Lookup argument cosmetics.

Signed-off-by: Daira Hopwood <daira@jacaranda.org>

book/src/design/proving-system/lookup.md

@@ -16,26 +16,26 @@ For ease of explanation, we'll first describe a simplified version of the argume
 ignores zero knowledge.
 
 We express lookups in terms of a "subset argument" over a table with $2^k$ rows (numbered
-from 0), and columns $A$ and $S$.
+from 0), and columns $A$ and $S.$
 
 The goal of the subset argument is to enforce that every cell in $A$ is equal to _some_
-cell in $S$. This means that more than one cell in $A$ can be equal to the _same_ cell in
-$S$, and some cells in $S$ don't need to be equal to any of the cells in $A$.
+cell in $S.$ This means that more than one cell in $A$ can be equal to the _same_ cell in
+$S,$ and some cells in $S$ don't need to be equal to any of the cells in $A.$
 
 - $S$ might be fixed, but it doesn't need to be. That is, we can support looking up values
   in either fixed or variable tables (where the latter includes advice columns).
 - $A$ and $S$ can contain duplicates. If the sets represented by $A$ and/or $S$ are not
-  naturally of size $2^k$, we extend $S$ with duplicates and $A$ with dummy values known
-  to be in $S$.
+  naturally of size $2^k,$ we extend $S$ with duplicates and $A$ with dummy values known
+  to be in $S.$
   - Alternatively we could add a "lookup selector" that controls which elements of the $A$
     column participate in lookups. This would modify the occurrence of $A(X)$ in the
     permutation rule below to replace $A$ with, say, $S_0$ if a lookup is not selected.
 
-Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i$, and $0$
+Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i,$ and $0$
 otherwise.
 
-We start by allowing the prover to supply permutation columns of $A$ and $S$. Let's call
-these $A'$ and $S'$, respectively. We can enforce that they are permutations using a
+We start by allowing the prover to supply permutation columns of $A$ and $S.$ Let's call
+these $A'$ and $S',$ respectively. We can enforce that they are permutations using a
 permutation argument with product column $Z$ with the rules:
 
 $$
@@ -53,7 +53,7 @@ Z_{2^k} = Z_0 = 1.
 $$
 
 This is a version of the permutation argument which allows $A'$ and $S'$ to be
-permutations of $A$ and $S$, respectively, but doesn't specify the exact permutations.
+permutations of $A$ and $S,$ respectively, but doesn't specify the exact permutations.
 $\beta$ and $\gamma$ are separate challenges so that we can combine these two permutation
 arguments into one without worrying that they might interfere with each other.
 
@@ -62,13 +62,13 @@ particular way:
 
 1. All the cells of column $A'$ are arranged so that like-valued cells are vertically
    adjacent to each other. This could be done by some kind of sorting algorithm, but all
-   that matters is that like-valued cells are on consecutive rows in column $A'$, and that
-   $A'$ is a permutation of $A$.
+   that matters is that like-valued cells are on consecutive rows in column $A',$ and that
+   $A'$ is a permutation of $A.$
 2. The first row in a sequence of like values in $A'$ is the row that has the
    corresponding value in $S'.$ Apart from this constraint, $S'$ is any arbitrary
-   permutation of $S$.
+   permutation of $S.$
 
-Now, we'll enforce that either $A'_i = S'_i$ or that $A'_i = A'_{i-1}$, using the rule
+Now, we'll enforce that either $A'_i = S'_i$ or that $A'_i = A'_{i-1},$ using the rule
 
 $$
 (A'(X) - S'(X)) \cdot (A'(X) - A'(\omega^{-1} X)) = 0
@@ -140,9 +140,9 @@ soundness is not affected.
 
 ## Cost
 
-* There is the original column $A$ and the fixed column $S$.
-* There is a permutation product column $Z$.
-* There are the two permutations $A'$ and $S'$.
+* There is the original column $A$ and the fixed column $S.$
+* There is a permutation product column $Z.$
+* There are the two permutations $A'$ and $S'.$
 * The gates are all of low degree.
 
 ## Generalizations
@@ -161,7 +161,7 @@ ways:
 - Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a
   subset argument, i.e. to constrain $\mathcal{R}(x, y, ...)$ in each row, consider
   $\mathcal{R}$ as a set of tuples $S$ (using the method of the previous point), and check
-  that $(x, y, ...) \in \mathcal{R}$.
+  that $(x, y, ...) \in \mathcal{R}.$
   - In the case where $\mathcal{R}$ represents a function, this implicitly also checks
     that the inputs are in the domain. This is typically what we want, and often saves an
     additional range check.