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fsub_mut.v
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fsub_mut.v
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(*
FSub (F<:) with Mutable References
T ::= Top | X | T -> T | Forall Z <: T. T^Z | Ref T
t ::= x | lambda x:T.t | Lambda X<:T.t | t t | t [T] | ret t | !t | t := t
*)
Require Export SfLib.
Require Export Arith.EqNat.
Require Export Arith.Le.
Require Import Coq.Program.Equality.
Require Import Omega.
Require Import NPeano.
(* ### Syntax ### *)
Definition id := nat.
Inductive ty : Type :=
| TTop : ty
| TFun : ty -> ty -> ty
| TAll : ty -> ty -> ty
| TVarF : id -> ty (* free type variable, in concrete environment *)
| TVarH : id -> ty (* free type variable, in abstract environment *)
| TVarB : id -> ty (* locally-bound type variable *)
| TCell : ty -> ty
.
Inductive tm : Type :=
| tvar : id -> tm
| tabs : ty -> tm -> tm
| tapp : tm -> tm -> tm
| ttabs : ty -> tm -> tm
| ttapp : tm -> ty -> tm
| tnew : tm -> tm
| tget : tm -> tm
| tset : tm -> tm -> tm
.
Inductive binding {X: Type} :=
| bind_tm : X -> binding
| bind_ty : X -> binding
.
Inductive vl : Type :=
(* a closure for a term abstraction *)
| vabs : list vl (*H*) -> ty -> tm -> vl
(* a closure for a type abstraction *)
| vtabs : list vl (*H*) -> ty -> tm -> vl
(* a closure over a type *)
| vty : list vl (*H*) -> ty -> vl
(* a location, for a mutable reference cell *)
| vloc : id -> vl
.
Definition tenv := list (@binding ty). (* Gamma environment: static *)
Definition venv := list vl. (* H environment: run-time *)
Definition aenv := list (venv*ty). (* J environment: abstract at run-time *)
(* ### Representation of Bindings ### *)
(* An environment is a list of values, indexed by decrementing ids. *)
Fixpoint indexr {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 indexr n l'
end.
Fixpoint update {X : Type} (n : id) (l : list X) (v: X) : list X :=
match l with
| [] => []
| a :: l' =>
if (beq_nat n (length l')) then v :: l' else a :: update n l' v
end.
Inductive closed: nat(*B*) -> nat(*H*) -> nat(*F*) -> ty -> Prop :=
| cl_top: forall i j k,
closed i j k TTop
| 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_all: forall i j k T1 T2,
closed i j k T1 ->
closed (S i) j k T2 ->
closed i j k (TAll T1 T2)
| cl_cell: forall i j k T,
closed i j k T ->
closed i j k (TCell T)
| cl_sel: forall i j k x,
k > x ->
closed i j k (TVarF x)
| cl_selh: forall i j k x,
j > x ->
closed i j k (TVarH x)
| cl_selb: forall i j k x,
i > x ->
closed i j k (TVarB x)
.
(* open define a locally-nameless encoding wrt to TVarB type variables. *)
(* substitute type u for all occurrences of (TVarB k) *)
Fixpoint open_rec (k: nat) (u: ty) (T: ty) { struct T }: ty :=
match T with
| TTop => TTop
| TFun T1 T2 => TFun (open_rec k u T1) (open_rec k u T2)
| TAll T1 T2 => TAll (open_rec k u T1) (open_rec (S k) u T2)
| TCell T1 => TCell (open_rec k u T1)
| TVarF x => TVarF x
| TVarH i => TVarH i
| TVarB i => if beq_nat k i then u else TVarB i
end.
Definition open u T := open_rec 0 u T.
(* Locally-nameless encoding with respect to varH variables. *)
Fixpoint subst (U : ty) (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TFun T1 T2 => TFun (subst U T1) (subst U T2)
| TAll T1 T2 => TAll (subst U T1) (subst U T2)
| TCell T1 => TCell (subst U T1)
| TVarB i => TVarB i
| TVarF i => TVarF i
| TVarH i => if beq_nat i 0 then U else TVarH (i-1)
end.
Definition liftb (f: ty -> ty) b :=
match b with
| bind_tm T => bind_tm (f T)
| bind_ty T => bind_ty (f T)
end.
Definition substb (U: ty) := liftb (subst U).
Fixpoint nosubst (T : ty) {struct T} : Prop :=
match T with
| TTop => True
| TFun T1 T2 => nosubst T1 /\ nosubst T2
| TAll T1 T2 => nosubst T1 /\ nosubst T2
| TCell T1 => nosubst T1
| TVarB i => True
| TVarF i => True
| TVarH i => i <> 0
end.
(* ### Static Subtyping ### *)
(*
The first env is for looking up varF variables.
The first env matches the concrete runtime environment, and is
extended during type assignment.
The second env is for looking up varH variables.
The second env matches the abstract runtime environment, and is
extended during subtyping.
*)
Inductive stp: tenv -> tenv -> ty -> ty -> Prop :=
| stp_top: forall G1 GH T1,
closed 0 (length GH) (length G1) T1 ->
stp G1 GH T1 TTop
| 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_sel1: forall G1 GH T T2 x,
indexr x G1 = Some (bind_ty T) ->
closed 0 0 (length G1) T ->
stp G1 GH T T2 ->
stp G1 GH (TVarF x) T2
| stp_selx: forall G1 GH v x,
(* This is a bit looser than just being able to select on TMem vars. *)
indexr x G1 = Some v ->
stp G1 GH (TVarF x) (TVarF x)
| stp_sela1: forall G1 GH T T2 x,
indexr x GH = Some (bind_ty T) ->
closed 0 x (length G1) T ->
stp G1 GH T T2 ->
stp G1 GH (TVarH x) T2
| stp_selax: forall G1 GH v x,
(* This is a bit looser than just being able to select on TMem vars. *)
indexr x GH = Some v ->
stp G1 GH (TVarH x) (TVarH x)
| stp_all: forall G1 GH T1 T2 T3 T4 x,
stp G1 GH T3 T1 ->
x = length GH ->
closed 1 (length GH) (length G1) T2 ->
closed 1 (length GH) (length G1) T4 ->
stp G1 ((bind_ty T3)::GH) (open (TVarH x) T2) (open (TVarH x) T4) ->
stp G1 GH (TAll T1 T2) (TAll T3 T4)
| stp_cell: forall G1 GH T1 T2,
stp G1 GH T1 T2 ->
stp G1 GH T2 T1 ->
stp G1 GH (TCell T1) (TCell T2)
.
(* ### Type Assignment ### *)
Inductive has_type : tenv -> tm -> ty -> Prop :=
| t_var: forall x env T1,
indexr x env = Some (bind_tm T1) ->
stp env [] T1 T1 ->
has_type env (tvar x) T1
| t_app: forall env f x T1 T2,
has_type env f (TFun T1 T2) ->
has_type env x T1 ->
has_type env (tapp f x) T2
| t_abs: forall env y T1 T2,
has_type (bind_tm T1::env) y T2 ->
stp env [] (TFun T1 T2) (TFun T1 T2) ->
has_type env (tabs T1 y) (TFun T1 T2)
| t_tapp: forall env f T11 T12 T,
has_type env f (TAll T11 T12) ->
T = open T11 T12 ->
has_type env (ttapp f T11) T
| t_tabs: forall env y T1 T2,
has_type (bind_ty T1::env) y (open (TVarF (length env)) T2) ->
stp env [] (TAll T1 T2) (TAll T1 T2) ->
has_type env (ttabs T1 y) (TAll T1 T2)
| t_new: forall env x T1,
has_type env x T1 ->
stp env [] T1 T1 ->
has_type env (tnew x) (TCell T1)
| t_get: forall env T1 x,
has_type env x (TCell T1) ->
has_type env (tget x) T1
| t_set: forall env T1 x y,
has_type env x (TCell T1) ->
has_type env y T1 ->
has_type env (tset x y) T1
| t_sub: forall env e T1 T2,
has_type env e T1 ->
stp env [] T1 T2 ->
has_type env e T2
.
(* ### Runtime Subtyping ### *)
(* H1 T1 <: H2 T2 -| J *)
Inductive stp2: bool (* whether the last rule may not be transitivity *) ->
venv -> ty -> venv -> ty -> aenv ->
nat (* derivation size *) ->
Prop :=
| stp2_top: forall G1 G2 GH T n,
closed 0 (length GH) (length G1) T ->
stp2 true G1 T G2 TTop GH (S n)
| stp2_fun: forall G1 G2 T1 T2 T3 T4 GH n1 n2,
stp2 false G2 T3 G1 T1 GH n1 ->
stp2 false G1 T2 G2 T4 GH n2 ->
stp2 true G1 (TFun T1 T2) G2 (TFun T3 T4) GH (S (n1 + n2))
(* concrete type variables *)
| stp2_sel1: forall G1 G2 GX TX x T2 GH n1,
indexr x G1 = Some (vty GX TX) ->
closed 0 0 (length GX) TX ->
stp2 true GX TX G2 T2 GH n1 ->
stp2 true G1 (TVarF x) G2 T2 GH (S n1)
| stp2_sel2: forall G1 G2 GX TX x T1 GH n1,
indexr x G2 = Some (vty GX TX) ->
closed 0 0 (length GX) TX ->
stp2 false G1 T1 GX TX GH n1 ->
stp2 true G1 T1 G2 (TVarF x) GH (S n1)
| stp2_selx: forall G1 G2 v x1 x2 GH n,
indexr x1 G1 = Some v ->
indexr x2 G2 = Some v ->
stp2 true G1 (TVarF x1) G2 (TVarF x2) GH (S n)
(* abstract type variables *)
(* X<:T, one sided *)
| stp2_sela1: forall G1 G2 GX TX x T2 GH n1,
indexr x GH = Some (GX, TX) ->
closed 0 x (length GX) TX ->
stp2 false GX TX G2 T2 GH n1 ->
stp2 true G1 (TVarH x) G2 T2 GH (S n1)
| stp2_selax: forall G1 G2 v x GH n,
indexr x GH = Some v ->
stp2 true G1 (TVarH x) G2 (TVarH x) GH (S n)
| stp2_all: forall G1 G2 T1 T2 T3 T4 x GH n1 n2,
stp2 false G2 T3 G1 T1 GH n1 ->
x = length GH ->
closed 1 (length GH) (length G1) T2 ->
closed 1 (length GH) (length G2) T4 ->
stp2 false G1 (open (TVarH x) T2) G2 (open (TVarH x) T4) ((G2, T3)::GH) n2 ->
stp2 true G1 (TAll T1 T2) G2 (TAll T3 T4) GH (S (n1 + n2))
| stp2_cell: forall G1 G2 T1 T2 GH n1 n2,
stp2 false G1 T1 G2 T2 GH n1 ->
stp2 false G2 T2 G1 T1 GH n2 ->
stp2 true G1 (TCell T1) G2 (TCell T2) GH (S (n1+n2))
| stp2_wrapf: forall G1 G2 T1 T2 GH n1,
stp2 true G1 T1 G2 T2 GH n1 ->
stp2 false G1 T1 G2 T2 GH (S n1)
| stp2_transf: forall G1 G2 G3 T1 T2 T3 GH n1 n2,
stp2 true G1 T1 G2 T2 GH n1 ->
stp2 false G2 T2 G3 T3 GH n2 ->
stp2 false G1 T1 G3 T3 GH (S (n1+n2))
.
(* consistent environment *)
Inductive wf_env : aenv -> venv -> tenv -> Prop :=
| wfe_nil : forall sto, wf_env sto nil nil
| wfe_cons : forall sto v t vs ts,
val_type sto (v::vs) v t ->
wf_env sto vs ts ->
wf_env sto (cons v vs) (cons t ts)
(* value type assignment *)
with val_type : aenv -> venv -> vl -> @binding ty -> Prop :=
| v_ty: forall sto env venv tenv T1 TE,
wf_env sto venv tenv ->
(exists n, stp2 true venv T1 env TE [] n) ->
val_type sto env (vty venv T1) (bind_ty TE)
| v_abs: forall sto env venv tenv x y T1 T2 TE,
wf_env sto venv tenv ->
has_type (bind_tm T1::tenv) y T2 ->
length venv = x ->
(exists n, stp2 true venv (TFun T1 T2) env TE [] n) ->
val_type sto env (vabs venv T1 y) (bind_tm TE)
| v_tabs: forall sto env venv tenv x y T1 T2 TE,
wf_env sto venv tenv ->
has_type (bind_ty T1::tenv) y (open (TVarF x) T2) ->
length venv = x ->
(exists n, stp2 true venv (TAll T1 T2) env TE [] n) ->
val_type sto env (vtabs venv T1 y) (bind_tm TE)
| v_loc: forall sto venv venv1 b T1 TE,
indexr b sto = Some (venv1,T1) ->
(exists n, stp2 true venv1 (TCell T1) venv TE [] n) ->
val_type sto venv (vloc b) (bind_tm TE)
.
Inductive wf_envh : venv -> aenv -> tenv -> Prop :=
| wfeh_nil : forall vvs, wf_envh vvs nil nil
| wfeh_cons : forall t vs vvs ts,
wf_envh vvs vs ts ->
wf_envh vvs (cons (vvs,t) vs) (cons (bind_ty t) ts)
.
Inductive valh_type : venv -> aenv -> (venv*ty) -> (@binding ty) -> Prop :=
| v_tya: forall aenv venv T1,
valh_type venv aenv (venv, T1) (bind_ty T1)
.
(* ### Evaluation (Big-Step Semantics) ### *)
(*
None means timeout
Some None means stuck
Some (Some v)) means result v
Could use do-notation to clean up syntax.
*)
Fixpoint teval(n: nat)(sto: venv)(env: venv)(t: tm){struct n}: (venv*option (option vl)) :=
match n with
| 0 => (sto, None)
| S n =>
match t with
| tvar x => (sto, Some (indexr x env))
| tabs T y => (sto, Some (Some (vabs env T y)))
| ttabs T y => (sto, Some (Some (vtabs env T y)))
| tnew ex =>
match teval n sto env ex with
| (sto1, None) => (sto1, None)
| (sto1, Some None) => (sto1, Some None)
| (sto1, Some (Some v)) => (v::sto1, Some (Some (vloc (length sto1))))
end
| tget ex =>
match teval n sto env ex with
| (sto1, None) => (sto1, None)
| (sto1, Some None) => (sto1, Some None)
| (sto1, Some (Some (vty _ _))) => (sto1, Some None)
| (sto1, Some (Some (vabs _ _ _))) => (sto1, Some None)
| (sto1, Some (Some (vtabs _ _ _))) => (sto1, Some None)
| (sto1, Some (Some (vloc i))) => (sto1, Some (indexr i sto1))
end
| tset ex ey =>
match teval n sto env ex with
| (sto1, None) => (sto1, None)
| (sto1, Some None) => (sto1, Some None)
| (sto1, Some (Some (vty _ _))) => (sto1, Some None)
| (sto1, Some (Some (vabs _ _ _))) => (sto1, Some None)
| (sto1, Some (Some (vtabs _ _ _))) => (sto1, Some None)
| (sto1, Some (Some (vloc i))) =>
match teval n sto1 env ey with
| (sto2, None) => (sto2, None)
| (sto2, Some None) => (sto2, Some None)
| (sto2, Some (Some v)) => (update i sto2 v, Some (Some v))
end
end
| tapp ef ex =>
match teval n sto env ex with
| (sto1, None) => (sto1, None)
| (sto1, Some None) => (sto1, Some None)
| (sto1, Some (Some vx)) =>
match teval n sto1 env ef with
| (sto2, None) => (sto2, None)
| (sto2, Some None) => (sto2, Some None)
| (sto2, Some (Some (vty _ _))) => (sto2, Some None)
| (sto2, Some (Some (vtabs _ _ _))) => (sto2, Some None)
| (sto2, Some (Some (vloc _))) => (sto2, Some None)
| (sto2, Some (Some (vabs env2 _ ey))) =>
teval n sto2 (vx::env2) ey
end
end
| ttapp ef ex =>
match teval n sto env ef with
| (sto1, None) => (sto1, None)
| (sto1, Some None) => (sto1, Some None)
| (sto1, Some (Some (vty _ _))) => (sto1, Some None)
| (sto1, Some (Some (vabs _ _ _))) => (sto1, Some None)
| (sto1, Some (Some (vloc _))) => (sto1, Some None)
| (sto1, Some (Some (vtabs env2 T ey))) =>
teval n sto1 ((vty env ex)::env2) ey
end
end
end.
(* automation *)
Hint Unfold venv.
Hint Unfold tenv.
Hint Unfold open.
Hint Unfold indexr.
Hint Unfold length.
Hint Constructors ty.
Hint Constructors tm.
Hint Constructors vl.
Hint Constructors closed.
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 Resolve ex_intro.
(* ############################################################ *)
(* Examples *)
(* ############################################################ *)
Ltac crush :=
try solve [eapply stp_selx; compute; eauto; crush];
try solve [eapply stp_selax; compute; eauto; crush];
try solve [econstructor; compute; eauto; crush];
try solve [eapply t_sub; crush].
(* define polymorphic identity function *)
Definition polyId := TAll TTop (TFun (TVarB 0) (TVarB 0)).
Example ex1: has_type [] (ttabs TTop (tabs (TVarF 0) (tvar 1))) polyId.
Proof.
crush.
Qed.
(* instantiate it to TTop *)
Example ex2: has_type [bind_tm polyId] (ttapp (tvar 0) TTop) (TFun TTop TTop).
Proof.
crush.
Qed.
(* ############################################################ *)
(* Proofs *)
(* ############################################################ *)
Fixpoint tsize(T: ty) :=
match T with
| TTop => 1
| TFun T1 T2 => S (tsize T1 + tsize T2)
| TAll T1 T2 => S (tsize T1 + tsize T2)
| TCell T1 => S (tsize T1)
| TVarF _ => 1
| TVarH _ => 1
| TVarB _ => 1
end.
Lemma open_preserves_size: forall T x j,
tsize T = tsize (open_rec j (TVarH x) T).
Proof.
intros T. induction T; intros; simpl; eauto.
- simpl. destruct (beq_nat j i); eauto.
Qed.
(* ## Extension, Regularity ## *)
Lemma wf_length : forall sto vs ts,
wf_env sto vs ts ->
(length vs = length ts).
Proof.
intros. induction H. auto.
compute. eauto.
Qed.
Hint Immediate wf_length.
Lemma wfh_length : forall vvs vs ts,
wf_envh vvs vs ts ->
(length vs = length ts).
Proof.
intros. induction H. auto.
compute. eauto.
Qed.
Hint Immediate wfh_length.
Lemma indexr_max : forall X vs n (T: X),
indexr n vs = Some T ->
n < length vs.
Proof.
intros X vs. induction vs.
- Case "nil". intros. inversion H.
- Case "cons".
intros. inversion H.
case_eq (beq_nat n (length vs)); intros E2.
+ SSCase "hit".
eapply beq_nat_true in E2. subst n. compute. eauto.
+ SSCase "miss".
rewrite E2 in H1.
assert (n < length vs). eapply IHvs. apply H1.
compute. eauto.
Qed.
Lemma le_xx : forall a b,
a <= b ->
exists E, le_lt_dec a b = left E.
Proof. intros.
case_eq (le_lt_dec a b). intros. eauto.
intros. omega.
Qed.
Lemma le_yy : forall a b,
a > b ->
exists E, le_lt_dec a b = right E.
Proof. intros.
case_eq (le_lt_dec a b). intros. omega.
intros. eauto.
Qed.
Lemma indexr_extend : forall X vs n x (T: X),
indexr n vs = Some T ->
indexr n (x::vs) = Some T.
Proof.
intros.
assert (n < length vs). eapply indexr_max. eauto.
assert (beq_nat n (length vs) = false) as E. eapply beq_nat_false_iff. omega.
unfold indexr. unfold indexr in H. rewrite H. rewrite E. reflexivity.
Qed.
(* splicing -- for stp_extend. *)
Fixpoint splice n (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TFun T1 T2 => TFun (splice n T1) (splice n T2)
| TAll T1 T2 => TAll (splice n T1) (splice n T2)
| TCell T1 => TCell (splice n T1)
| TVarF i => TVarF i
| TVarB i => TVarB i
| TVarH i => if le_lt_dec n i then TVarH (i+1) else TVarH i
end.
Definition spliceb n := liftb (splice n).
Definition spliceat n (V: (venv*ty)) :=
match V with
| (G,T) => (G,splice n T)
end.
Lemma splice_open_permute: forall {X} (G0:list X) T2 n j,
(open_rec j (TVarH (n + S (length G0))) (splice (length G0) T2)) =
(splice (length G0) (open_rec j (TVarH (n + length G0)) T2)).
Proof.
intros X G T. induction T; intros; simpl; eauto;
try rewrite IHT1; try rewrite IHT2; try rewrite IHT; eauto.
case_eq (le_lt_dec (length G) i); intros E LE; simpl; eauto.
case_eq (beq_nat j i); intros E; simpl; eauto.
case_eq (le_lt_dec (length G) (n + length G)); intros EL LE.
assert (n + S (length G) = n + length G + 1). omega.
rewrite H. eauto.
omega.
Qed.
Lemma indexr_splice_hi: forall G0 G2 x0 v1 T,
indexr x0 (G2 ++ G0) = Some T ->
length G0 <= x0 ->
indexr (x0 + 1) (map (splice (length G0)) G2 ++ v1 :: G0) = Some (splice (length G0) T).
Proof.
intros G0 G2. induction G2; intros.
- eapply indexr_max in H. simpl in H. omega.
- simpl in H.
case_eq (beq_nat x0 (length (G2 ++ G0))); intros E.
+ rewrite E in H. inversion H. subst. simpl.
rewrite app_length in E.
rewrite app_length. rewrite map_length. simpl.
assert (beq_nat (x0 + 1) (length G2 + S (length G0)) = true). {
eapply beq_nat_true_iff. eapply beq_nat_true_iff in E. omega.
}
rewrite H1. eauto.
+ rewrite E in H. eapply IHG2 in H. eapply indexr_extend. eapply H. eauto.
Qed.
Lemma indexr_spliceb_hi: forall G0 G2 x0 v1 b,
indexr x0 (G2 ++ G0) = Some b ->
length G0 <= x0 ->
indexr (x0 + 1) (map (spliceb (length G0)) G2 ++ v1 :: G0) =
Some (spliceb (length G0) b).
Proof.
intros G0 G2. induction G2; intros.
- eapply indexr_max in H. simpl in H. omega.
- simpl in H.
case_eq (beq_nat x0 (length (G2 ++ G0))); intros E.
+ rewrite E in H. inversion H. subst. simpl.
rewrite app_length in E.
rewrite app_length. rewrite map_length. simpl.
assert (beq_nat (x0 + 1) (length G2 + S (length G0)) = true). {
eapply beq_nat_true_iff. eapply beq_nat_true_iff in E. omega.
}
rewrite H1. eauto.
+ rewrite E in H. eapply IHG2 in H. eapply indexr_extend. eapply H. eauto.
Qed.
Lemma indexr_spliceat_hi: forall G0 G2 x0 v1 G T,
indexr x0 (G2 ++ G0) = Some (G, T) ->
length G0 <= x0 ->
indexr (x0 + 1) (map (spliceat (length G0)) G2 ++ v1 :: G0) =
Some (G, splice (length G0) T).
Proof.
intros G0 G2. induction G2; intros.
- eapply indexr_max in H. simpl in H. omega.
- simpl in H. destruct a.
case_eq (beq_nat x0 (length (G2 ++ G0))); intros E.
+ rewrite E in H. inversion H. subst. simpl.
rewrite app_length in E.
rewrite app_length. rewrite map_length. simpl.
assert (beq_nat (x0 + 1) (length G2 + S (length G0)) = true). {
eapply beq_nat_true_iff. eapply beq_nat_true_iff in E. omega.
}
rewrite H1. eauto.
+ rewrite E in H. eapply IHG2 in H. eapply indexr_extend. eapply H. eauto.
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 indexr_splice_lo0: forall {X} G0 G2 x0 (T:X),
indexr x0 (G2 ++ G0) = Some T ->
x0 < length G0 ->
indexr x0 G0 = Some T.
Proof.
intros X G0 G2. induction G2; intros.
- simpl in H. apply H.
- simpl in H.
case_eq (beq_nat x0 (length (G2 ++ G0))); intros E.
+ eapply beq_nat_true_iff in E. subst.
rewrite app_length in H0. apply plus_lt_contra in H0. inversion H0.
+ rewrite E in H. apply IHG2. apply H. apply H0.
Qed.
Lemma indexr_extend_mult: forall {X} G0 G2 x0 (T:X),
indexr x0 G0 = Some T ->
indexr 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 indexr_max in H. subst.
rewrite app_length in H. apply plus_lt_contra in H. inversion H.
+ apply IHG2. assumption.
Qed.
Lemma indexr_splice_lo: forall G0 G2 x0 v1 T f,
indexr x0 (G2 ++ G0) = Some T ->
x0 < length G0 ->
indexr x0 (map (splice f) G2 ++ v1 :: G0) = Some T.
Proof.
intros.
assert (indexr x0 G0 = Some T). eapply indexr_splice_lo0; eauto.
eapply indexr_extend_mult. eapply indexr_extend. eauto.
Qed.
Lemma indexr_spliceb_lo: forall G0 G2 x0 v1 T f,
indexr x0 (G2 ++ G0) = Some T ->
x0 < length G0 ->
indexr x0 (map (spliceb f) G2 ++ v1 :: G0) = Some T.
Proof.
intros.
assert (indexr x0 G0 = Some T). eapply indexr_splice_lo0; eauto.
eapply indexr_extend_mult. eapply indexr_extend. eauto.
Qed.
Lemma indexr_spliceat_lo: forall G0 G2 x0 v1 G T f,
indexr x0 (G2 ++ G0) = Some (G, T) ->
x0 < length G0 ->
indexr x0 (map (spliceat f) G2 ++ v1 :: G0) = Some (G, T).
Proof.
intros.
assert (indexr x0 G0 = Some (G, T)). eapply indexr_splice_lo0; eauto.
eapply indexr_extend_mult. eapply indexr_extend. eauto.
Qed.
Lemma closed_splice: forall i j k T n,
closed i j k T ->
closed i (S j) k (splice n T).
Proof.
intros. induction H; simpl; eauto.
case_eq (le_lt_dec n x); intros E LE.
apply cl_selh. omega.
apply cl_selh. omega.
Qed.
Lemma map_splice_length_inc: forall G0 G2 v1,
(length (map (splice (length G0)) G2 ++ v1 :: G0)) = (S (length (G2 ++ G0))).
Proof.
intros. rewrite app_length. rewrite map_length. induction G2.
- simpl. reflexivity.
- simpl. eauto.
Qed.
Lemma map_spliceb_length_inc: forall G0 G2 v1,
(length (map (spliceb (length G0)) G2 ++ v1 :: G0)) = (S (length (G2 ++ G0))).
Proof.
intros. rewrite app_length. rewrite map_length. induction G2.
- simpl. reflexivity.
- simpl. eauto.
Qed.
Lemma map_spliceat_length_inc: forall G0 G2 v1,
(length (map (spliceat (length G0)) G2 ++ v1 :: G0)) = (S (length (G2 ++ G0))).
Proof.
intros. rewrite app_length. rewrite map_length. induction G2.
- simpl. reflexivity.
- simpl. eauto.
Qed.
Lemma closed_inc_mult: forall i j k T,
closed i j k T ->
forall i' j' k',
i' >= i -> j' >= j -> k' >= k ->
closed i' j' k' T.
Proof.
intros i j k T H. induction H; intros; eauto; try solve [constructor; omega].
- apply cl_all. apply IHclosed1; omega. apply IHclosed2; omega.
Qed.
Lemma closed_inc: forall i j k T,
closed i j k T ->
closed i (S j) k T.
Proof.
intros. apply (closed_inc_mult i j k T H i (S j) k); omega.
Qed.
Lemma closed_splice_idem: forall i j k T n,
closed i j k T ->
n >= j ->
splice n T = T.
Proof.
intros. induction H; eauto.
- (* TFun *) simpl.
rewrite IHclosed1. rewrite IHclosed2.
reflexivity.
assumption. assumption.
- (* TAll *) simpl.
rewrite IHclosed1. rewrite IHclosed2.
reflexivity.
assumption. assumption.
- (* TCell *) simpl.
rewrite IHclosed.
reflexivity.
assumption.
- (* TVarH *) simpl.
case_eq (le_lt_dec n x); intros E LE. omega. reflexivity.
Qed.
Ltac ev := repeat match goal with
| H: exists _, _ |- _ => destruct H
| H: _ /\ _ |- _ => destruct H
end.
Lemma stp_closed : forall G GH T1 T2,
stp G GH T1 T2 ->
closed 0 (length GH) (length G) T1 /\ closed 0 (length GH) (length G) T2.
Proof.
intros. induction H;
try solve [repeat ev; split; eauto using indexr_max].
Qed.
Lemma stp_closed2 : forall G1 GH T1 T2,
stp G1 GH T1 T2 ->
closed 0 (length GH) (length G1) T2.
Proof.
intros. apply (proj2 (stp_closed G1 GH T1 T2 H)).
Qed.
Lemma stp_closed1 : forall G1 GH T1 T2,
stp G1 GH T1 T2 ->
closed 0 (length GH) (length G1) T1.
Proof.
intros. apply (proj1 (stp_closed G1 GH T1 T2 H)).
Qed.
Lemma stp2_closed: forall G1 G2 T1 T2 GH m n,
stp2 m G1 T1 G2 T2 GH n ->
closed 0 (length GH) (length G1) T1 /\ closed 0 (length GH) (length G2) T2.
intros. induction H;
try solve [repeat ev; split; eauto using indexr_max].
Qed.
Lemma stp2_closed2 : forall G1 G2 T1 T2 GH m n,
stp2 m G1 T1 G2 T2 GH n ->
closed 0 (length GH) (length G2) T2.
Proof.
intros. apply (proj2 (stp2_closed G1 G2 T1 T2 GH m n H)).
Qed.
Lemma stp2_closed1 : forall G1 G2 T1 T2 GH m n,
stp2 m G1 T1 G2 T2 GH n ->
closed 0 (length GH) (length G1) T1.
Proof.
intros. apply (proj1 (stp2_closed G1 G2 T1 T2 GH m n H)).
Qed.
Lemma closed_upgrade: forall i j k i' T,
closed i j k T ->
i' >= i ->
closed i' j k T.
Proof.
intros. apply (closed_inc_mult i j k T H i' j k); omega.
Qed.
Lemma closed_upgrade_free: forall i j k j' T,
closed i j k T ->
j' >= j ->
closed i j' k T.
Proof.
intros. apply (closed_inc_mult i j k T H i j' k); omega.
Qed.
Lemma closed_upgrade_freef: forall i j k k' T,
closed i j k T ->
k' >= k ->
closed i j k' T.
Proof.
intros. apply (closed_inc_mult i j k T H i j k'); omega.
Qed.
Lemma closed_open: forall i j k TX T, closed (i+1) j k T -> closed i j k TX ->
closed i j k (open_rec i TX T).
Proof.
intros. generalize dependent i.
induction T; intros; inversion H;
try econstructor;
try eapply IHT1; eauto; try eapply IHT2; eauto; try eapply IHT; eauto.
eapply closed_upgrade. eauto. eauto.
- Case "TVarB". simpl.
case_eq (beq_nat i0 i); intros E. eauto.
econstructor. eapply beq_nat_false_iff in E. omega.
Qed.
Lemma indexr_has: forall X (G: list X) x,
length G > x ->
exists v, indexr x G = Some v.
Proof.
intros. remember (length G) as n.
generalize dependent x.
generalize dependent G.
induction n; intros; try omega.
destruct G; simpl.
- simpl in Heqn. inversion Heqn.
- simpl in Heqn. inversion Heqn. subst.
case_eq (beq_nat x (length G)); intros E.
+ eexists. reflexivity.
+ apply beq_nat_false in E. apply IHn; eauto.
omega.
Qed.
Lemma stp_refl_aux: forall n T G GH,
closed 0 (length GH) (length G) T ->
tsize T < n ->
stp G GH T T.
Proof.
intros n. induction n; intros; try omega.
inversion H; subst; eauto;
try solve [omega];
try solve [simpl in H0; constructor; apply IHn; eauto; try omega];
try solve [apply indexr_has in H1; destruct H1; eauto].
- simpl in H0.
eapply stp_all.
eapply IHn; eauto; try omega.
reflexivity.
assumption.
assumption.
apply IHn; eauto.
simpl. apply closed_open; auto using closed_inc.
unfold open. rewrite <- open_preserves_size. omega.
Qed.
Lemma stp_refl: forall T G GH,
closed 0 (length GH) (length G) T ->
stp G GH T T.
Proof.
intros. apply stp_refl_aux with (n:=S (tsize T)); eauto.
Qed.
Definition stpd2 m G1 T1 G2 T2 GH := exists n, stp2 m G1 T1 G2 T2 GH n.
Ltac ep := match goal with
| [ |- stp2 ?M ?G1 ?T1 ?G2 ?T2 ?GH ?N ] =>
assert (exists (n:nat), stp2 M G1 T1 G2 T2 GH n) as EEX
end.
Ltac eu := match goal with
| H: stpd2 _ _ _ _ _ _ |- _ =>
destruct H as [? H]
end.
Hint Unfold stpd2.
Lemma stp2_refl_aux: forall n T G GH,
closed 0 (length GH) (length G) T ->
tsize T < n ->
stpd2 true G T G T GH.
Proof.
intros n. induction n; intros; try omega.
inversion H; subst; eauto; try omega; try simpl in H0.
- destruct (IHn T1 G GH) as [n1 IH1]; eauto; try omega.
destruct (IHn T2 G GH) as [n2 IH2]; eauto; try omega.
eexists; constructor; try constructor; eauto.
- destruct (IHn T1 G GH) as [n1 IH1]; eauto; try omega.
destruct (IHn (open (TVarH (length GH)) T2) G ((G,T1)::GH)); eauto; try omega.
simpl. apply closed_open; auto using closed_inc.
unfold open. rewrite <- open_preserves_size. omega.
eexists; econstructor; try constructor; eauto.
- destruct (IHn T0 G GH) as [n0 IH0]; eauto; try omega.
eexists; econstructor; eauto.
- eapply indexr_has in H1. destruct H1 as [v HI].
eexists; eapply stp2_selx; eauto.
- eapply indexr_has in H1. destruct H1 as [v HI].
eexists; eapply stp2_selax; eauto.
Grab Existential Variables. apply 0. apply 0. apply 0.
Qed.
Lemma stp2_refl: forall T G GH,
closed 0 (length GH) (length G) T ->
stpd2 true G T G T GH.
Proof.
intros. apply stp2_refl_aux with (n:=S (tsize T)); eauto.
Qed.
Lemma stp_splice : forall GX G0 G1 T1 T2 v1,
stp GX (G1++G0) T1 T2 ->
stp GX ((map (spliceb (length G0)) G1) ++ v1::G0)
(splice (length G0) T1) (splice (length G0) T2).
Proof.
intros GX G0 G1 T1 T2 v1 H. remember (G1++G0) as G.
revert G0 G1 HeqG.
induction H; intros; subst GH; simpl; eauto.
- Case "top".
eapply stp_top.
rewrite map_spliceb_length_inc.
apply closed_splice.
assumption.
- Case "sel1".
eapply stp_sel1. apply H. assumption.
assert (splice (length G0) T=T) as A. {
eapply closed_splice_idem. eassumption. omega.
}
rewrite <- A. apply IHstp. reflexivity.
- Case "sela1".
case_eq (le_lt_dec (length G0) x); intros E LE.
+ apply stp_sela1 with (T:=(splice (length G0) T)).
assert (bind_ty (splice (length G0) T)=(spliceb (length G0) (bind_ty T))) as B by auto.
rewrite B. apply indexr_spliceb_hi. eauto. eauto.
eapply closed_splice in H0. assert (S x = x +1) as A by omega.
rewrite <- A. eapply H0.
eapply IHstp. eauto.
+ eapply stp_sela1. eapply indexr_spliceb_lo. eauto. eauto. eauto. eauto.
assert (splice (length G0) T=T) as A. {
eapply closed_splice_idem. eassumption. omega.
}
rewrite <- A. eapply IHstp. eauto.
- Case "selax".
case_eq (le_lt_dec (length G0) x); intros E LE.
+ eapply stp_selax.
eapply indexr_spliceb_hi. eassumption. assumption.