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Main.v
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Require Import Arith List.
Require Import "Lattice".
Require Import "Environment".
Inductive type : Type :=
| Bool : label -> type
| Arrow : type -> type -> label -> type.
Inductive expr : Type :=
| TT : label -> expr
| FF : label -> expr
| Cond : expr -> expr -> expr -> expr
| Var : nat -> expr
| Abs : nat -> type -> expr -> label -> expr
| App : expr -> expr -> expr.
Inductive value : expr -> Prop :=
| value_true :
forall l,
value (TT l)
| value_false :
forall l,
value (FF l)
| value_abs :
forall x t e l,
value (Abs x t e l).
Definition value_with_label v l :=
value v /\
(v = (TT l) \/ v = (FF l) \/ (exists x t e, v = Abs x t e l)).
Definition stamp expr label :=
match expr with
| TT l => TT (join l label)
| FF l => FF (join l label)
| Abs x t e l => Abs x t e (join l label)
| _ => expr
end.
Lemma stamp_preserves_value :
forall e l,
value (stamp e l) <->
value e.
Proof.
intros e l.
split; intro H.
destruct e; intros; try inversion H.
apply value_true.
apply value_false.
apply value_abs.
destruct e; intros; try inversion H; subst; simpl.
apply value_true.
apply value_false.
apply value_abs.
Qed.
Lemma stamp_value_is_join :
forall v l l',
value_with_label v l ->
value_with_label (stamp v l') (join l l').
Proof.
intros v l l' Hv; unfold value_with_label.
destruct Hv.
destruct H0.
subst.
simpl.
split.
apply value_true.
left.
reflexivity.
destruct H0.
subst.
split.
apply value_false.
right.
left.
reflexivity.
destruct H0.
destruct H0.
destruct H0.
subst.
split.
simpl.
apply value_abs.
right.
right.
exists x.
exists x0.
exists x1.
reflexivity.
Qed.
Lemma stamp_high_is_high :
forall v l,
value_with_label v l ->
value_with_label (stamp v High) High.
Proof.
intros.
apply (stamp_value_is_join v l High) in H.
rewrite join_high_r in H.
apply H.
Qed.
Lemma stamp_low_is_neutral :
forall v l,
value_with_label v l ->
value_with_label (stamp v Low) l.
Proof.
intros.
apply (stamp_value_is_join v l Low) in H.
rewrite join_low_r in H.
apply H.
Qed.
Fixpoint sub (v : expr) (x : nat) (e : expr) :=
match e with
| TT l =>
TT l
| FF l =>
FF l
| Cond e1 e2 e3 =>
Cond (sub v x e1) (sub v x e2) (sub v x e3)
| Var y =>
if beq_nat x y then v else e
| Abs y t f l =>
if beq_nat x y then e else Abs y t (sub v x f) l
| App e1 e2 =>
App (sub v x e1) (sub v x e2)
end.
Inductive big_step : expr -> expr -> Prop :=
| big_step_val :
forall v,
value v ->
big_step v v
| big_step_cond_true :
forall e1 l e2 e3 v,
big_step e1 (TT l) ->
big_step e2 v ->
big_step (Cond e1 e2 e3) (stamp v l)
| big_step_cond_false :
forall e1 l e2 e3 v,
big_step e1 (FF l) ->
big_step e2 v ->
big_step (Cond e1 e2 e3) (stamp v l)
| big_step_app :
forall e1 x s e l e2 v v',
big_step e1 (Abs x s e l) ->
big_step e2 v ->
big_step (sub v x e) v' ->
big_step (App e1 e2) (stamp v' l).
Lemma strong_big_step_ind :
forall P : expr -> expr -> Prop,
(forall v : expr, value v -> P v v) ->
(forall (e1 : expr) (l : label) (e2 e3 v : expr),
big_step e1 (TT l) ->
P e1 (TT l) ->
big_step e2 v ->
P e2 v ->
P (Cond e1 e2 e3) (stamp v l)) ->
(forall (e1 : expr) (l : label) (e2 e3 v : expr),
big_step e1 (FF l) ->
P e1 (FF l) ->
big_step e2 v ->
P e2 v ->
P (Cond e1 e2 e3) (stamp v l)) ->
(forall (e1 : expr) (x : nat) (s : type) (e : expr) (l : label) (e2 v v' : expr),
big_step e1 (Abs x s e l) ->
P e1 (Abs x s e l) ->
big_step e2 v ->
P e2 v ->
big_step (sub v x e) v' ->
P (sub v x e) v'
-> P (App e1 e2) (stamp v' l)) ->
forall e e0 : expr, big_step e e0 -> P e e0.
Proof.
exact big_step_ind.
Qed.
Lemma big_steps_to_value :
forall e v,
big_step e v ->
value v.
Proof.
intros e v Hstep.
induction Hstep; subst; auto.
apply stamp_preserves_value with (l := l); auto.
apply stamp_preserves_value; auto.
apply stamp_preserves_value; auto.
Qed.
Inductive subtype : type -> type -> Prop :=
| TFunSub :
forall s1' s1 s2 s2' l l',
subtype s1' s1 ->
subtype s2 s2' ->
flows_to l l' ->
subtype (Arrow s1 s2 l) (Arrow s1' s2' l')
| TBoolSub :
forall l l',
flows_to l l' ->
subtype (Bool l) (Bool l').
Lemma subtype_refl :
forall t,
subtype t t.
Proof.
induction t.
apply TBoolSub.
apply flows_to_refl.
apply TFunSub.
apply IHt1.
apply IHt2.
apply flows_to_refl.
Qed.
Lemma subtype_antisym :
forall t t',
subtype t t' ->
subtype t' t ->
t = t'.
Proof.
intros t t' Hsub.
induction Hsub; intros.
inversion H0.
subst.
rewrite IHHsub1; auto.
rewrite IHHsub2; auto.
rewrite (flows_to_antisym l l'); auto.
inversion H0.
subst.
rewrite (flows_to_antisym l l'); auto.
Qed.
Lemma subtype_bool_left :
forall l s,
subtype (Bool l) s ->
exists l',
s = (Bool l') /\ flows_to l l'.
Proof.
destruct s.
intros.
exists l0.
inversion H; subst.
split.
reflexivity.
apply H2.
intros.
inversion H.
Qed.
Lemma subtype_bool_right :
forall l s,
subtype s (Bool l) ->
exists l',
s = (Bool l') /\ flows_to l' l.
Proof.
destruct s.
intros.
exists l0.
inversion H; subst.
split.
reflexivity.
apply H2.
intros.
inversion H.
Qed.
Lemma subtype_arrow_left :
forall l s1 s2 s,
subtype (Arrow s1 s2 l) s ->
exists s1' s2' l',
s = (Arrow s1' s2' l') /\ flows_to l l' /\ subtype s1' s1 /\ subtype s2 s2'.
Proof.
destruct s.
intros.
inversion H.
intros.
exists s3. exists s4. exists l0.
inversion H; subst.
split.
reflexivity.
split.
assumption.
split; assumption.
Qed.
Lemma subtype_arrow_right :
forall l s1 s2 s,
subtype s (Arrow s1 s2 l) ->
exists s1' s2' l',
s = (Arrow s1' s2' l') /\ flows_to l' l /\ subtype s1 s1' /\ subtype s2' s2.
Proof.
destruct s.
intros.
inversion H.
intros.
exists s3. exists s4. exists l0.
inversion H; subst.
split.
reflexivity.
split.
assumption.
split; assumption.
Qed.
Lemma subtype_trans :
forall t t' t'',
subtype t t' ->
subtype t' t'' ->
subtype t t''.
Proof.
intros t t'.
revert t.
induction t' as [l'| s1' IHs1' s2' IHs2' l'].
intros t t'' H H0.
apply subtype_bool_left in H0.
destruct H0 as [l'' [H0 H0']].
subst.
apply subtype_bool_right in H.
destruct H as [l [H H']].
subst.
apply TBoolSub.
apply flows_to_trans with (l' := l'); assumption.
intros t t'' H0 H.
apply subtype_arrow_left in H.
destruct H as [s1'' [s2'' [l'' [H5 [H6 [H7 H8]]]]]].
subst.
apply subtype_arrow_right in H0.
destruct H0 as [s1 [s2 [l [H5 [H9 [H10 H11]]]]]].
subst.
apply TFunSub.
apply IHs1'; assumption.
apply IHs2'; assumption.
apply flows_to_trans with (l' := l'); assumption.
Qed.
Definition stamp_type t l :=
match t with
| Bool l' => Bool (join l' l)
| Arrow t1 t2 l' => Arrow t1 t2 (join l' l)
end.
Definition type_with_label t l :=
match t with
| Bool l' => l = l'
| Arrow _ _ l' => l = l'
end.
Lemma all_types_have_label :
forall t, exists l,
type_with_label t l.
Proof.
destruct t.
exists l; auto.
reflexivity.
exists l.
reflexivity.
Qed.
Lemma stamp_type_is_join :
forall t l l',
type_with_label t l ->
type_with_label (stamp_type t l') (join l l').
Proof.
intros.
destruct t.
simpl.
simpl in H.
subst.
reflexivity.
simpl.
simpl in H.
subst.
reflexivity.
Qed.
Lemma stamp_preserves_subtype :
forall s l s',
subtype (stamp_type s l) s' ->
subtype s s'.
Proof.
induction s; intros.
simpl in H.
apply subtype_bool_left in H.
destruct H.
destruct H.
subst.
apply TBoolSub.
apply flows_to_trans with (l' := (join l l0)).
apply join_is_upper_bound.
apply H0.
simpl in H.
apply subtype_arrow_left in H.
destruct H.
destruct H.
destruct H.
destruct H.
destruct H0.
destruct H1.
subst.
apply TFunSub.
apply H1.
apply H2.
apply flows_to_trans with (l' := (join l l0)).
apply join_is_upper_bound.
apply H0.
Qed.
Lemma stamp_l_preserves_subtype :
forall s l s',
subtype s s' ->
subtype (stamp_type s l) (stamp_type s' l).
Proof.
induction s; intros.
apply subtype_bool_left in H.
destruct H.
destruct H.
subst.
simpl.
apply TBoolSub.
destruct (join_is_upper_bound x l0).
apply join_is_least_upper_bound; auto.
apply flows_to_trans with (l' := x); auto.
apply subtype_arrow_left in H.
destruct H.
destruct H.
destruct H.
destruct H.
destruct H0.
destruct H1.
subst.
apply TFunSub; auto.
destruct (join_is_upper_bound x1 l0); auto.
apply join_is_least_upper_bound; auto.
apply flows_to_trans with (l' := x1); auto.
Qed.
Inductive typing : environment type -> expr -> type -> Prop :=
| typing_true :
forall c l l',
flows_to l l' ->
typing c (TT l) (Bool l')
| typing_false :
forall c l l',
flows_to l l' ->
typing c (FF l) (Bool l')
| typing_cond :
forall c s s' e1 e2 e3 l,
typing c e1 (Bool l) ->
typing c e2 s ->
typing c e3 s ->
subtype (stamp_type s l) s' ->
typing c (Cond e1 e2 e3) s'
| typing_app :
forall c e1 e2 s2 s2' s s' l,
typing c e1 (Arrow s2 s l) ->
typing c e2 s2' ->
subtype s2' s2 ->
subtype (stamp_type s l) s' ->
typing c (App e1 e2) s'
| typing_abs :
forall c x e s1' s1 s2 s2' l l',
typing (Extend type x s1 c) e s2 ->
subtype s1' s1 ->
subtype s2 s2' ->
flows_to l l' ->
typing c (Abs x s1 e l) (Arrow s1' s2' l')
| typing_var :
forall x c s s',
lookup x c = Some s ->
subtype s s' ->
typing c (Var x) s'.
Lemma typing_inversion_true :
forall c l s,
typing c (TT l) s ->
exists l',
s = (Bool l') /\ flows_to l l'.
Proof.
intros.
inversion H; subst.
exists l'; auto.
Qed.
Lemma typing_inversion_false :
forall c l s,
typing c (FF l) s ->
exists l',
s = (Bool l') /\ flows_to l l'.
Proof.
intros.
inversion H; subst.
exists l'; auto.
Qed.
Lemma typing_inversion_cond :
forall c e1 e2 e3 s l,
typing c (Cond e1 e2 e3) s ->
type_with_label s l ->
exists l' s',
typing c e1 (Bool l') /\
typing c e2 s' /\
typing c e3 s' /\
subtype (stamp_type s' l') s.
Proof.
intros.
inversion H; subst.
exists l0.
exists s0.
split; auto.
Qed.
Lemma typing_inversion_var :
forall c x s,
typing c (Var x) s ->
exists s',
lookup x c = Some s' /\ subtype s' s.
Proof.
intros.
inversion H; subst.
exists s0.
split; auto.
Qed.
Lemma typing_inversion_app :
forall c e1 e2 s l,
typing c (App e1 e2) s ->
type_with_label s l ->
exists l' s1 s1' s2,
typing c e1 (Arrow s1 s2 l') /\
typing c e2 s1' /\
subtype s1' s1 /\
subtype (stamp_type s2 l') s.
Proof.
intros.
inversion H; subst.
exists l0.
exists s2.
exists s2'.
exists s0.
split; auto.
Qed.
Lemma typing_inversion_abs :
forall c x s1' e l' u,
typing c (Abs x s1' e l') u ->
exists s1 s2 s2' l,
u = (Arrow s1 s2 l) /\
typing (Extend _ x s1' c) e s2' /\
subtype s1 s1' /\
subtype s2' s2 /\
flows_to l' l.
Proof.
intros.
inversion H; subst.
exists s1'0.
exists s2'.
exists s2.
exists l'0; auto.
Qed.
Lemma typing_is_context_invariant :
forall c c' e s,
env_equiv c c' ->
typing c e s ->
typing c' e s.
Proof.
intros c c' e s Hequiv Htype.
revert c' Hequiv.
induction Htype; intros c' Hequiv.
apply typing_true; auto.
apply typing_false; auto.
apply typing_cond with (s := s)(l := l); auto.
apply typing_app with (s2 := s2)(s := s)(l := l)(s2' := s2'); auto.
apply typing_abs with (s2 := s2); auto.
apply IHHtype.
apply env_equiv_extend_eq.
apply Hequiv.
apply typing_var with (s := s); auto.
rewrite <- H.
symmetry.
apply Hequiv.
Qed.
Inductive free_in : nat -> expr -> Prop :=
| free_in_cond :
forall x e1 e2 e3,
(free_in x e1 \/ free_in x e2 \/ free_in x e3) ->
free_in x (Cond e1 e2 e3)
| free_in_app :
forall x e1 e2,
(free_in x e1 \/ free_in x e2) ->
free_in x (App e1 e2)
| free_in_var :
forall x,
free_in x (Var x)
| free_in_abs :
forall x y s e l,
x <> y ->
free_in x e ->
free_in x (Abs y s e l).
Lemma free_in_dec :
forall x e,
{ free_in x e} + { ~ free_in x e }.
Proof.
unfold not.
induction e.
right.
intros.
inversion H.
right.
intros.
inversion H.
destruct IHe1.
left.
apply free_in_cond.
left.
assumption.
destruct IHe2.
left.
apply free_in_cond.
right.
left.
assumption.
destruct IHe3.
left.
apply free_in_cond.
right.
right.
apply f1.
right.
intros.
inversion H; subst.
destruct H2.
auto.
destruct H0; auto.
case_eq (beq_nat x n).
intros.
apply beq_nat_true in H.
subst.
left.
apply free_in_var.
right.
intros.
inversion H0.
subst.
rewrite <- beq_nat_refl in H.
inversion H.
destruct IHe.
case_eq (beq_nat x n); intros.
apply beq_nat_true in H.
subst.
right.
intros.
inversion H.
auto.
left.
apply beq_nat_false in H.
apply free_in_abs; assumption.
right.
intros.
inversion H; auto.
destruct IHe1.
left.
apply free_in_app; auto.
destruct IHe2.
left.
apply free_in_app; auto.
right.
intros.
inversion H.
destruct H2; auto.
Qed.
Definition ctxt_approx c c' :=
forall x t,
(lookup x c = Some t ->
exists t',
lookup x c' = Some t' /\ subtype t t').
Definition ctxt_equiv c c' :=
ctxt_approx c c' /\ ctxt_approx c' c.
Lemma ctxt_equiv_sound :
forall e c c',
ctxt_equiv c c' ->
(forall x, free_in x e ->
lookup x c = lookup x c').
Proof.
unfold ctxt_equiv.
unfold ctxt_approx.
intros.
destruct H.
case_eq (lookup x c); intros.
case_eq (lookup x c'); intros.
specialize (H x t H2).
specialize (H1 x t0 H3).
destruct H.
destruct H.
destruct H1.
destruct H1.
rewrite H1 in H2.
inversion H2.
subst.
clear H2.
rewrite H in H3.
inversion H3.
subst.
clear H3.
assert (t = t0).
apply subtype_antisym; auto.
subst.
reflexivity.
specialize (H x t H2).
destruct H.
destruct H.
rewrite H in H3.
inversion H3.
case_eq (lookup x c'); intros.
specialize (H1 x t H3).
destruct H1.
destruct H1.
rewrite H2 in H1.
inversion H1.
reflexivity.
Qed.
Lemma env_approx :
forall e c c' s s' l l',
typing c e s ->
type_with_label s l ->
ctxt_approx c c' ->
typing c' e s' ->
type_with_label s' l' ->
subtype s (stamp_type s' l).
Proof.
intros e c c' s s' l l' Htyp Hlab Happx Htyp'.
generalize dependent c.
generalize dependent s.
revert l.
revert l'.
induction Htyp'; intros l'' l''' s'' Htwl c' Htype Happx Htwl'.
simpl in Htwl.
apply typing_inversion_true in Htype.
destruct Htype.
destruct H0.
subst.
simpl in Htwl.
subst.
apply TBoolSub.
apply join_is_upper_bound.
simpl in Htwl.
apply typing_inversion_false in Htype.
destruct Htype.
destruct H0.
subst.
simpl in Htwl.
subst.
apply TBoolSub.
apply join_is_upper_bound.
apply typing_inversion_cond with (l := l) in Htype.
destruct Htype as [l' H0].
destruct H0 as [t H0].
destruct H0.
destruct H1.
destruct H2.
Abort.
Lemma env_invariance :
forall e c c' s,
typing c e s ->
(forall x, free_in x e -> lookup x c = lookup x c') ->
typing c' e s.
Proof.
intros e c c' s Htype.
revert c'.
induction Htype; intros.
apply typing_true.
assumption.
apply typing_false.
assumption.
apply typing_cond with (s := s)(l := l); auto.
apply (IHHtype1 c'); auto.
intros.
apply H0.
apply free_in_cond; auto.
apply (IHHtype2 c'); auto; intros.
apply H0.
apply free_in_cond; auto.
apply (IHHtype3 c'); auto.
intros.
apply H0.
apply free_in_cond; auto.
apply typing_app with (s2 := s2)(s := s)(l := l)(s2' := s2'); auto.
apply IHHtype1; intros.
apply H1.
apply free_in_app; auto.
apply IHHtype2; intros.
apply H1.
apply free_in_app; auto.
apply typing_abs with (s2 := s2); auto.
apply IHHtype.
intros.
simpl.
case_eq (beq_nat x0 x); intro Heq.
reflexivity.
apply H2.
apply free_in_abs.
apply beq_nat_false in Heq.
assumption.
apply H3.
apply typing_var with (s := s); auto.
rewrite <- H.
symmetry.
apply H1.
apply free_in_var.
Qed.
Lemma free_in_env :
forall x e c s,
typing c e s ->
free_in x e ->
exists s', lookup x c = Some s'.
Proof.
intros x e c s Htype.
revert x.
induction Htype; intros x' Hfree.
inversion Hfree.
inversion Hfree.
inversion Hfree.
destruct H2.
apply IHHtype1.
assumption.
destruct H2.
apply IHHtype2.
assumption.
apply IHHtype3.
assumption.
inversion Hfree.
destruct H3.
apply IHHtype1.
assumption.
apply IHHtype2.
assumption.
inversion Hfree; subst.
specialize (IHHtype x' H8).
destruct IHHtype.
apply beq_nat_false_iff in H5.
simpl in H2.
rewrite H5 in H2.
exists x0.
assumption.
inversion Hfree; subst.
exists s.
assumption.
Qed.
Lemma subsumption :
forall c e s s',
typing c e s ->
subtype s s' ->
typing c e s'.
Proof.
intros c e s s' Htyp.
revert s'.
induction Htyp.
intros s' Htyp.
apply subtype_bool_left in Htyp.
destruct Htyp.
destruct H0.
subst.
apply typing_true.
apply flows_to_trans with (l' := l'); assumption.
intros s' Htyp.
apply subtype_bool_left in Htyp.
destruct Htyp.
destruct H0.
subst.
apply typing_false.
apply flows_to_trans with (l' := l'); assumption.
intros s'' Hsub.
apply typing_cond with (s := s)(l := l).
apply Htyp1.
apply Htyp2.
apply Htyp3.
apply subtype_trans with (t' := s').
apply H.
apply Hsub.
intros s'' Hsub.
apply typing_app with (s2 := s2)(s := s) (l := l)(s2' := s2'); auto.
apply subtype_trans with (t' := s'); auto.
intros s' Hsub.
apply subtype_arrow_left in Hsub.
destruct Hsub.
destruct H2.
destruct H2.
destruct H2.
destruct H3.
destruct H4.
subst.
apply typing_abs with (s2 := s2).
apply Htyp.
apply subtype_trans with (t' := s1'); assumption.
apply subtype_trans with (t' := s2'); assumption.
apply flows_to_trans with (l' := l'); assumption.
intros s'' Hsub.
apply typing_var with (s := s).
assumption.
apply subtype_trans with (t' := s'); assumption.
Qed.
Lemma stamp_idemp :
forall s l,
type_with_label s l ->
s = (stamp_type s l).
Proof.
destruct s; intros; simpl in H; subst; simpl; rewrite <- join_idempotent; reflexivity.
Qed.
Lemma canonical_form_bool :
forall c v l,
value v ->
typing c v (Bool l) ->
exists l',
(v = (TT l') \/ v = (FF l')) /\ flows_to l' l.
Proof.
intros c e l' Hval.
inversion Hval; subst.
intros Htype.
inversion Htype; subst.
exists l.
split.
left.
reflexivity.
apply H2.
intros Htype.
inversion Htype; subst.
exists l.
split.
right.
reflexivity.
apply H2.
intros Htype.
inversion Htype.
Qed.
Lemma canonical_form_arrow :
forall c v s1 s2 l,
value v ->
typing c v (Arrow s1 s2 l) ->
exists x s1' e l',
v = (Abs x s1' e l') /\ subtype s1 s1' /\ flows_to l' l.
Proof.