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AbstractInterpretation: optimized execution engine some more, finishing loopy

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Adam Chlipala 2016-03-05 16:34:45 -05:00
parent 2068f7691a
commit 5ae0e6641e
1 changed files with 59 additions and 24 deletions
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83
AbstractInterpretation.v
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@ -243,12 +243,13 @@ Module SimpleAbstractInterpreter.
Inductive oneStepClosure a : astates a -> astates a -> Prop :=
| OscNil :
oneStepClosure $0 $0
| OscCons : forall ss c s ss',
| OscCons : forall ss c s ss' ss'',
oneStepClosure ss ss'
-> oneStepClosure (ss $+ (c, s)) (match absint_step s c (fun x => x) with
| None => ss'
| Some ss'' => merge_astates ss'' ss'
end).
-> match absint_step s c (fun x => x) with
| None => ss'
| Some ss'' => merge_astates ss'' ss'
end = ss''
-> oneStepClosure (ss $+ (c, s)) ss''.
Definition subsumed a (s1 s2 : astate a) :=
forall x, match s1 $? x with
@ -305,8 +306,8 @@ Module SimpleAbstractInterpreter.
cases (command_equal c c0); subst; simplify.
invert H1.
invert H2.
invert H3.
rewrite H5.
unfold merge_astates; simplify.
rewrite H7.
@ -319,11 +320,11 @@ Module SimpleAbstractInterpreter.
invert H1; eauto.
eauto.
apply IHoneStepClosure in H2; auto.
invert H2; propositional.
apply IHoneStepClosure in H3; auto.
invert H3; propositional.
cases (absint_step s c (fun x => x)); eauto.
unfold merge_astates; simplify.
rewrite H2.
rewrite H3.
cases (a0 $? c'); eauto.
eexists; propositional.
unfold subsumed; simplify.
@ -717,14 +718,6 @@ Module SimpleAbstractInterpreter.
exists x0; ring.
Qed.
Definition loopy :=
"n" <- 100;;
"a" <- 0;;
while "n" loop
"a" <- "a" + "n";;
"n" <- "n" - 2
done.
Lemma merge_astates_fok : forall x : option (astate parity_absint),
match x with Some x' => Some x' | None => None end = x.
Proof.
@ -765,14 +758,17 @@ Module SimpleAbstractInterpreter.
invert H1; eauto.
Qed.
Ltac interpret1 := eapply InterpretStep; [ repeat (apply OscNil || apply OscCons)
| unfold merge_astates, merge_astate;
simplify; repeat simplify_map ].
Ltac interpret_simpl := unfold merge_astates, merge_astate;
simplify; repeat simplify_map.
Ltac interpret_done := eapply InterpretDone; [
repeat (apply OscNil || apply OscCons)
| unfold merge_astates, merge_astate; simplify; repeat simplify_map;
repeat (apply subsumeds_add_left || apply subsumeds_empty); (simplify; equality) ].
Ltac oneStepClosure := apply OscNil
|| (eapply OscCons; [ oneStepClosure
| interpret_simpl; reflexivity ]).
Ltac interpret1 := eapply InterpretStep; [ oneStepClosure | interpret_simpl ].
Ltac interpret_done := eapply InterpretDone; [ oneStepClosure
| repeat (apply subsumeds_add_left || apply subsumeds_empty); (simplify; equality) ].
Lemma final_even : forall (s s' : astate parity_absint) v x,
compatible1 s v
@ -824,4 +820,43 @@ Module SimpleAbstractInterpreter.
eapply final_even; eauto; simplify; equality.
Qed.
Definition loopy :=
"n" <- 100;;
"a" <- 0;;
while "n" loop
"a" <- "a" + "n";;
"n" <- "n" - 2
done.
Theorem loopy_even : forall v,
invariantFor (trsys_of v loopy) (fun p => snd p = Skip
-> exists n, fst p $? "n" = Some n /\ isEven n).
Proof.
simplify.
eapply invariant_weaken.
unfold loopy.
eapply invariant_simulates.
apply absint_simulates with (a := parity_absint).
apply parity_sound.
apply interpret_sound.
apply parity_sound.
interpret1.
interpret1.
interpret1.
interpret1.
interpret1.
interpret1.
interpret1.
interpret_done.
invert 1.
first_order.
invert H0; simplify.
invert H1.
eapply final_even; eauto; simplify; equality.
Qed.
End SimpleAbstractInterpreter.