Combine continuation tables

docs/arithmetization/tables.tex

@@ -6,7 +6,6 @@ \section{Tables}
 \input{tables/byte-packing}
 \input{tables/logic}
 \input{tables/memory}
-\input{tables/mem-before}
-\input{tables/mem-after}
+\input{tables/mem-continuations}
 \input{tables/keccak-f}
 \input{tables/keccak-sponge}

docs/arithmetization/tables/mem-continuations.tex

@@ -0,0 +1,15 @@
+\subsection{Memory continuations}
+\label{mem-continuations}
+
+The MemBefore (resp. MemAfter) table holds the content of the memory before (resp. after) the execution of the current segment.
+For consistency, the MemAfter trace of a segment must be identical to the MemAfter trace of the next segment.
+Each row of these tables contains:
+
+\begin{enumerate}
+  \item $a$, the memory cell address,
+  \item $v$, the initial value of the cell.
+\end{enumerate}
+The tables should be ordered by $(a, \tau)$. Since they only hold values, there are no constraints between the rows.
+
+A CTL copies all of the MemBefore values in the memory trace as reads, at timestamp $\tau = 0$.
+Another CTL copies the final values from memory to MemAfter. For more details on which values are propagated, consult \ref{final-memory}.

docs/arithmetization/tables/memory.tex

@@ -85,6 +85,7 @@ \subsubsection{Memory initialization}
 \end{itemize}
 
 \subsubsection{Final memory}
+\label{final-memory}
 
 The final value of each cell of the memory must be propagated to the MemAfter table. Since memory operations are ordered by address and by timestamps, this is
 easy to do: the last value of an address is the value of the last row touching this address. In other words, we propagate values of rows before the address changes.