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AbstractInterpretation: analyzed one program
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Adam Chlipala
parent
e892c8dbab
commit
c303dc02c9
2 changed files with 99 additions and 70 deletions
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@ -250,16 +250,6 @@ Module SimpleAbstractInterpreter.
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| Some ss'' => merge_astates ss'' ss'
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end).
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Inductive interpret a : astates a -> astates a -> Prop :=
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| InterpretDone : forall ss,
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oneStepClosure ss ss
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-> interpret ss ss
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| InterpretStep : forall ss ss' ss'',
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oneStepClosure ss ss'
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-> interpret (merge_astates ss ss') ss''
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-> interpret ss ss''.
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Definition subsumed a (s1 s2 : astate a) :=
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forall x, match s1 $? x with
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| None => s2 $? x = None
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@ -291,6 +281,17 @@ Module SimpleAbstractInterpreter.
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Hint Resolve subsumeds_refl.
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Inductive interpret a : astates a -> astates a -> astates a -> Prop :=
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| InterpretDone : forall ss1 any ss2,
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oneStepClosure ss1 ss2
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-> subsumeds ss2 ss1
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-> interpret ss1 any ss1
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| InterpretStep : forall ss worklist ss' ss'',
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oneStepClosure worklist ss'
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-> interpret (merge_astates ss ss') ss' ss''
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-> interpret ss worklist ss''.
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Lemma oneStepClosure_sound : forall a, absint_sound a
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-> forall ss ss' : astates a, oneStepClosure ss ss'
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-> forall c s s' c', ss $? c = Some s
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@ -474,25 +475,25 @@ Module SimpleAbstractInterpreter.
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Qed.
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Lemma interpret_sound' : forall c a, absint_sound a
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-> forall ss ss' : astates a, interpret ss ss'
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-> forall ss worklist ss' : astates a, interpret ss worklist ss'
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-> ss $? c = Some $0
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-> invariantFor (absint_trsys a c) (fun p => exists s, ss' $? snd p = Some s
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/\ subsumed (fst p) s).
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Proof.
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induct 2; simplify.
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induct 2; simplify; subst.
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apply invariant_induction; simplify; propositional; subst; simplify; eauto.
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invert H2; propositional.
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invert H3; propositional.
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cases s.
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cases s'.
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simplify.
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eapply abs_step_monotone in H3; eauto.
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invert H3; propositional.
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eapply oneStepClosure_sound in H3; eauto.
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invert H3; propositional.
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eexists; propositional.
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eassumption.
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eapply abs_step_monotone in H4; eauto.
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invert H4; propositional.
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eapply oneStepClosure_sound in H4; eauto.
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invert H4; propositional.
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eapply H1 in H4.
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invert H4; propositional.
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eauto using subsumed_trans.
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apply IHinterpret.
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@ -506,7 +507,7 @@ Module SimpleAbstractInterpreter.
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Theorem interpret_sound : forall c a (ss : astates a),
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absint_sound a
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-> interpret ($0 $+ (c, $0)) ss
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-> interpret ($0 $+ (c, $0)) ($0 $+ (c, $0)) ss
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-> invariantFor (absint_trsys a c) (fun p => exists s, ss $? snd p = Some s
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/\ subsumed (fst p) s).
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Proof.
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@ -781,9 +782,9 @@ Module SimpleAbstractInterpreter.
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Ltac simplify_map :=
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match goal with
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| [ |- context[?m $+ (?k, ?v)] ] =>
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| [ |- context[@add ?A ?B ?m ?k ?v] ] =>
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simplify_map' (m $+ (k, v)) tt ltac:(fun m' =>
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replace (m $+ (k, v)) with m' by maps_equal)
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replace (@add A B m k v) with m' by maps_equal)
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end.
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Definition easy :=
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@ -792,9 +793,59 @@ Module SimpleAbstractInterpreter.
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"n" <- "n" - 2
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done.
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Ltac interpret1 := eapply InterpretStep; [ repeat (apply OscNil || apply OscCons)
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| unfold merge_astates, merge_astate;
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simplify; repeat simplify_map ].
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Lemma subsumeds_empty : forall a (ss : astates a),
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subsumeds $0 ss.
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Proof.
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unfold subsumeds; simplify.
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equality.
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Qed.
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Lemma subsumeds_add_left : forall a (ss1 ss2 : astates a) c s,
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ss2 $? c = Some s
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-> subsumeds ss1 ss2
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-> subsumeds (ss1 $+ (c, s)) ss2.
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Proof.
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unfold subsumeds; simplify.
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cases (command_equal c c0); subst; simplify; eauto.
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invert H1; eauto.
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Qed.
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Ltac interpret_done := eapply InterpretDone; [
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repeat (apply OscNil || apply OscCons)
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| unfold merge_astates, merge_astate; simplify; repeat simplify_map;
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repeat (apply subsumeds_add_left || apply subsumeds_empty); (simplify; equality) ].
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Lemma final_even : forall (s s' : astate parity_absint) v x,
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compatible1 s v
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-> subsumed s s'
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-> s' $? x = Some Even
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-> exists n, v $? x = Some n /\ isEven n.
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Proof.
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unfold compatible1, subsumed; simplify.
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specialize (H x); specialize (H0 x).
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cases (s $? x); simplify.
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rewrite Heq in *.
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assert (Some t = Some t) by equality.
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apply H in H2.
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first_order.
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eapply H0 in H1.
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invert H1.
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eauto.
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assumption.
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rewrite Heq in *.
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equality.
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Qed.
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Theorem easy_even : forall v n,
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isEven n
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-> invariantFor (trsys_of v easy) (fun p => exists n, fst p $? "n" = Some n /\ isEven n).
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-> invariantFor (trsys_of v easy) (fun p => snd p = Skip
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-> exists n, fst p $? "n" = Some n /\ isEven n).
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Proof.
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simplify.
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eapply invariant_weaken.
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@ -807,52 +858,18 @@ Module SimpleAbstractInterpreter.
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apply interpret_sound.
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apply parity_sound.
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Ltac interpret1 := eapply InterpretStep; [ repeat (apply OscNil || apply OscCons)
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| unfold merge_astates, merge_astate;
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simplify; repeat simplify_map ].
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interpret1.
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interpret1.
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interpret1.
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interpret1.
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interpret1.
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interpret_done.
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interpret1.
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interpret1.
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interpret1.
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invert 1.
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first_order.
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invert H1; simplify.
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invert H2.
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eapply final_even; eauto; simplify; equality.
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Qed.
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eapply InterpretStep.
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repeat (apply OscNil || apply OscCons).
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simplify.
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unfold merge_astates, merge_astate.
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simplify.
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interpret1.
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simplify.
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simplify_map.
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Ltac interpret1 :=
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eapply InterpretStep; [ (repeat (apply OscNil || eapply OscCons);
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invert 1; repeat (match goal with
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| [ H : step _ _ |- _ ] => invert H
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| [ _ : match ?E with
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| Some n => n
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| None => 0
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end = 0 |- _ ] => cases E; try linear_arithmetic
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end; simplify); simplify;
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try solve [ eapply compatible_skip; eassumption
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| eapply compatible_skip2; eassumption || congruence
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| eapply compatible_const; eassumption ]) | ];
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repeat match goal with
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| [ |- context[@add ?A ?B ?m _ _] ] =>
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match m with
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| @empty _ _ => fail 1
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| _ => unify m (@empty A B)
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end
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end.
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interpret1.
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unfold merge_astates, merge_astate; simplify.
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interpret1.
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unfold merge_astates, merge_astate; simplify.
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simplify_map.
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interpret1.
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unfold merge_astates, merge_astate; simplify.
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simplify_map.
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interpret1.
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End SimpleAbstractInterpreter.
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12
Map.v
12
Map.v
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@ -139,6 +139,10 @@ Module Type S.
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end).
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Hint Extern 3 (_ = _) => maps_equal.
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Axiom lookup_split : forall A B (m : fmap A B) k v k' v',
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(m $+ (k, v)) $? k' = Some v'
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-> (k' <> k /\ m $? k' = Some v') \/ (k' = k /\ v' = v).
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End S.
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Module M : S.
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@ -388,6 +392,14 @@ Module M : S.
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| [ _ : context[if ?E then _ else _] |- _ ] => destruct E
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end; intuition congruence).
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Qed.
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Lemma lookup_split : forall A B (m : fmap A B) k v k' v',
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lookup (add m k v) k' = Some v'
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-> (k' <> k /\ lookup m k' = Some v') \/ (k' = k /\ v' = v).
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Proof.
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unfold lookup, add; simpl; intros.
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destruct (decide (k' = k)); intuition congruence.
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Qed.
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End M.
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Export M.
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