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dsubsup.v
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dsubsup.v
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(*
DSub (D<:) + Bot
T ::= Top | Bot | x.Type | { Type: S..U } | (z: T) -> T^z
t ::= x | { Type = T } | lambda x: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.
(* term variables occurring in types *)
Inductive var : Type :=
| varF : id -> var (* free, in concrete environment *)
| varH : id -> var (* free, in abstract environment *)
| varB : id -> var (* locally-bound variable *)
.
Inductive ty : Type :=
| TTop : ty
| TBot : ty
(* (z: T) -> T^z *)
| TAll : ty -> ty -> ty
(* x.Type *)
| TSel : var -> ty
(* { Type: S..U } *)
| TMem : ty(*S*) -> ty(*U*) -> ty
.
Inductive tm : Type :=
(* x -- free variable, matching concrete environment *)
| tvar : id -> tm
(* { Type = T } *)
| ttyp : ty -> tm
(* lambda x:T.t *)
| tabs : ty -> tm -> tm
(* t t *)
| tapp : tm -> tm -> tm
.
Inductive vl : Type :=
(* a closure for a lambda abstraction *)
| vabs : list vl (*H*) -> ty -> tm -> vl
(* a closure for a first-class type *)
| vty : list vl (*H*) -> ty -> vl
.
Definition tenv := list 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.
Inductive closed: nat(*B*) -> nat(*H*) -> nat(*F*) -> ty -> Prop :=
| cl_top: forall i j k,
closed i j k TTop
| cl_bot: forall i j k,
closed i j k TBot
| 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_sel: forall i j k x,
k > x ->
closed i j k (TSel (varF x))
| cl_selh: forall i j k x,
j > x ->
closed i j k (TSel (varH x))
| cl_selb: forall i j k x,
i > x ->
closed i j k (TSel (varB x))
| 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)
.
(* open define a locally-nameless encoding wrt to TVarB type variables. *)
(* substitute var u for all occurrences of (varB k) *)
Fixpoint open_rec (k: nat) (u: var) (T: ty) { struct T }: ty :=
match T with
| TTop => TTop
| TBot => TBot
| TAll T1 T2 => TAll (open_rec k u T1) (open_rec (S k) u T2)
| TSel (varF x) => TSel (varF x)
| TSel (varH i) => TSel (varH i)
| TSel (varB i) => if beq_nat k i then TSel u else TSel (varB i)
| TMem T1 T2 => TMem (open_rec k u T1) (open_rec k u T2)
end.
Definition open u T := open_rec 0 u T.
(* Locally-nameless encoding with respect to varH variables. *)
Fixpoint subst (U : var) (T : ty) {struct T} : ty :=
match T with
| TTop => TTop
| TBot => TBot
| TAll T1 T2 => TAll (subst U T1) (subst U T2)
| TSel (varB i) => TSel (varB i)
| TSel (varF i) => TSel (varF i)
| TSel (varH i) => if beq_nat i 0 then TSel U else TSel (varH (i-1))
| TMem T1 T2 => TMem (subst U T1) (subst U T2)
end.
Fixpoint nosubst (T : ty) {struct T} : Prop :=
match T with
| TTop => True
| TBot => True
| TAll T1 T2 => nosubst T1 /\ nosubst T2
| TSel (varB i) => True
| TSel (varF i) => True
| TSel (varH i) => i <> 0
| TMem T1 T2 => nosubst T1 /\ nosubst T2
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_bot: forall G1 GH T2,
closed 0 (length GH) (length G1) T2 ->
stp G1 GH TBot T2
| stp_mem: forall G1 GH S1 U1 S2 U2,
stp G1 GH U1 U2 ->
stp G1 GH S2 S1 ->
stp G1 GH (TMem S1 U1) (TMem S2 U2)
| stp_sel1: forall G1 GH TX T2 x,
indexr x G1 = Some TX ->
closed 0 0 (length G1) TX ->
stp G1 GH TX (TMem TBot T2) ->
stp G1 GH (TSel (varF x)) T2
| stp_sel2: forall G1 GH TX T1 x,
indexr x G1 = Some TX ->
closed 0 0 (length G1) TX ->
stp G1 GH TX (TMem T1 TTop) ->
stp G1 GH T1 (TSel (varF x))
| 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 (TSel (varF x)) (TSel (varF x))
| stp_sela1: forall G1 GH TX T2 x,
indexr x GH = Some TX ->
closed 0 x (length G1) TX ->
stp G1 GH TX (TMem TBot T2) ->
stp G1 GH (TSel (varH x)) T2
| stp_sela2: forall G1 GH TX T1 x,
indexr x GH = Some TX ->
closed 0 x (length G1) TX ->
stp G1 GH TX (TMem T1 TTop) ->
stp G1 GH T1 (TSel (varH x))
| 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 (TSel (varH x)) (TSel (varH 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 (T3::GH) (open (varH x) T2) (open (varH x) T4) ->
stp G1 GH (TAll T1 T2) (TAll T3 T4)
.
(* ### Type Assignment ### *)
Inductive has_type : tenv -> tm -> ty -> Prop :=
| t_var: forall x env T1,
indexr x env = Some T1 ->
stp env [] T1 T1 ->
has_type env (tvar x) T1
| t_typ: forall env T1,
closed 0 0 (length env) T1 ->
has_type env (ttyp T1) (TMem T1 T1)
| t_app: forall env f x T1 T2,
has_type env f (TAll T1 T2) ->
has_type env x T1 ->
closed 0 0 (length env) T2 ->
has_type env (tapp f x) T2
| t_dapp:forall env f x T1 T2 T,
has_type env f (TAll T1 T2) ->
has_type env (tvar x) T1 ->
T = open (varF x) T2 ->
closed 0 0 (length env) T ->
has_type env (tapp f (tvar x)) T
| t_abs: forall env y T1 T2,
has_type (T1::env) y (open (varF (length env)) T2) ->
closed 0 0 (length env) (TAll T1 T2) ->
has_type env (tabs T1 y) (TAll T1 T2)
| t_sub: forall env e T1 T2,
has_type env e T1 ->
stp env [] T1 T2 ->
has_type env e T2
.
Definition base (v:vl): venv :=
match v with
| vty GX _ => GX
| vabs GX _ _ => GX
end.
(* ### Runtime Subtyping ### *)
(* H1 T1 <: H2 T2 -| J *)
Inductive stp2: bool (* whether selections are precise *) ->
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 s n,
closed 0 (length GH) (length G1) T ->
stp2 s true G1 T G2 TTop GH (S n)
| stp2_bot: forall G1 G2 GH T s n,
closed 0 (length GH) (length G2) T ->
stp2 s true G1 TBot G2 T GH (S n)
| stp2_mem: forall G1 G2 S1 U1 S2 U2 GH s n1 n2,
stp2 s s G1 U1 G2 U2 GH n1 ->
stp2 s false G2 S2 G1 S1 GH n2 ->
stp2 s true G1 (TMem S1 U1) G2 (TMem S2 U2) GH (S (n1+n2))
(* concrete type variables *)
(* precise/invertible bounds *)
(* vty already marks binding as type binding, so no need for additional TMem marker *)
| stp2_strong_sel1: forall G1 G2 GX TX x T2 GH n1,
indexr x G1 = Some (vty GX TX) ->
val_type GX (vty GX TX) (TMem TX TX) -> (* for downgrade *)
closed 0 0 (length GX) TX ->
stp2 true true GX TX G2 T2 GH n1 ->
stp2 true true G1 (TSel (varF x)) G2 T2 GH (S n1)
| stp2_strong_sel2: forall G1 G2 GX TX x T1 GH n1,
indexr x G2 = Some (vty GX TX) ->
val_type GX (vty GX TX) (TMem TX TX) -> (* for downgrade *)
closed 0 0 (length GX) TX ->
stp2 true false G1 T1 GX TX GH n1 ->
stp2 true true G1 T1 G2 (TSel (varF x)) GH (S n1)
(* imprecise type *)
| stp2_sel1: forall G1 G2 v TX x T2 GH n1,
indexr x G1 = Some v ->
val_type (base v) v TX ->
closed 0 0 (length (base v)) TX ->
stp2 false false (base v) TX G2 (TMem TBot T2) GH n1 ->
stp2 false true G1 (TSel (varF x)) G2 T2 GH (S n1)
| stp2_sel2: forall G1 G2 v TX x T1 GH n1,
indexr x G2 = Some v ->
val_type (base v) v TX ->
closed 0 0 (length (base v)) TX ->
stp2 false false (base v) TX G1 (TMem T1 TTop) GH n1 ->
stp2 false true G1 T1 G2 (TSel (varF x)) GH (S n1)
| stp2_selx: forall G1 G2 v x1 x2 GH s n,
indexr x1 G1 = Some v ->
indexr x2 G2 = Some v ->
stp2 s true G1 (TSel (varF x1)) G2 (TSel (varF x2)) GH (S n)
(* abstract type variables *)
| 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 false GX TX G2 (TMem TBot T2) GH n1 ->
stp2 false true G1 (TSel (varH x)) G2 T2 GH (S n1)
| stp2_sela2: forall G1 G2 GX T1 TX x GH n1,
indexr x GH = Some (GX, TX) ->
closed 0 x (length GX) TX ->
stp2 false false GX TX G1 (TMem T1 TTop) GH n1 ->
stp2 false true G1 T1 G2 (TSel (varH x)) GH (S n1)
| stp2_selax: forall G1 G2 v x GH s n,
indexr x GH = Some v ->
stp2 s true G1 (TSel (varH x)) G2 (TSel (varH x)) GH (S n)
| stp2_all: forall G1 G2 T1 T2 T3 T4 x GH s n1 n2,
stp2 false 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 false G1 (open (varH x) T2) G2 (open (varH x) T4) ((G2, T3)::GH) n2 ->
stp2 s true G1 (TAll T1 T2) G2 (TAll T3 T4) GH (S (n1 + n2))
| stp2_wrapf: forall G1 G2 T1 T2 GH s n1,
stp2 s true G1 T1 G2 T2 GH n1 ->
stp2 s false G1 T1 G2 T2 GH (S n1)
| stp2_transf: forall G1 G2 G3 T1 T2 T3 GH s n1 n2,
stp2 s true G1 T1 G2 T2 GH n1 ->
stp2 s false G2 T2 G3 T3 GH n2 ->
stp2 s false G1 T1 G3 T3 GH (S (n1+n2))
(* consistent environment *)
with wf_env : venv -> tenv -> Prop :=
| wfe_nil : wf_env nil nil
| wfe_cons : forall v t vs ts,
val_type (v::vs) v t ->
wf_env vs ts ->
wf_env (cons v vs) (cons t ts)
(* value type assignment *)
with val_type : venv -> vl -> ty -> Prop :=
| v_ty: forall env venv tenv T1 TE,
wf_env venv tenv ->
(exists n, stp2 true true venv (TMem T1 T1) env TE [] n) ->
val_type env (vty venv T1) TE
| v_abs: forall env venv tenv x y T1 T2 TE,
wf_env venv tenv ->
has_type (T1::tenv) y (open (varF x) T2) ->
length venv = x ->
(exists n, stp2 true true venv (TAll T1 T2) env TE [] n) ->
val_type env (vabs venv T1 y) 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 t ts)
.
Inductive valh_type : venv -> aenv -> (venv*ty) -> ty -> Prop :=
| v_tya: forall aenv venv T1,
valh_type venv aenv (venv, T1) 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)(env: venv)(t: tm){struct n}: option (option vl) :=
match n with
| 0 => None
| S n =>
match t with
| tvar x => Some (indexr x env)
| ttyp T => Some (Some (vty env T))
| tabs T y => Some (Some (vabs env T y))
| tapp ef ex =>
match teval n env ex with
| None => None
| Some None => Some None
| Some (Some vx) =>
match teval n env ef with
| None => None
| Some None => Some None
| Some (Some (vty _ _)) => Some None
| Some (Some (vabs env2 _ ey)) =>
teval n (vx::env2) ey
end
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 wf_envh.
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 (TMem TBot TTop) (TAll (TSel (varB 0)) (TSel (varB 1))).
Example ex1: has_type [] (tabs (TMem TBot TTop) (tabs (TSel (varF 0)) (tvar 1))) polyId.
Proof.
crush.
Qed.
(* instantiate it to TTop *)
Example ex2: has_type [polyId] (tapp (tvar 0) (ttyp TTop)) (TAll TTop TTop).
Proof.
crush.
Qed.
(* ############################################################ *)
(* Proofs *)
(* ############################################################ *)
Fixpoint tsize(T: ty) :=
match T with
| TTop => 1
| TBot => 1
| TAll T1 T2 => S (tsize T1 + tsize T2)
| TSel _ => 1
| TMem T1 T2 => S (tsize T1 + tsize T2)
end.
Lemma open_preserves_size: forall T x j,
tsize T = tsize (open_rec j (varH x) T).
Proof.
intros T. induction T; intros; simpl; eauto.
- destruct v; simpl; destruct (beq_nat j i); eauto.
Qed.
(* ## Extension, Regularity ## *)
Lemma wf_length : forall vs ts,
wf_env 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
| TBot => TBot
| TAll T1 T2 => TAll (splice n T1) (splice n T2)
| TSel (varF i) => TSel (varF i)
| TSel (varB i) => TSel (varB i)
| TSel (varH i) => if le_lt_dec n i then TSel (varH (i+1)) else TSel (varH i)
| TMem T1 T2 => TMem (splice n T1) (splice n T2)
end.
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 (varH (n + S (length G0))) (splice (length G0) T2)) =
(splice (length G0) (open_rec j (varH (n + length G0)) T2)).
Proof.
intros X G T. induction T; intros; simpl; eauto;
try rewrite IHT1; try rewrite IHT2; try rewrite IHT; eauto;
destruct v; eauto.
case_eq (le_lt_dec (length G) i); intros E LE; simpl; eauto.
rewrite LE. eauto.
rewrite LE. eauto.
case_eq (beq_nat j i); intros E; simpl; eauto.
case_eq (le_lt_dec (length G) (n + length G)); intros EL LE.
rewrite E.
assert (n + S (length G) = n + length G + 1). omega.
rewrite H. eauto.
omega.
rewrite E. eauto.
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_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_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_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.
- (* TAll *) simpl.
rewrite IHclosed1. rewrite IHclosed2.
reflexivity.
assumption. assumption.
- (* TVarH *) simpl.
case_eq (le_lt_dec n x); intros E LE. omega. reflexivity.
- (* TMem *) simpl.
rewrite IHclosed1. rewrite IHclosed2.
reflexivity.
assumption. assumption.
Qed.
Ltac ev := repeat match goal with
| H: exists _, _ |- _ => destruct H
| H: _ /\ _ |- _ => destruct H
end.
Ltac inv_mem := match goal with
| H: closed 0 (length ?GH) (length ?G) (TMem ?T1 ?T2) |-
closed 0 (length ?GH) (length ?G) ?T2 => inversion H; subst; eauto
| H: closed 0 (length ?GH) (length ?G) (TMem ?T1 ?T2) |-
closed 0 (length ?GH) (length ?G) ?T1 => inversion H; subst; eauto
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; try inv_mem; 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 s m n,
stp2 s 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; try inv_mem; eauto using indexr_max].
Qed.
Lemma stp2_closed2 : forall G1 G2 T1 T2 GH s m n,
stp2 s 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 s m n H)).
Qed.
Lemma stp2_closed1 : forall G1 G2 T1 T2 GH s m n,
stp2 s 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 s 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 V T, closed (i+1) j k T -> closed i j k (TSel V) ->
closed i j k (open_rec i V 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 i x); 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 s m G1 T1 G2 T2 GH := exists n, stp2 s m G1 T1 G2 T2 GH n.
Ltac ep := match goal with
| [ |- stp2 ?S ?M ?G1 ?T1 ?G2 ?T2 ?GH ?N ] =>
assert (exists (n:nat), stp2 S 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 s,
closed 0 (length GH) (length G) T ->
tsize T < n ->
stpd2 s 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 false) as [n1 IH1]; eauto; try omega.
destruct (IHn (open (varH (length GH)) T2) G ((G,T1)::GH) false); eauto; try omega.
simpl. apply closed_open; auto using closed_inc.
unfold open. rewrite <- open_preserves_size. omega.
eexists; econstructor; try constructor; 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.
- destruct (IHn T1 G GH s) as [n1 IH1]; eauto; try omega.
destruct (IHn T2 G GH s) as [n2 IH2]; eauto; try omega.
destruct s; eexists; econstructor; try constructor; eauto.
Grab Existential Variables. apply 0. apply 0. apply 0. apply 0.
Qed.
Lemma stp2_refl: forall T G GH s,
closed 0 (length GH) (length G) T ->
stpd2 s true G T G T GH.
Proof.
intros. apply stp2_refl_aux with (n:=S (tsize T)); eauto.
Qed.
Lemma concat_same_length: forall {X} (GU: list X) (GL: list X) (GH1: list X) (GH0: list X),
GU ++ GL = GH1 ++ GH0 ->
length GU = length GH1 ->
GU=GH1 /\ GL=GH0.
Proof.
intros. generalize dependent GH1. induction GU; intros.
- simpl in H0. induction GH1. rewrite app_nil_l in H. rewrite app_nil_l in H.
split. reflexivity. apply H.
simpl in H0. omega.
- simpl in H0. induction GH1. simpl in H0. omega.
simpl in H0. inversion H0. simpl in H. inversion H. specialize (IHGU GH1 H4 H2).
destruct IHGU. subst. split; reflexivity.
Qed.
Lemma concat_same_length': forall {X} (GU: list X) (GL: list X) (GH1: list X) (GH0: list X),
GU ++ GL = GH1 ++ GH0 ->
length GL = length GH0 ->
GU=GH1 /\ GL=GH0.
Proof.
intros.
assert (length (GU ++ GL) = length (GH1 ++ GH0)) as A. {
rewrite H. reflexivity.
}
rewrite app_length in A. rewrite app_length in A.
rewrite H0 in A. apply NPeano.Nat.add_cancel_r in A.
apply concat_same_length; assumption.
Qed.
Lemma exists_GH1L: forall {X} (GU: list X) (GL: list X) (GH1: list X) (GH0: list X) x0,
length GL = x0 ->
GU ++ GL = GH1 ++ GH0 ->
length GH0 <= x0 ->
exists GH1L, GH1 = GU ++ GH1L /\ GL = GH1L ++ GH0.
Proof.
intros X GU. induction GU; intros.
- eexists. rewrite app_nil_l. split. reflexivity. simpl in H0. assumption.
- induction GH1.
simpl in H0.
assert (length (a :: GU ++ GL) = length GH0) as Contra. {
rewrite H0. reflexivity.
}
simpl in Contra. rewrite app_length in Contra. omega.
simpl in H0. inversion H0.
specialize (IHGU GL GH1 GH0 x0 H H4 H1).
destruct IHGU as [GH1L [IHA IHB]].
exists GH1L. split. simpl. rewrite IHA. reflexivity. apply IHB.
Qed.
Lemma exists_GH0U: forall {X} (GH1: list X) (GH0: list X) (GU: list X) (GL: list X) x0,
length GL = x0 ->
GU ++ GL = GH1 ++ GH0 ->
x0 < length GH0 ->
exists GH0U, GH0 = GH0U ++ GL.
Proof.
intros X GH1. induction GH1; intros.
- simpl in H0. exists GU. symmetry. assumption.
- induction GU.
simpl in H0.
assert (length GL = length (a :: GH1 ++ GH0)) as Contra. {
rewrite H0. reflexivity.
}
simpl in Contra. rewrite app_length in Contra. omega.
simpl in H0. inversion H0.
specialize (IHGH1 GH0 GU GL x0 H H4 H1).
destruct IHGH1 as [GH0U IH].
exists GH0U. apply IH.
Qed.
Lemma stp_splice : forall GX G0 G1 T1 T2 v1,
stp GX (G1++G0) T1 T2 ->
stp GX ((map (splice (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_splice_length_inc.
apply closed_splice.
assumption.
- Case "bot".
eapply stp_bot.
rewrite map_splice_length_inc.
apply closed_splice.
assumption.
- Case "sel1".
eapply stp_sel1. apply H. assumption.
assert (splice (length G0) TX=TX) as A. {
eapply closed_splice_idem. eassumption. omega.
}
rewrite <- A. apply IHstp. reflexivity.
- Case "sel2".
eapply stp_sel2. apply H. assumption.
assert (splice (length G0) TX=TX) 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.
+ eapply stp_sela1.
apply indexr_splice_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_splice_lo. eauto. eauto. eauto.
assert (splice (length G0) TX=TX) as A. {
eapply closed_splice_idem. eassumption. omega.
}
rewrite <- A. eapply IHstp. eauto.
- Case "sela2".