### section 5

parent 85467b42
 ... ... @@ -86,10 +86,10 @@ Let $\varphi \equiv x < y: \hls{}(x, y; 3)$ and $s$ be a store such that $s(x)= The following lemma (proved in the appendix) states that the effective upper bounds of chunks in heap lists atoms of$\varphi$with respect to stacks %\mihaela{r3: not clear} can be captured by an {\EPbA} formula. can be captured by a {\qfpa} formula. \begin{lemma}\label{lem-eub} For an {\slah} formula$\varphi \equiv \Pi: \hls{}(t'_1, t'_2; t'_3)$, a formula$\xi_{eub,\varphi}(z)$can be constructed in linear time such that for every store$s$satisfying$s \models \abs(\varphi)$, we have$s[z \gets \eub_\varphi(s)] \models \xi_{eub,\varphi}(z)$. For an {\slah} formula$\varphi \equiv \Pi: \hls{}(t'_1, t'_2; t'_3)$, a {\qfpa} formula$\xi_{eub,\varphi}(z)$can be constructed in linear time such that for every store$s$satisfying$s \models \abs(\varphi)$, we have$s[z \gets \eub_\varphi(s)] \models \xi_{eub,\varphi}(z)$. \end{lemma} The following lemma (proof in the appendix) provides the correct test used ... ... @@ -101,10 +101,11 @@ Then$\varphi \models_\preceq \psi$iff %$C_\preceq \wedge \abs(\varphi) \models \exists z.\ \xi_{eub, \varphi}(z) \wedge z \le t_3$.$C_\preceq \wedge \abs(\varphi) \models \forall z.\ \xi_{eub, \varphi}(z) \rightarrow z \le t_3$. \end{lemma} From Lemma~\ref{lem-hls-hls}, it follows that the entailment of From Lemma~\ref{lem-hls-hls}, it follows that$\varphi \equiv \Pi: \hls{}(t'_1, t'_2; t'_3) \models_\preceq \hls{}(t_1, t_2; t_3)$is invalid iff$C_\preceq \wedge \abs(\varphi) \wedge \exists z.\ \xi_{eub, \varphi}(z) \wedge \neg z \le t_3$is satisfiable, which is an {\EPbA} formula. Therefore, this special case of the ordered entailment problem is in coNP. \medskip %\vspace{-4mm} \subsubsection{At least two atoms in the antecedent:} Recall that \noindent {\bf At least two atoms in the antecedent:} Recall that$\varphi\equiv C_\preceq \wedge \Pi: a_1 \sepc \cdots \sepc a_m$; a case analysis on the form of the first atom of the antecedent,$a_1\$, follows. ... ...
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