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<h2id="committing-to-the-circuit-assignments"><aclass="header"href="#committing-to-the-circuit-assignments">Committing to the circuit assignments</a></h2>
<p>At the start of proof creation, the prover has a table of cell assignments that it claims
satisfy the constraint system. The table has <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.849108em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.849108em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> rows, and is broken into advice,
instance, and fixed columns. We define <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.969438em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.13889em;">F</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.311664em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">i</span><spanclass="mpunct mtight">,</span><spanclass="mord mathnormal mtight"style="margin-right:0.05724em;">j</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> as the assignment in the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.85396em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05724em;">j</span></span></span></span>th row of
the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.65952em;vertical-align:0em;"></span><spanclass="mord mathnormal">i</span></span></span></span>th fixed column. Without loss of generality, we'll similarly define <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.969438em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.311664em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight">i</span><spanclass="mpunct mtight">,</span><spanclass="mord mathnormal mtight"style="margin-right:0.05724em;">j</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> to
<p>To commit to these assignments, we construct Lagrange polynomials of degree <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.66666em;vertical-align:-0.08333em;"></span><spanclass="mord mathnormal">n</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> for
each column, over an evaluation domain of size <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span> (where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">ω</span></span></span></span> is the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">n</span></span></span></span>th primitive
<p><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">F</span></span></span></span> is constructed as part of key generation, using a blinding factor of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>.
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">A</span></span></span></span> is constructed by the prover and sent to the verifier.</p>
<h2id="committing-to-the-lookup-permutations"><aclass="header"href="#committing-to-the-lookup-permutations">Committing to the lookup permutations</a></h2>
<p>The verifier starts by sampling <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.02778em;">θ</span></span></span></span>, which is used to keep individual columns within
lookups independent. Then, the prover commits to the permutations for each lookup as
<p>The prover then permutes <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.036108em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.3361079999999999em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord text mtight"><spanclass="mord mtight">compressed</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.036108em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.3361079999999999em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord text mtight"><spanclass="mord mtight">compressed</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> according
to the <ahref="lookup.html">rules of the lookup argument</a>, obtaining <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span>.</p>
<p>After the verifier receives <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">A</span></span></span></span>, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">F</span></span></span></span>, and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">L</span></span></span></span>, it samples
challenges <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05278em;">β</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05556em;">γ</span></span></span></span> that will be used in the permutation argument and the
remainder of the lookup argument below. (These challenges can be reused because the
arguments are independent.)</p>
<h2id="committing-to-the-equality-constraint-permutation"><aclass="header"href="#committing-to-the-equality-constraint-permutation">Committing to the equality constraint permutation</a></h2>
<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">c</span></span></span></span> be the number of columns that are enabled for equality constraints.</p>
<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">m</span></span></span></span> be the maximum number of columns that can accomodated by a
<ahref="permutation.html#spanning-a-large-number-of-columns">column set</a> without exceeding
the PLONK configuration's polynomial degree bound.</p>
<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">u</span></span></span></span> be the number of “usable” rows as defined in the
<p>The prover constructs a vector <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">P</span></span></span></span> of length <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span><spanclass="mord mathnormal">u</span></span></span></span> such that for each
column set <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.78041em;vertical-align:-0.13597em;"></span><spanclass="mord">0</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.5782em;vertical-align:-0.0391em;"></span><spanclass="mord mathnormal">a</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel"><</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span></span></span></span> and each row <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.78041em;vertical-align:-0.13597em;"></span><spanclass="mord">0</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.85396em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05724em;">j</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel"><</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal">u</span><spanclass="mpunct">,</span></span></span></span></p>
<p>The prover then computes a running product of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">P</span></span></span></span>, starting at <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>,
and a vector of polynomials <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.969438em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.3361079999999999em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.13889em;">P</span><spanclass="mpunct mtight">,</span><spanclass="mord mtight">0..</span><spanclass="mord mathnormal mtight">b</span><spanclass="mbin mtight">−</span><spanclass="mord mtight">1</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> that each have a Lagrange basis
representation corresponding to a <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">u</span></span></span></span>-sized slice of this running product, as
described in the <ahref="permutation.html#argument-specification">Permutation argument</a>
section.</p>
<p>The prover creates blinding commitments to each <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.969438em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.328331em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.13889em;">P</span><spanclass="mpunct mtight">,</span><spanclass="mord mathnormal mtight">a</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> polynomial:</p>
<h2id="committing-to-the-lookup-permutation-product-columns"><aclass="header"href="#committing-to-the-lookup-permutation-product-columns">Committing to the lookup permutation product columns</a></h2>
<li>The prover constructs a polynomial <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.83333em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.32833099999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">L</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> which has a Lagrange basis representation
corresponding to a running product of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.13889em;">P</span></span></span></span>, starting at <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.32833099999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">L</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>.</li>
<p><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05278em;">β</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05556em;">γ</span></span></span></span> are used to combine the permutation arguments for <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span>
while keeping them independent. The important thing here is that the verifier samples
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05278em;">β</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05556em;">γ</span></span></span></span> after the prover has created <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">A</span></span></span></span>, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">F</span></span></span></span>, and
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68611em;vertical-align:0em;"></span><spanclass="mord mathbf">L</span></span></span></span> (and thus committed to all the cell values used in lookup columns, as well
as <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.001892em;vertical-align:-0.25em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> for each lookup).</p>