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Co-authored-by: Daira-Emma Hopwood <[email protected]>

src/relation.md

@@ -34,39 +34,44 @@ If the proof system is knowledge sound, then the prover must have knowledge of t
 
 ### Instances
 
-The relation $\mathcal{R}_{\mathsf{plonkish}}$ takes instances of the following form:
-* Global instances
+The relation $\mathcal{R}_{\mathsf{plonkish}}$ takes instances with the following elements:
+
+* Global instance elements
 
 | Instance element | Description                               |
 |------------------|-------------------------------------------|
 | $\mathbb{F}$     | A prime field.                            |
 | $n > 0$          | Number of rows for the witness matrix.    |
 | $m > 0$          | Number of columns for the witness matrix. |
+| $t$              | Length of the instance vector. |
 
-* Instances for [Fixed constraints](#fixed-constraints)
+* The instance vector
+
+| Instance element | Description                               |
+|------------------|-------------------------------------------|
+| $\phi$           | The instance vector $\phi \mathrel{⦂} \mathbb{F}^t$. |
+* Instance elements for [Fixed constraints](#fixed-constraints)
 
 | Instance element | Description |
 |------------------| -------- |
 | $m_f \leq m$     | Number of columns that are fixed. |
 | $f$              | The fixed content of the first $m_f$ columns, $f \mathrel{⦂} \mathbb{F}^{m_f \times n}$. |
 
-* Instances for [Copy constraints](#copy-constraints)
+* Instance elements for [Copy constraints](#copy-constraints)
 
 | Instance element | Description |
 |------------------| -------- |
 | $\equiv$         | An equivalence relation on $[0,m) \times [0,n)$ indicating which witness entries are equal to each other. |
-| $t$              | Length of the instance vector. |
-| $\phi$           | The instance vector $\phi \mathrel{⦂} \mathbb{F}^t$. |
 | $S$              | A set $S \subseteq ([0,m) \times [0,n)) \times [0,t)$ indicating which witness entries are equal to instance vector entries. |
 
-* Instances for [Custom constraints](#custom-constraints)
+* Instance elements for [Custom constraints](#custom-constraints)
 
 | Instance element | Description |
 |------------------| -------- |
 | $p_u$            | Custom multivariate polynomials $p_u \mathrel{⦂} \mathbb{F}^m \rightarrow \mathbb{F}$. |
 | $\mathsf{CUS}_u$ | Sets $\mathsf{CUS}_u \subseteq [0,n)$ indicating rows on which the custom polynomials $p_u$ are constrained to evaluate to 0. |
 
-* Instances for [Lookup constraints](#lookup-constraints)
+* Instance elements for [Lookup constraints](#lookup-constraints)
 
 | Instance element | Description                                                                                                                                   |
 |------------------|-----------------------------------------------------------------------------------------------------------------------------------------------|