### aplas: fix typos from atva

parent b3066f8e
 ... ... @@ -107,7 +107,9 @@ to obtain in polynomial time an equi-satisfiable {\qfpa} abstraction for a symbo We introduce some notations first. % Given a formula $\varphi\equiv \Pi : \Sigma$, $\atoms(\varphi)$ denotes the set of spatial atoms in $\Sigma$, and $\patoms(\varphi)$ denotes the set of predicate atoms in $\Sigma$. $\atoms(\varphi)$ denotes the set of spatial atoms in $\Sigma$, and $\patoms(\varphi)$ denotes the set of predicate atoms in $\Sigma$. We also denote $\overline{\patoms}(\varphi)$ for $\atoms(\varphi)\setminus\patoms(\varphi)$. \begin{definition}{(Presburger abstraction of \slah\ formula)} Let $\varphi\equiv \Pi : \Sigma$ be a \slah\ formula. ... ... @@ -139,13 +141,13 @@ where \quad (\atomtail(a_j) \le \atomhead(a_i)\lor \atomtail(a_i) \le \atomhead(a_j)) \end{array} \\ \phi_2 & \triangleq & \bigwedge\limits_{a_i \in \patoms, a_j \not \in \patoms} \phi_2 & \triangleq & \bigwedge\limits_{a_i \in \patoms(\varphi), a_j \in \overline{\patoms}(\varphi)} \begin{array}[t]{l} (\isnonemp_{a_i}) \limp \\ \quad (\atomtail(a_j) \le \atomhead(a_i) \lor \atomtail(a_i) \le \atomhead(a_j)) \end{array} \\ \phi_3 & \triangleq & \bigwedge\limits_{a_i, a_j \not \in \patoms(\varphi), i < j} \phi_3 & \triangleq & \bigwedge\limits_{a_i, a_j \in \overline{\patoms}(\varphi), i < j} \atomtail(a_j) \le \atomhead(a_i) \lor \atomtail(a_i) \le \atomhead(a_j) \end{eqnarray} } ... ...
 ... ... @@ -61,7 +61,7 @@ The general case is dealt with in Section~\ref{ssec:ent-all}; %At first, it is easy to observe that if $\abs(\varphi)$ is unsatisfiable, %then the entailment holds. Moreover, if $\abs(\varphi) \not\models \abs(\psi)$, %the the entailment $\varphi \models \psi$ does not hold. In the sequel, we assume that $\abs(\varphi)$ is satisfiable and $\abs(\varphi) \models \abs(\psi)$. Otherwise, the entailment is trivially unsatisfiable. In the sequel, we assume that $\abs(\varphi)$ is satisfiable and $\abs(\varphi) \models \abs(\psi)$. Otherwise, the entailment is trivially invalid. \subsection{Decomposition into ordered entailments} \label{ssec:order} ... ...
 ... ... @@ -139,7 +139,7 @@ $s,h \models \exists\vec{z}\cdot\Pi:\Sigma$ iff $\exists \vec{n}\in\NN^{|\vec{z \end{definition} %\vspace{-2mm} We write$A \models B$for$A$and$B$(sub-)formula in \slah\ for We write$A \models B$where$A$and$B$are (sub-)formula in \slah\ for$A$entails$B$, i.e., that for any model$(s,h)$such that$s,h\models A$then$s,h\models B$. ... ...  ... ... @@ -177,7 +177,7 @@ when the heap it denotes is not empty is$(v=2 \land \exists k\cdot k > 0 \land 2k = y - x) \lor (2 < v \land 2 < y-x)$, i.e., either all chunks have size 2 and the heap-list has an even size or$v$and the size of the heap-list are strictly greater than 2. both$v$and the size of the heap-list are strictly greater than 2. For the empty case, the summary is trivially$x=y$. %For$v=\infty$, the summary is$2 < y-x$. The other spatial atoms$a$(e.g.,$x \pto v$and$\blk(x,y)\$) ... ...
 ... ... @@ -101,7 +101,7 @@ \newcommand \patoms {{\tt PAtoms}} \newcommand \atoms {{\tt Atoms}} \newcommand \atomhead {{\tt head}} \newcommand \atomtail {{\tt tail}} \newcommand \atomtail {{\tt next}} \newcommand \vars {{\tt Vars}} \newcommand{\addr}{\mathcal{A}} ... ...
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