1 T2,
stpd2 GH G1 T1 T2 ->
stpd2 GH G1 T2 T2.
Proof. intros. eapply stpd2_refl. eapply stpd2_closed2. 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
| H1: TAnd _ _ = _ |- _ => 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: stp2 _ true _ _ (TAnd _ _) (TVar _ _) _ |- _ => inversion H1
| _ => idtac
end.
Definition substt x T := (subst (TVar true x) T).
Hint Immediate substt.
Lemma closed_no_open: forall T x k l j,
closed l k j T ->
T = open j (TVar false 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].
Case "TSelB".
simpl.
assert (k <> x0). omega.
apply beq_nat_false_iff in H0.
rewrite H0. auto.
Qed.
Lemma closed_no_subst: forall T j k TX,
closed 0 j k T ->
subst TX T = T.
Proof.
intros T. induction T; intros; inversion H; simpl; eauto;
try rewrite (IHT j (S k) TX); eauto;
(* try rewrite (IHT2 (S j) TX); eauto; *)
try rewrite (IHT j k TX); eauto;
try rewrite (IHT1 j k TX); eauto;
try rewrite (IHT2 j k TX); eauto.
subst. inversion H4.
eapply closed_upgrade. eauto. eauto.
Qed.
Lemma closed_subst: forall j n k V T, closed (n+1) k j T -> closed n k 0 V -> closed n k j (subst V T).
Proof.
intros. generalize dependent j. induction T; intros; inversion H; try econstructor; try eapply IHT1; eauto; try eapply IHT2; eauto; try eapply IHT; eauto.
- Case "TSelH". simpl.
case_eq (beq_nat i 0); intros E. eapply closed_upgrade. eapply closed_upgrade_gh. eauto. eauto. omega. econstructor. subst.
assert (i > 0). eapply beq_nat_false_iff in E. omega. omega.
Qed.
(* not used? *)
Lemma subst_open_commute_m: forall j k n m V T2, closed (n+1) k (j+1) T2 -> closed m k 0 V ->
subst V (open j (TVar false (n+1)) T2) = open j (TVar false n) (subst V T2).
Proof.
intros. generalize dependent j. generalize dependent n.
induction T2; intros; inversion H; simpl; eauto;
try rewrite IHT2_1; try rewrite IHT2_2; try rewrite IHT2; eauto.
simpl. case_eq (beq_nat i 0); intros E.
eapply closed_no_open. eapply closed_upgrade. eauto. omega.
simpl. eauto.
simpl. case_eq (beq_nat j i); intros E.
simpl. case_eq (beq_nat (n+1) 0); intros E2. eapply beq_nat_true_iff in E2. omega.
assert (n+1-1 = n) as A. omega. rewrite A. eauto.
eauto.
Qed.
(* not used? *)
Lemma subst_open_commute: forall j k n V T2, closed (n+1) k (j+1) T2 -> closed 0 k 0 V ->
subst V (open j (TVar false (n+1)) T2) = open j (TVar false n) (subst V T2).
Proof.
intros. eapply subst_open_commute_m; eauto.
Qed.
Lemma subst_open_commute0: forall T0 n j TX,
closed 0 n (j+1) T0 ->
(subst TX (open j (TVar false 0) T0)) = open j TX T0.
Proof.
intros T0 n. induction T0; intros.
eauto. eauto. eauto.
simpl. inversion H. rewrite IHT0_1. rewrite IHT0_2. eauto. eauto. eauto.
simpl. inversion H. rewrite IHT0_1. rewrite IHT0_2. eauto. eauto. eauto.
simpl. inversion H. omega. eauto.
simpl. inversion H. subst. destruct i. case_eq (beq_nat j 0); intros E; simpl; eauto.
case_eq (beq_nat j (S i)); intros E; simpl; eauto.
simpl. inversion H. rewrite IHT0. eauto. eauto.
simpl. inversion H. rewrite IHT0. eauto. subst. eauto.
simpl. inversion H. rewrite IHT0_1. rewrite IHT0_2. eauto. eauto. eauto.
Qed.
Lemma subst_open_commute1: forall T0 x x0 j,
(open j (TVar true x0) (subst (TVar true x) T0))
= (subst (TVar true x) (open j (TVar true x0) T0)).
Proof.
induction T0; intros.
eauto. eauto. eauto.
simpl. rewrite IHT0_1. rewrite IHT0_2. eauto. eauto. eauto.
simpl. rewrite IHT0_1. rewrite IHT0_2. eauto. eauto. eauto.
simpl. destruct b. simpl. eauto.
case_eq (beq_nat i 0); intros E. simpl. eauto. simpl. eauto.
simpl. case_eq (beq_nat j i); intros E. simpl. eauto. simpl. eauto.
simpl. rewrite IHT0. eauto.
simpl. rewrite IHT0. eauto.
simpl. rewrite IHT0_1. rewrite IHT0_2. eauto.
Qed.
Lemma subst_closed_id: forall x j k T2,
closed 0 j k T2 ->
substt x T2 = T2.
Proof. intros. eapply closed_no_subst. eauto. Qed.
Lemma closed_subst0: forall i j k x T2,
closed (i + 1) j k T2 -> x < j ->
closed i j k (substt x T2).
Proof. intros. eapply closed_subst. eauto. econstructor. eauto. Qed.
Lemma closed_subst1: forall i j k x T2,
closed i j k T2 -> x < j -> i <> 0 ->
closed (i-1) j k (substt x T2).
Proof.
intros. eapply closed_subst.
assert ((i - 1 + 1) = i) as R. omega.
rewrite R. eauto. econstructor. eauto.
Qed.
Lemma index_subst: forall GH TX T0 T3 x,
index (length (GH ++ [TX])) (T0 :: GH ++ [TX]) = Some T3 ->
index (length GH) (map (substt x) (T0 :: GH)) = Some (substt x T3).
Proof.
intros GH. induction GH; intros; inversion H.
- eauto.
- rewrite beq_nat_true_eq in H1. inversion H1. subst. simpl.
rewrite map_length. rewrite beq_nat_true_eq. eauto.
Qed.
Lemma index_subst1: forall GH TX T3 x x0,
index x0 (GH ++ [TX]) = Some T3 -> x0 <> 0 ->
index (x0-1) (map (substt x) GH) = Some (substt x T3).
Proof.
intros GH. induction GH; intros; inversion H.
- eapply beq_nat_false_iff in H0. rewrite H0 in H2. inversion H2.
- simpl.
assert (beq_nat (x0 - 1) (length (map (substt x) GH)) = beq_nat x0 (length (GH ++ [TX]))). {
case_eq (beq_nat x0 (length (GH ++ [TX]))); intros E.
eapply beq_nat_true_iff. rewrite map_length. eapply beq_nat_true_iff in E. subst x0.
rewrite app_length. simpl. omega.
eapply beq_nat_false_iff. eapply beq_nat_false_iff in E.
rewrite app_length in E. simpl in E. rewrite map_length.
destruct x0. destruct H0. reflexivity. omega.
}
rewrite H1. case_eq (beq_nat x0 (length (GH ++ [TX]))); intros E; rewrite E in H2.
inversion H2. subst. eauto. eauto.
Qed.
Lemma index_hit0: forall (GH:tenv) TX T2,
index 0 (GH ++ [TX]) = Some T2 -> T2 = TX.
Proof.
intros GH. induction GH; intros; inversion H.
- eauto.
- rewrite app_length in H1. simpl in H1.
remember (length GH + 1) as L. destruct L. omega. eauto.
Qed.
Lemma subst_open: forall TX n x j,
(substt x (open j (TVar false (n+1)) TX)) =
(open j (TVar false n) (substt x TX)).
Proof.
intros TX. induction TX; intros; eauto.
- unfold substt. simpl. unfold substt in IHTX1. unfold substt in IHTX2. erewrite <-IHTX1. erewrite <-IHTX2. eauto.
- unfold substt. simpl. unfold substt in IHTX1. unfold substt in IHTX2. erewrite <-IHTX1. erewrite <-IHTX2. eauto.
- unfold substt. simpl. destruct b. eauto.
case_eq (beq_nat i 0); intros E. eauto. eauto.
- unfold substt. simpl.
case_eq (beq_nat j i); intros E. simpl.
assert (beq_nat (n + 1) 0 = false). eapply beq_nat_false_iff. omega.
assert ((n + 1 - 1 = n)). omega.
rewrite H. rewrite H0. eauto. eauto.
- unfold substt. simpl. unfold substt in IHTX. erewrite <-IHTX. eauto.
- unfold substt. simpl. unfold substt in IHTX. erewrite <-IHTX. eauto.
- unfold substt. simpl. unfold substt in IHTX1. unfold substt in IHTX2. erewrite <-IHTX1. erewrite <-IHTX2. eauto.
Qed.
Lemma subst_open3: forall TX0 (GH:tenv) TX x,
(substt x (open 0 (TVar false (length (GH ++ [TX]))) TX0)) =
(open 0 (TVar false (length GH)) (substt x TX0)).
Proof. intros. rewrite app_length. simpl. eapply subst_open. Qed.
Lemma subst_open4: forall T0 (GH:tenv) TX x,
substt x (open 0 (TVar false (length (GH ++ [TX]))) T0) =
open 0 (TVar false (length (map (substt x) GH))) (substt x T0).
Proof. intros. rewrite map_length. eapply subst_open3. Qed.
Lemma subst_open5: forall (GH:tenv) T0 x xi,
xi <> 0 -> substt x (open 0 (TVar false xi) T0) =
open 0 (TVar false (xi-1)) (substt x T0).
Proof.
intros. remember (xi-1) as n. assert (xi=n+1) as R. omega. rewrite R.
eapply subst_open.
Qed.
Lemma subst_open_commute0b: forall x T1 n,
substt x (open n (TVar false 0) T1) = open n (TVar true x) (substt x T1).
Proof.
unfold substt.