make invariant more readable

papers/icfp2021/figure-queue-invariant.tex

 \begin{figure}
 \[\begin{array}{rcl}
-% public state:
-  \gqueue
-  & : &
-  \authm(\exm(\typeView \times \typeView \times \typeList(\typeVal \times \typeView)))
-\\
-  \isQueue \tview \hview \elemViewList
-  &\eqdef&
-  \ownGhost\gqueue{\authfrag \tuple{\tview, \hview, \elemViewList}}
-\\
-\hline
-% monotonicity of statuses:
-  \gmonos
-  & : &
-  \listm(\authm(\statusViewBotLat))
+
+  % public state:
+
+    \gqueue
+  & :
+  & \authm(\exm(\typeView \times \typeView \times \typeList(\typeVal \times \typeView)))
 \\
-  \monoWitness{\idx}{\pair\status\sview}
-  &\eqdef&
-  \ownGhost\gmonos{\mapsingleton{\idx}{\authfrag \pair\status\sview}}
+    \isQueue \tview \hview \elemViewList
+  & \eqdef
+  & \ownGhost\gqueue{\authfrag \tuple{\tview, \hview, \elemViewList}}
+
+\\ \hline
+
+  % monotonicity of statuses:
+
+    \gmonos
+  & :
+  & \listm(\authm(\statusViewBotLat))
 \\
-\hline
-% tokens:
-  \gtokens
-  & : &
-  \authm(\finmapm{\Z}{\exm(\typeUnit) + \exm(\typeVal \times \typeView)})
+    \monoWitness{\idx}{\pair\status\sview}
+  & \eqdef
+  & \ownGhost\gmonos{\mapsingleton{\idx}{\authfrag \pair\status\sview}}
+
+\\ \hline
+
+  % tokens:
+
+    \gtokens
+  & :
+  & \authm(\finmapm{\Z}{\exm(\typeUnit) + \exm(\typeVal \times \typeView)})
 \\
-  \tokenR\idx
-  &\eqdef&
-  \ownGhost\gtokens{\authfrag \mapsingleton{\idx}{\Unit}}
+    \tokenR\idx
+  & \eqdef
+  & \ownGhost\gtokens{\authfrag \mapsingleton{\idx}{\Unit}}
 \\
-  \tokenW\idx{\pair\elem\eview}
-  &\eqdef&
-  \ownGhost\gtokens{\authfrag \mapsingleton{\idx}{\pair\elem\eview}}
-  % hide the injection \exinj and left/right injections
+    \tokenW\idx{\pair\elem\eview}
+  & \eqdef
+  & \ownGhost\gtokens{\authfrag \mapsingleton{\idx}{\pair\elem\eview}}
+    % hide the injection \exinj and left/right injections
+
+\\ \hline
+
+  % invariant (outer):
+
+    \queueInv
+  & \eqdef
+  & \Exists \gmonos, \gtokens. \knowInv{}{\queueInvInner}
+
 \\
-\hline
-% invariant:
-  \queueInv
-  &\eqdef&
-  \left\{
-  \begin{array}{l}
-    \Exists \gmonos, \gtokens, \reftail, \refhead, \refstatuses, \refelements.
-  \\\quad
-    \ISep{%
-          \queue = \Tuple{\reftail, \refhead, \refstatuses, \refelements}
-      \cr \knowInv{}{\queueInvInner}
-    }% end of \ISep
-  \end{array}
-  \right.
+
+  % invariant (inner):
+
+    \queueInvInner
+  & \eqdef
 \\
-  \queueInvInner
-  &\eqdef&
-  \left\{
-  \begin{array}{l}
-    \Exists \tail, \head, \tview, \hview, \elemViewListWithLenInv,
-  \\ \phantom{\exists}\statusViewListWithLen.
-  \\\quad
-    \ISep{%
-      % pure assertions:
-          0 \le \tail \le \head \le \tail + \capacity
-      %\cr \length{\statusViewList} = \capacity
-      %\cr \length{\elemViewList} = \head - \tail
-      % pointsto assertions:
-      \cr \pointstoAT\reftail{\mkval\tail\tview}
-      \cr \pointstoAT\refhead{\mkval\head\hview}
-      \cr \arraypointstoAT\refstatuses\statusViewList
-      % authoritative ghost state assertions:
-      \cr \ownGhost\gqueue{\authfull \tuple{\tview, \hview, \elemViewListInv}}
-      \cr \ownGhost\gmonos{\mkDotList{\ANON{\authfull \nthStatusView{#1}}}\idx\capacity}
-      \cr \ownGhost\gtokens{%
-            \authfull
-              \left(\begin{array}{lll}
-                % authority for read tokens:
-                & \{
-                    \mapbinding \idx \Unit
-                &   \mid
-                    \head-c \leq \idx < \tail
-                  \}
-              \\  \uplus
-                % authority for write tokens:
-                & \{
-                    \mapbinding \idx {\nthElemView{\idx-\tail}}
-                &   \mid
-                    \tail \leq \idx < \head
-                  \}
-              \end{array}\right)
-          }
-      % cells which are empty or being emptied:
-      \cr \displaystyle\Sep_{\head-\capacity \leq \idx < \tail}
-              % cell bearing no value, available for next round:
-              \emptyCell \idx {\nthStatusView{\modcap\idx}}
-            \lor
-              % cell being emptied by a dequeuer:
-              \status_{\modcap\idx} = 2\idx + 1
-              % NOTE: here we can add 0 ≤ i to the invariant, but it is unneeded
-      % cells which are full or being filled:
-      \cr \displaystyle\Sep_{\tail \leq \idx < \head}
-              % cell bearing a value:
-              \fullCell \idx {\nthElemView{\idx-\tail}} {\nthStatusView{\modcap\idx}}
-            \lor
-              % cell being filled with a value by an enqueuer:
-              \status_{\modcap\idx} = 2\idx
-    }%end of the main \ISep of the invariant
-  \end{array}
-  \right.
+\multicolumn{3}{r}{%
+
+      \left\{\begin{array}{l}
+          \Exists \reftail, \refhead, \refstatuses, \refelements.
+      \\  \Exists \tail, \tview, \head, \hview, \elemViewListWithLenInv, \statusViewListWithLen.
+      \\  \quad \ISep{%
+            % pure assertions:
+                \queue = \Tuple{\reftail, \refhead, \refstatuses, \refelements}
+            \cr 0 \le \tail \le \head \le \tail + \capacity
+            %\cr \length{\statusViewList} = \capacity
+            %\cr \length{\elemViewList} = \head - \tail
+            % pointsto assertions:
+            \cr \pointstoAT\reftail{\mkval\tail\tview}
+            \cr \pointstoAT\refhead{\mkval\head\hview}
+            \cr \arraypointstoAT\refstatuses\statusViewList
+            % authoritative ghost state assertions:
+            \cr \ownGhost\gqueue{\authfull \tuple{\tview, \hview, \elemViewListInv}}
+            \cr \ownGhost\gmonos{\mkDotList{\ANON{\authfull \nthStatusView{##1}}}\idx\capacity}
+            \cr \ownGhost\gtokens{%
+                  \authfull
+                    \left(\begin{array}{lllr@{\;}l}
+                      % authority for read tokens:
+                      & \{
+                          \mapbinding \idx \Unit
+                          &\mid&
+                          \head-\capacity &\leq \idx < \tail
+                        \}
+                    \\  \uplus
+                      % authority for write tokens:
+                      & \{
+                          \mapbinding \idx {\nthElemView{\idx-\tail}}
+                          &\mid&
+                          \tail &\leq \idx < \head
+                        \}
+                    \end{array}\right)
+                }
+            \cr \begin{array}{@{\!\!}c@{}l@{\;}c@{\;}l} % array for aligning the formulas of the two big-stars
+            % cells which are empty or being emptied:
+                \displaystyle\Sep_{\head-\capacity \leq \idx < \tail}&
+                    % cell bearing no value, available for next round:
+                    \emptyCell \idx {\nthStatusView{\modcap\idx}}
+                  &\lor&
+                    % cell being emptied by a dequeuer:
+                    \status_{\modcap\idx} = 2\idx + 1
+                    % NOTE: here we can add 0 ≤ i to the invariant, but it is unneeded
+            % cells which are full or being filled:
+            \\  \displaystyle\Sep_{\tail \leq \idx < \head}&
+                    % cell bearing a value:
+                    \fullCell \idx {\nthElemView{\idx-\tail}} {\nthStatusView{\modcap\idx}}
+                  &\lor&
+                    % cell being filled with a value by an enqueuer:
+                    \status_{\modcap\idx} = 2\idx
+            \end{array}% end of array used to align the two big-stars
+          }%end of the main \ISep of the invariant
+      \end{array}\right.% end of the definition of \queueInvInner
+
+}% end of \multicolumn
+
 \\
-% predicate for empty cells:
-  \emptyCell \idx {\pair\status\sview}
-  &\eqdef&
-  \ISep{
-        \status = 2(\idx + \capacity)
-    \cr (\nthpointstoNA\refelements{\modcap\idx}-) \opat \sview
-    \cr \tokenR\idx
-  }
+
+  % predicate for available cells:
+
+    \emptyCell \idx {\pair\status\sview}
+  & \eqdef
+  & \ISep{%
+          \status = 2(\idx + \capacity)
+      \cr (\nthpointstoNA\refelements{\modcap\idx}-) \opat \sview
+      \cr \tokenR\idx
+    }
+
 \\
-% predicate for full cells:
-  \fullCell \idx {\pair\elem\eview} {\pair\status\sview}
-  &\eqdef&
-  \ISep{
-        \status = 2\idx + 1
-    \cr (\nthpointstoNA\refelements{\modcap\idx}{\elem}) \opat \sview
-    \cr \tokenW\idx{\pair\elem\eview}
-    \cr \eview \viewleq \sview
-  }
-\end{array}\]
 
+  % predicate for occupied cells:
+
+    \fullCell \idx {\pair\elem\eview} {\pair\status\sview}
+  & \eqdef
+  & \ISep{%
+          \status = 2\idx + 1
+      \cr (\nthpointstoNA\refelements{\modcap\idx}{\elem}) \opat \sview
+      \cr \tokenW\idx{\pair\elem\eview}
+      \cr \eview \viewleq \sview
+    }
+
+\end{array}\]
 \Description{Internal invariant of the bounded queue.}
 \caption{Internal invariant of the bounded queue}
 \label{fig:queue:invariant}

papers/icfp2021/notations.tex

@@ -619,9 +619,9 @@
 \newcommand{\isQueueSC}[1]{\ISQUEUE\;\gqueue\;{#1}}
 \newcommand{\isQueue}[3]{\ISQUEUE\;\gqueue\;{#1}\;{#2}\;{#3}}
 \newcommand{\queueInv}{\fname{queueInv}\;\queue\;\gqueue}
-\newcommand{\queueInvInner}{\fname{queueInvInner}\;\gqueue\;\gmonos\;\gtokens\;…}%\reftail\;\refhead\;\refstatuses\;\refelements}
-\newcommand{\emptyCell}[2]{\fname{available}\;{#1}\;{#2}}
-\newcommand{\fullCell}[3]{\fname{occupied}\;{#1}\;{#2}\;{#3}}
+\newcommand{\queueInvInner}{\fname{queueInvInner}\;\queue\;\gqueue\;\gmonos\;\gtokens}
+\newcommand{\emptyCell}[2]{\fname{available}\;\refelements\;\gtokens\;{#1}\;{#2}}
+\newcommand{\fullCell}[3]{\fname{occupied}\;\refelements\;\gtokens\;{#1}\;{#2}\;{#3}}
 
 % canonical variable names:
 \newcommand{\capacity}{\ensuremath{C}}