Finish proving wp_correctness
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wp.v
182
wp.v
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@ -441,20 +441,30 @@ Qed.
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(***** Weakest precondition **************************************************)
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Fixpoint wp (instr: Instr) (cond: Assert) : Assert := match instr with
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| skip =>
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cond
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Fixpoint wp (instr: Instr) (condOpt: option Assert) : option Assert :=
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match condOpt with
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| None => None
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| Some cond => match instr with
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| skip =>
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Some cond
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| abort =>
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assertTop
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| assign x expr =>
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cond [[ x <- expr expr ]]
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| seq s1 s2 =>
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wp s1 (wp s2 cond)
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| ifelse guard sIf sElse =>
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((assertOfExpr guard -> wp sIf cond)
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/\ (~ (assertOfExpr guard) -> wp sElse cond)) % assert
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| while guard body => assertTop
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end.
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Some assertTop
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| assign x expr =>
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Some (cond [[ x <- expr expr ]])
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| seq s1 s2 =>
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wp s1 (wp s2 condOpt)
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| ifelse guard sIf sElse =>
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match (wp sIf condOpt, wp sElse condOpt) with
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| (None, _) | (_, None) => None
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| (Some wpIf, Some wpElse) =>
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Some (
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((assertOfExpr guard -> wpIf)
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/\ (~ (assertOfExpr guard) -> wpElse)) % assert)
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end
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| while guard body =>
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None
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end
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end.
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Lemma assertImplElim {a b: Assert} :
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forall (m: Mem), (assertImpl a b) m -> a m -> b m.
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@ -472,42 +482,120 @@ Proof.
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unfold assertImplLogical. intros mem x. assumption.
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Qed.
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Theorem wp_correctness_provable (instr: Instr) :
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forall post, ( |- [| wp instr post |] instr [| post |] ) % assert.
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Definition whatever_or_none (whatever: Assert -> Instr -> Assert -> Prop)
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(pre: option Assert) (instr: Instr) (post: option Assert) : Prop :=
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match (pre, post) with
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| (Some _, None) => False
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| (None, _) => True
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| (Some pre0, Some post0) => whatever pre0 instr post0
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end.
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Definition provable_or_none := whatever_or_none hoare_provability.
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Notation "|-opt [| pre |] instr [| post |]" :=
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(provable_or_none pre instr post) (at level 30) : assert.
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Definition consequence_or_none := whatever_or_none hoare_consequence.
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Notation "|=opt [| pre |] instr [| post |]" :=
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(consequence_or_none pre instr post) (at level 30) : assert.
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Lemma postnone_is_okay {instr post}:
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(forall post0, (|-opt [|wp instr (Some post0)|] instr [|Some post0|])%assert)
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-> (|-opt [|wp instr post|] instr [|post|])%assert.
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Proof.
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induction instr; intros post; simpl.
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* apply (H_skip post).
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* apply (H_abort assertTop post).
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* apply (H_assign post v e).
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* remember (wp instr2 post) as mid eqn:midRel.
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intros prf. destruct post.
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- apply prf.
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- unfold provable_or_none; unfold whatever_or_none.
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unfold wp; destruct instr; trivial.
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Qed.
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Theorem wp_correctness_provable (instr: Instr) :
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forall post, ( |-opt [| wp instr post |] instr [| post |] ) % assert.
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Proof.
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induction instr; intros post; apply postnone_is_okay; intros post0.
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* apply (H_skip post0).
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* apply (H_abort assertTop post0).
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* apply (H_assign post0 v e).
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* remember (wp instr2 (Some post0)) as mid eqn:midRel.
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remember (wp instr1 mid) as pre eqn:preRel.
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specialize IHinstr2 with post.
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specialize IHinstr1 with mid.
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rewrite <- midRel in IHinstr2.
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rewrite <- preRel in IHinstr1.
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apply (H_seq pre mid post instr1 instr2).
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assumption. assumption.
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* remember ((assertOfExpr e -> wp instr1 post)
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/\ (~ assertOfExpr e -> wp instr2 post)) % assert
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as pre eqn:preRel.
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apply (H_if pre post e instr1 instr2).
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- apply (H_conseq
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(pre /\ assertOfExpr e)%assert post
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(wp instr1 post) post instr1
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(IHinstr1 post)).
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+ rewrite preRel. unfold assertImplLogical.
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intros mem. intros [ [disjunctIf disjunctElse] isIf].
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apply (assertImplElim mem disjunctIf isIf).
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+ apply (assertImplSelf post).
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- apply (H_conseq
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(pre /\ ~ assertOfExpr e)%assert post
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(wp instr2 post) post instr2
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(IHinstr2 post)).
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+ rewrite preRel. unfold assertImplLogical.
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intros mem. intros [ [disjunctIf disjunctElse] isElse].
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apply (assertImplElim mem disjunctElse isElse).
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+ apply (assertImplSelf post).
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*
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simpl; rewrite <- midRel; rewrite <- preRel.
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specialize IHinstr2 with (Some post0) as IHpost.
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specialize IHinstr1 with mid as IHmid.
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rewrite <- midRel in IHpost; rewrite <- preRel in IHmid.
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destruct mid as [mid0 | ] eqn:mid0Rel.
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destruct pre as [pre0 | ] eqn: pre0Rel.
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apply (H_seq pre0 mid0 post0).
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apply IHmid. apply IHpost.
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+ unfold whatever_or_none; trivial.
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+ destruct pre.
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- unfold provable_or_none in IHmid; unfold whatever_or_none in IHmid.
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exfalso. apply IHmid.
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- unfold whatever_or_none; trivial.
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* specialize IHinstr1 with (Some post0);
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specialize IHinstr2 with (Some post0).
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destruct (wp instr1 (Some post0)) as [preIf | ] eqn:preIfRel;
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destruct (wp instr2 (Some post0)) as [preElse | ] eqn:preElseRel.
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remember ((assertOfExpr e -> preIf)
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/\ (~ assertOfExpr e -> preElse)) % assert
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as pre0 eqn:pre0Rel.
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assert (wp (ifelse e instr1 instr2) (Some post0) = (Some pre0)).
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{
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rewrite pre0Rel; simpl; rewrite preIfRel; rewrite preElseRel; congruence.
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}
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{
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rewrite H.
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apply (H_if pre0 post0 e instr1 instr2).
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- apply (H_conseq
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(pre0 /\ assertOfExpr e)%assert post0
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preIf post0 instr1
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IHinstr1).
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+ rewrite pre0Rel. unfold assertImplLogical.
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intros mem [ [disjunctIf disjunctElse] isIf].
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apply (assertImplElim mem disjunctIf isIf).
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+ apply (assertImplSelf post0).
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- apply (H_conseq
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(pre0 /\ ~ assertOfExpr e)%assert post0
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preElse post0 instr2
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IHinstr2).
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+ rewrite pre0Rel. unfold assertImplLogical.
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intros mem [ [disjunctIf disjunctElse] isElse].
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apply (assertImplElim mem disjunctElse isElse).
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+ apply (assertImplSelf post0).
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}
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- unfold provable_or_none; simpl; rewrite preElseRel;
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rewrite preIfRel; trivial.
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- unfold provable_or_none; simpl; rewrite preElseRel;
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rewrite preIfRel; trivial.
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- unfold provable_or_none; simpl; rewrite preElseRel;
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rewrite preIfRel; trivial.
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* unfold wp; unfold provable_or_none; unfold whatever_or_none; trivial.
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Qed.
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Lemma provable_opt_implies_provable {pre instr post} :
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(|-opt [|Some pre|] instr [|Some post|])%assert
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-> (|- [|pre|] instr [|post|])%assert.
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Proof.
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intros prf.
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unfold provable_or_none in prf; unfold whatever_or_none in prf. assumption.
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Qed.
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Theorem wp_correctness (instr: Instr) :
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forall post, ( |=opt [| wp instr post |] instr [| post |] ) % assert.
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Proof.
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intros post.
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destruct post as [post0 | ] eqn:post0Rel.
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remember (wp instr (Some post0)) as pre eqn:preRel.
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destruct pre as [ pre0 | ] eqn:pre0Rel.
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- unfold consequence_or_none; unfold whatever_or_none.
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apply hoare_provability_implies_consequence.
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apply provable_opt_implies_provable.
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rewrite preRel.
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apply wp_correctness_provable.
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- unfold consequence_or_none; unfold whatever_or_none; trivial.
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- unfold wp.
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destruct instr; unfold consequence_or_none; unfold
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whatever_or_none; trivial.
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Qed.
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(* vim: ts=2 sw=2
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*)
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