tex: Improve proof of example

tex/simple_barrier.tex

@@ -105,30 +105,30 @@ This makes it possible to call $\signalName$ and $\waitName$ on the returned bar
         \RES{x \mapsto \False} \\
         \COMMENT{Allocate $x$} \\
         \CODE{\Let b = \newbarrier{\TT} in} \\
-        \COMMENT{Use \ruleref{New Barrier-Spec} with $P \eqdef \Exists b'. b' \in \Bool \ast x \mapsto b'$} \\
-        \RES{x \mapsto \False \ast \send{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'} \ast \recv{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'}} \\
+        \COMMENT{Use \ruleref{New Barrier-Spec} with $P \eqdef \Exists b' \in \Bool. x \mapsto b'$} \\
+        \RES{x \mapsto \False \ast \send{b}{\Exists b' \in \Bool. x \mapsto b'} \ast \recv{b}{\Exists b' \in \Bool. x \mapsto b'}} \\
         \COMMENT{Give points-to and $\sendName$ predicate to left thread, $\recvName$ predicate to right thread} \\
         $\left(\begin{array}{l||l}
             \begin{array}{@{}l@{}}
                 \RES{\begin{array}{@{}l@{}}
                     x \mapsto \False \\
-                    \quad\ast\spac \send{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'}
+                    \quad\ast\spac \send{b}{\Exists b' \in \Bool. x \mapsto b'}
                 \end{array}} \\
                 \CODE{x \gets \expensiveBoolName\spac \TT;} \\
                 \COMMENT{Specification of $\expensiveBoolName$} \\
                 \RES{\begin{array}{@{}l@{}}
                     b' \in \Bool \ast x \mapsto b' \\
-                    \quad\ast\spac \send{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'}
+                    \quad\ast\spac \send{b}{\Exists b' \in \Bool. x \mapsto b'}
                 \end{array}} \\
                 \CODE{\signal{b}} \\
                 \COMMENT{Use \ruleref{Signal-Spec}} \\
                 \RES{\Ret v. v = \TT}
             \end{array} &
             \begin{array}{@{}l@{}}
-                \RES{\recv{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'}} \\
+                \RES{\recv{b}{\Exists b' \in \Bool. x \mapsto b'}} \\
                 \CODE{\Let n = \expensiveIntName\spac \TT in} \\
                 \COMMENT{Specification of $\expensiveIntName$} \\
-                \RES{n \in \Z \ast \recv{b}{\Exists b'. b' \in \Bool \ast x \mapsto b'}} \\
+                \RES{n \in \Z \ast \recv{b}{\Exists b' \in \Bool. x \mapsto b'}} \\
                 \CODE{\wait{b}} \\
                 \COMMENT{Use \ruleref{Wait-Spec}} \\
                 \RES{n \in \Z \ast b' \in \Bool \ast x \mapsto b'} \\
@@ -165,7 +165,7 @@ Before we can use \ruleref{New Barrier-Spec}, we should decide which resource we
 We can see that the right thread needs access to the points-to predicate of $x$ and that a boolean should be stored.
 We will thus use the following as our resource:
 \begin{gather*}
-    \Exists b'. b' \in \Bool \ast x \mapsto b'
+    \Exists b' \in \Bool. x \mapsto b'
 \end{gather*}
 We can now create a new barrier. We give the points-to and $\sendName$ predicates to the left thread and the $\recvName$ predicate to the right.
 In the left thread we first compute $\expensiveBoolName$ and store it in $x$.