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TypesAndMutation: finish lambda-ref soundness proof

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Adam Chlipala 2016-03-22 14:17:40 -04:00
parent c279d3d610
commit cf9062fa4e
1 changed files with 43 additions and 21 deletions
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64
TypesAndMutation.v
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@ -181,9 +181,6 @@ Module Rlc.
| [ H : hasty _ _ ?e _, H' : value ?e |- _ ] => (invert H'; invert H); [] | [ H : hasty _ _ ?e _, H' : value ?e |- _ ] => (invert H'; invert H); []
| [ H : hasty _ _ _ _ |- _ ] => invert1 H | [ H : hasty _ _ _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => invert1 H | [ H : plug _ _ _ |- _ ] => invert1 H
| [ H : forall l t, ?h $? l = Some t -> _,
H' : ?h $? _ = Some _ |- _ ] => apply H in H'
end; subst. end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7. Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
@ -197,6 +194,8 @@ Module Rlc.
\/ exists he', step (h, e) he'. \/ exists he', step (h, e) he'.
Proof. Proof.
induct 2; t. induct 2; t.
apply H2 in H8; t.
apply H1 in H8; t.
Qed. Qed.
Lemma weakening_override : forall (G G' : fmap var type) x t, Lemma weakening_override : forall (G G' : fmap var type) x t,
@ -288,7 +287,9 @@ Module Rlc.
-> forall H1 t, hasty H1 $0 e1 t -> forall H1 t, hasty H1 $0 e1 t
-> heapty H1 h1 -> heapty H1 h1
-> exists H2, hasty H2 $0 e2 t -> exists H2, hasty H2 $0 e2 t
/\ heapty H2 h2. /\ heapty H2 h2
/\ (forall l t, H1 $? l = Some t
-> H2 $? l = Some t).
Proof. Proof.
invert 1; t. invert 1; t.
@ -297,31 +298,51 @@ Module Rlc.
econstructor. econstructor.
simplify. simplify.
auto. auto.
eauto. eauto 6.
apply H3 in H9; t.
rewrite H1 in H2. rewrite H1 in H2.
invert H2. invert H2.
eauto. eauto.
rewrite H1 in H2. assert (H0 $? l = Some t) by assumption.
invert H2. apply H3 in H8.
exists H0; propositional. invert H8; propositional.
Admitted. rewrite H1 in H5.
invert H5.
eexists; propositional.
eauto.
exists bound; propositional.
cases (l ==n l0); simplify; eauto.
subst.
rewrite H in H2; invert H2.
eauto.
apply H4 in H2.
cases (l ==n l0); simplify; equality.
assumption.
Qed.
Hint Resolve preservation0. Hint Resolve preservation0.
Lemma generalize_plug : forall e1 C e1', Lemma generalize_plug : forall H e1 C e1',
plug C e1 e1' plug C e1 e1'
-> forall e2 e2', plug C e2 e2' -> forall t, hasty H $0 e1' t
-> (forall H t, hasty H $0 e1 t -> hasty H $0 e2 t) -> exists t0, hasty H $0 e1 t0
-> (forall H t, hasty H $0 e1' t -> hasty H $0 e2' t). /\ (forall e2 e2' H',
hasty H' $0 e2 t0
-> plug C e2 e2'
-> (forall l t, H $? l = Some t -> H' $? l = Some t)
-> hasty H' $0 e2' t).
Proof. Proof.
induct 1; t. Ltac applyIn := match goal with
| [ H : forall x, _, H' : _ |- _ ] =>
apply H in H'; clear H; invert H'; propositional
end.
induct 1; t; (try applyIn; eexists; t).
Qed. Qed.
Hint Resolve generalize_plug. Lemma preservation : forall h1 e1 h2 e2,
Lemma preservation : forall h1 e1 h2 e2,
step (h1, e1) (h2, e2) step (h1, e1) (h2, e2)
-> forall H1 t, hasty H1 $0 e1 t -> forall H1 t, hasty H1 $0 e1 t
-> heapty H1 h1 -> heapty H1 h1
@ -329,11 +350,12 @@ Module Rlc.
/\ heapty H2 h2. /\ heapty H2 h2.
Proof. Proof.
invert 1; simplify. invert 1; simplify.
eapply preservation0 in H6. eapply generalize_plug in H; eauto.
invert H; propositional.
eapply preservation0 in H6; eauto.
invert H6; propositional. invert H6; propositional.
exists x; propositional. eauto.
3: eauto. Qed.
Admitted.
Hint Resolve progress preservation. Hint Resolve progress preservation.