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<tr><th>Doc. no.:</th>    <td>P0608R3</td></tr>
<tr><th>Date:</th>    <td>2018-10-03</td></tr>
<tr><th>Audience:</th>    <td>LWG</td></tr>
<tr><th>Reply-to:</th>    <td>Zhihao Yuan &lt;zy at miator dot net&gt;</td></tr>
</tbody></table><h1 id="A-sane-variant-converting-constructor" style=""><a class="anchor hidden-xs" href="#A-sane-variant-converting-constructor" title="A-sane-variant-converting-constructor"><span class="octicon octicon-link"></span></a>A sane variant converting constructor</h1><h2 id="Changes-since-R1" style=""><a class="anchor hidden-xs" href="#Changes-since-R1" title="Changes-since-R1"><span class="octicon octicon-link"></span></a>Changes since R1</h2><ul>
<li>Discuss whether to add special cases for unambiguous conversions.</li>
<li>Take the user-defined types that trigger standard conversions into account.</li>
<li>Remove an unhelpful example.</li>
</ul><h2 id="Problems-to-solve" style=""><a class="anchor hidden-xs" href="#Problems-to-solve" title="Problems-to-solve"><span class="octicon octicon-link"></span></a>Problems to solve</h2><ol>
<li><code>variant</code> constructs entirely unintended alternatives.</li>
</ol><pre><code class="c++ hljs">variant&lt;<span class="hljs-built_in">string</span>, <span class="hljs-keyword">bool</span>&gt; x = <span class="hljs-string">"abc"</span>;             <span class="hljs-comment">// holds bool</span>
</code></pre><p>The above holds <code>string</code> with the proposed fix.</p><ol start="2">
<li><code>variant</code> prefers constructions with information losses.</li>
</ol><pre><code class="c++ hljs">variant&lt;char, optional&lt;char16_t&gt;&gt; x = <span class="hljs-string">u'\u2043'</span>;  // holds char = <span class="hljs-string">'C'</span>
double d = <span class="hljs-number">3.14</span>;
variant&lt;int, reference_wrapper&lt;double&gt;&gt; y = d;    // holds int = <span class="hljs-number">3</span>
</code></pre><p>The above preserves the input value in <code>optional&lt;char16_t&gt;</code> and <code>reference_wrapper&lt;double&gt;</code>, respectively, with the proposed fix.</p><ol start="3">
<li><code>variant</code> performs unstable constructions.</li>
</ol><pre><code class="c++ hljs"><span class="hljs-keyword">using</span> T = variant&lt;<span class="hljs-keyword">float</span>, <span class="hljs-keyword">int</span>&gt;;
T v;
v = <span class="hljs-number">0</span>;    <span class="hljs-comment">// switches to int</span>
</code></pre><ul>
<li>When <code>T</code> is upgraded to <code>variant&lt;float, long&gt;</code>,</li>
</ul><pre><code class="c++ hljs"><span class="hljs-keyword">using</span> T = variant&lt;<span class="hljs-keyword">float</span>, <span class="hljs-keyword">long</span>&gt;;
T v;
v = <span class="hljs-number">0</span>;    <span class="hljs-comment">// error</span>
</code></pre><ul>
<li>When <code>T</code> is upgraded to <code>variant&lt;float, big_int&lt;256&gt;&gt;</code>,</li>
</ul><pre><code class="c++ hljs"><span class="hljs-keyword">using</span> T = variant&lt;<span class="hljs-keyword">float</span>, big_int&lt;<span class="hljs-number">256</span>&gt;&gt;;
T v;
v = <span class="hljs-number">0</span>;    <span class="hljs-comment">// holds 0.f</span>
</code></pre><p>In both cases, the proposed fix consistently constructs with the second alternative.</p><p>As shown, the problems equally apply to the converting constructor and the converting assignment operator.</p><p>See also <a href="https://issues.isocpp.org/show_bug.cgi?id=227" target="_blank">LEWG 227</a>.</p><h2 id="Proposed-resolution" style=""><a class="anchor hidden-xs" href="#Proposed-resolution" title="Proposed-resolution"><span class="octicon octicon-link"></span></a>Proposed resolution</h2><p>This paper proposes to constrain the <code>variant</code> converting constructor and the converting assignment operator to prevent narrowing conversions and conversions to <code>bool</code>.  This section explains what exactly this change brings.</p><p><strong>Lemma 1.</strong> Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>X</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 1.026em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.862em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.86em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: MathJax_Math; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">X</script></span> be a sum type <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo>+</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4" style="width: 3.128em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.694em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1002.69em, 2.722em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-5"><span class="mi" id="MathJax-Span-6" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-7" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-8" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.158em; border-left: 0px solid; width: 0px; height: 1.012em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>+</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">T+U</script></span>.  Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>A</mi></mrow><mo>=</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-9" style="width: 10.186em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.782em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.513em, 1008.72em, 2.836em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-10"><span class="texatom" id="MathJax-Span-11"><span class="mrow" id="MathJax-Span-12"><span class="mi" id="MathJax-Span-13" style="font-family: MathJax_Main;">A</span></span></span><span class="mo" id="MathJax-Span-14" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mo" id="MathJax-Span-15" style="font-family: MathJax_Main; padding-left: 0.278em;">{</span><span class="msubsup" id="MathJax-Span-16"><span style="display: inline-block; position: relative; width: 1.013em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-17" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.584em;"><span class="mn" id="MathJax-Span-18" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-19" style="font-family: MathJax_Main;">,</span><span class="msubsup" id="MathJax-Span-20" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.013em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-21" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.584em;"><span class="mn" id="MathJax-Span-22" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-23" style="font-family: MathJax_Main;">,</span><span class="msubsup" id="MathJax-Span-24" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.013em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-25" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.584em;"><span class="mn" id="MathJax-Span-26" style="font-size: 70.7%; font-family: MathJax_Main;">3</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-27" style="font-family: MathJax_Main;">,</span><span class="mo" id="MathJax-Span-28" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-29" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-30" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-31" style="font-family: MathJax_Main; padding-left: 0.167em;">}</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-3">{\rm A}=\{T_1, T_2, T_3, ...\}</script></span> be a set of types that are convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-32" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-33"><span class="mi" id="MathJax-Span-34" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">T</script></span>, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>B</mi></mrow><mo>=</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><msub><mi>U</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-35" style="width: 10.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.052em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.513em, 1008.99em, 2.836em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-36"><span class="texatom" id="MathJax-Span-37"><span class="mrow" id="MathJax-Span-38"><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Main;">B</span></span></span><span class="mo" id="MathJax-Span-40" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mo" id="MathJax-Span-41" style="font-family: MathJax_Main; padding-left: 0.278em;">{</span><span class="msubsup" id="MathJax-Span-42"><span style="display: inline-block; position: relative; width: 1.112em; height: 0px;"><span style="position: absolute; clip: rect(3.142em, 1000.77em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-43" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.683em;"><span class="mn" id="MathJax-Span-44" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-45" style="font-family: MathJax_Main;">,</span><span class="msubsup" id="MathJax-Span-46" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.112em; height: 0px;"><span style="position: absolute; clip: rect(3.142em, 1000.77em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.683em;"><span class="mn" id="MathJax-Span-48" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-49" style="font-family: MathJax_Main;">,</span><span class="msubsup" id="MathJax-Span-50" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.112em; height: 0px;"><span style="position: absolute; clip: rect(3.142em, 1000.77em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.683em;"><span class="mn" id="MathJax-Span-52" style="font-size: 70.7%; font-family: MathJax_Main;">3</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-53" style="font-family: MathJax_Main;">,</span><span class="mo" id="MathJax-Span-54" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-55" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-56" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-57" style="font-family: MathJax_Main; padding-left: 0.167em;">}</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><msub><mi>U</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-5">{\rm B}=\{U_1, U_2, U_3, ...\}</script></span> be a set of types that are convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-58" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-59"><span class="mi" id="MathJax-Span-60" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-6">U</script></span>.  Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Y</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-61" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.58em, 1000.74em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-62"><span class="texatom" id="MathJax-Span-63"><span class="mrow" id="MathJax-Span-64"><span class="mi" id="MathJax-Span-65" style="font-family: MathJax_Main;">Y</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Y</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-7">{\rm Y}</script></span> be a set of types that are convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>X</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-66" style="width: 1.026em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.862em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.86em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-67"><span class="mi" id="MathJax-Span-68" style="font-family: MathJax_Math; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-8">X</script></span>.  <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Y</mi></mrow><mo>=</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>A</mi></mrow><mo>&amp;#x2296;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>B</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-69" style="width: 5.498em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.741em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.547em, 1004.68em, 2.669em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-70"><span class="texatom" id="MathJax-Span-71"><span class="mrow" id="MathJax-Span-72"><span class="mi" id="MathJax-Span-73" style="font-family: MathJax_Main;">Y</span></span></span><span class="mo" id="MathJax-Span-74" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="texatom" id="MathJax-Span-75" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-76"><span class="mi" id="MathJax-Span-77" style="font-family: MathJax_Main;">A</span></span></span><span class="mo" id="MathJax-Span-78" style="font-family: MathJax_Main; padding-left: 0.222em;">⊖</span><span class="texatom" id="MathJax-Span-79" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-80"><span class="mi" id="MathJax-Span-81" style="font-family: MathJax_Main;">B</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.159em; border-left: 0px solid; width: 0px; height: 1.052em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Y</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>⊖</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-9">{\rm Y}={\rm A}\ominus {\rm B}</script></span> (symmetric difference) rather than <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>A</mi></mrow><mo>&amp;#x222A;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>B</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-82" style="width: 3.02em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.586em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.547em, 1002.53em, 2.608em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-83"><span class="texatom" id="MathJax-Span-84"><span class="mrow" id="MathJax-Span-85"><span class="mi" id="MathJax-Span-86" style="font-family: MathJax_Main;">A</span></span></span><span class="mo" id="MathJax-Span-87" style="font-family: MathJax_Main; padding-left: 0.222em;">∪</span><span class="texatom" id="MathJax-Span-88" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-89"><span class="mi" id="MathJax-Span-90" style="font-family: MathJax_Main;">B</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.981em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>∪</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-10">{\rm A}\cup {\rm B}</script></span>, because each <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msub><mi>T</mi><mi>i</mi></msub><mo>=</mo><msub><mi>U</mi><mi>j</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-91" style="width: 3.828em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.287em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.58em, 1003.29em, 2.88em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-92"><span class="msubsup" id="MathJax-Span-93"><span style="display: inline-block; position: relative; width: 0.903em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-94" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.584em;"><span class="mi" id="MathJax-Span-95" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-96" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-97" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.049em; height: 0px;"><span style="position: absolute; clip: rect(3.142em, 1000.77em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-98" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.683em;"><span class="mi" id="MathJax-Span-99" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">j</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.404em; border-left: 0px solid; width: 0px; height: 1.259em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>i</mi></msub><mo>=</mo><msub><mi>U</mi><mi>j</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-11">T_i=U_j</script></span> causes an ambiguity.</p><p><strong>Theorem 1.</strong> If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>A</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-100" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.547em, 1000.72em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-101"><span class="texatom" id="MathJax-Span-102"><span class="mrow" id="MathJax-Span-103"><span class="mi" id="MathJax-Span-104" style="font-family: MathJax_Main;">A</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.956em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-12">{\rm A}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>B</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-105" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.58em, 1000.64em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-106"><span class="texatom" id="MathJax-Span-107"><span class="mrow" id="MathJax-Span-108"><span class="mi" id="MathJax-Span-109" style="font-family: MathJax_Main;">B</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-13">{\rm B}</script></span> are extended to include a type <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>A</mi></mrow><mo>&amp;#x2229;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>B</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-110" style="width: 5.013em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.31em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.601em, 1004.25em, 2.68em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-111"><span class="mi" id="MathJax-Span-112" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span><span class="mo" id="MathJax-Span-113" style="font-family: MathJax_Main; padding-left: 0.278em;">∈</span><span class="texatom" id="MathJax-Span-114" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-115"><span class="mi" id="MathJax-Span-116" style="font-family: MathJax_Main;">A</span></span></span><span class="mo" id="MathJax-Span-117" style="font-family: MathJax_Main; padding-left: 0.222em;">∩</span><span class="texatom" id="MathJax-Span-118" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-119"><span class="mi" id="MathJax-Span-120" style="font-family: MathJax_Main;">B</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.109em; border-left: 0px solid; width: 0px; height: 1.002em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-14">\tau \in {\rm A}\cap {\rm B}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Y</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-121" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.58em, 1000.74em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-122"><span class="texatom" id="MathJax-Span-123"><span class="mrow" id="MathJax-Span-124"><span class="mi" id="MathJax-Span-125" style="font-family: MathJax_Main;">Y</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Y</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-15">{\rm Y}</script></span> is shrunk.</p><p>In short, constraining <code>variant&lt;</code><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-126" style="width: 2.697em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.317em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1002.24em, 2.65em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-127"><span class="mi" id="MathJax-Span-128" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-129" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-130" style="font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-131" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-132" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.922em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-16">Ts...</script></span><code>&gt;</code> (with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>&amp;gt;</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-133" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.94em, 2.626em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-134"><span class="munderover" id="MathJax-Span-135"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-136"><span class="mi" id="MathJax-Span-137" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-138" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-139" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-140" style="font-family: MathJax_Main; padding-left: 0.278em;">&gt;</span><span class="mn" id="MathJax-Span-141" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.109em; border-left: 0px solid; width: 0px; height: 1.342em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>&gt;</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-17">\overline{Ts}>1</script></span>) converting constructor may enable more types to be convertible to a variant.</p><p><strong>Definition 1.</strong> For type <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-142" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-143"><span class="mi" id="MathJax-Span-144" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-18">T</script></span> that is convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-145" style="width: 1.242em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.504em, 1001.08em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-146"><span class="msup" id="MathJax-Span-147"><span style="display: inline-block; position: relative; width: 1.059em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-148" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-149" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.005em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>′</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-19">T'</script></span>, let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>P</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-150" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.58em, 1000.64em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-151"><span class="texatom" id="MathJax-Span-152"><span class="mrow" id="MathJax-Span-153"><span class="mi" id="MathJax-Span-154" style="font-family: MathJax_Main;">P</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-20">{\rm P}</script></span> be a set of all the possible values for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-155" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-156"><span class="mi" id="MathJax-Span-157" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-21">T</script></span>, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Q</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-158" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1000.7em, 2.779em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-159"><span class="texatom" id="MathJax-Span-160"><span class="mrow" id="MathJax-Span-161"><span class="mi" id="MathJax-Span-162" style="font-family: MathJax_Main;">Q</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.286em; border-left: 0px solid; width: 0px; height: 1.167em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-22">{\rm Q}</script></span> be a set of all the possible values for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-163" style="width: 1.242em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.504em, 1001.08em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-164"><span class="msup" id="MathJax-Span-165"><span style="display: inline-block; position: relative; width: 1.059em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-166" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-167" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.005em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>′</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-23">T'</script></span>.  If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>P</mi></mrow><mo>&amp;#x2286;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Q</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-168" style="width: 3.289em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.802em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1002.75em, 2.779em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-169"><span class="texatom" id="MathJax-Span-170"><span class="mrow" id="MathJax-Span-171"><span class="mi" id="MathJax-Span-172" style="font-family: MathJax_Main;">P</span></span></span><span class="mo" id="MathJax-Span-173" style="font-family: MathJax_Main; padding-left: 0.278em;">⊆</span><span class="texatom" id="MathJax-Span-174" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-175"><span class="mi" id="MathJax-Span-176" style="font-family: MathJax_Main;">Q</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.286em; border-left: 0px solid; width: 0px; height: 1.167em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow><mo>⊆</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-24">{\rm P}\subseteq{\rm Q}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-177" style="width: 1.242em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.504em, 1001.08em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-178"><span class="msup" id="MathJax-Span-179"><span style="display: inline-block; position: relative; width: 1.059em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-180" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-181" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.005em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>′</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-25">T'</script></span> is denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>+</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-182" style="width: 1.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.401em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.488em, 1001.4em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-183"><span class="msubsup" id="MathJax-Span-184"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-185" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-186" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-26">T^+</script></span>.  Otherwise, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>P</mi></mrow><mo>&amp;#x2288;</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mi mathvariant=&quot;normal&quot;>Q</mi></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-187" style="width: 3.289em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.802em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.677em, 1002.75em, 3.105em, -1000em); top: -2.64em; left: 0em;"><span class="mrow" id="MathJax-Span-188"><span class="texatom" id="MathJax-Span-189"><span class="mrow" id="MathJax-Span-190"><span class="mi" id="MathJax-Span-191" style="font-family: MathJax_Main;">P</span></span></span><span class="mo" id="MathJax-Span-192" style="font-family: MathJax_AMS; padding-left: 0.278em;">⊈</span><span class="texatom" id="MathJax-Span-193" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-194"><span class="mi" id="MathJax-Span-195" style="font-family: MathJax_Main;">Q</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.64em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.414em; border-left: 0px solid; width: 0px; height: 1.406em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow><mo>⊈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-27">{\rm P}\nsubseteq{\rm Q}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-196" style="width: 1.242em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.504em, 1001.08em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-197"><span class="msup" id="MathJax-Span-198"><span style="display: inline-block; position: relative; width: 1.059em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-199" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-200" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.005em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>′</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-28">T'</script></span> is denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-201" style="width: 1.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.401em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.488em, 1001.4em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-202"><span class="msubsup" id="MathJax-Span-203"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-204" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-205" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-29">T^-</script></span>.  The conversion from <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-206" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-207"><span class="mi" id="MathJax-Span-208" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-30">T</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-209" style="width: 1.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.401em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.488em, 1001.4em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-210"><span class="msubsup" id="MathJax-Span-211"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-212" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-213" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-31">T^-</script></span> (denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-214" style="width: 4.259em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.664em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1003.66em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-215"><span class="mi" id="MathJax-Span-216" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-217" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="msubsup" id="MathJax-Span-218" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-219" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-220" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-32">T\rightharpoonup T^-</script></span>) is a <em>potentially unrepresentable</em> conversion.</p><p>In this paper, narrowing conversions (considering only the types) and boolean conversions assemble the <em>potentially unrepresentable</em> conversions in C++.</p><p><strong>Lemma 2.</strong> <em>Potentially unrepresentable</em> conversions in C++ have Conversion rank.</p><p>The proof is left as an exercise for the reader.</p><p>Let <code>X</code> be <code>variant&lt;</code><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-221" style="width: 2.697em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.317em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1002.24em, 2.65em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-222"><span class="mi" id="MathJax-Span-223" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-224" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-225" style="font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-226" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-227" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.922em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-33">Ts...</script></span><code>&gt;</code>, <code>r</code> be a value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-228" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-229"><span class="mi" id="MathJax-Span-230" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">T</script></span>.</p><p>When <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-231" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.94em, 2.596em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-232"><span class="munderover" id="MathJax-Span-233"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-234"><span class="mi" id="MathJax-Span-235" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-236" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-237" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-238" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-239" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.308em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-35">\overline{Ts}=1</script></span>, without loss of generality, <code>X</code> is <code>variant&lt;</code><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-240" style="width: 1.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.401em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.488em, 1001.4em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-241"><span class="msubsup" id="MathJax-Span-242"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-243" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-244" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-36">T^-</script></span><code>&gt;</code>.  Effects of the proposed resolution can be summarized as follows:</p><table>
<thead>
<tr>
<th><code>X v = r;</code></th>
<th>before</th>
<th>after</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>variant&lt;float&gt; v = 0;</code></td>
<td>holds <code>.0f</code></td>
<td>ill-formed</td>
</tr>
<tr>
<td><code>variant&lt;float&gt; v = INT_MAX;</code></td>
<td>holds <code>INT_MAX + 1</code></td>
<td>ill-formed</td>
</tr>
</tbody>
</table><p>However, <code>variant&lt;V&gt;</code> is such a rare variant, as you can hardly say that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>V</mi><mo>+</mo><mi mathvariant=&quot;normal&quot;>&amp;#x22A5;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-245" style="width: 3.182em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.748em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1002.69em, 2.722em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-246"><span class="mi" id="MathJax-Span-247" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.186em;"></span></span><span class="mo" id="MathJax-Span-248" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-249" style="font-family: MathJax_Main; padding-left: 0.222em;">⊥</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.158em; border-left: 0px solid; width: 0px; height: 1.012em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>+</mo><mi mathvariant="normal">⊥</mi></math></span></span><script type="math/tex" id="MathJax-Element-37">V+\bot</script></span> is a sum type.</p><p>When <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>=</mo><mn>2</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-250" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.97em, 2.596em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-251"><span class="munderover" id="MathJax-Span-252"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-253"><span class="mi" id="MathJax-Span-254" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-255" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-256" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-257" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-258" style="font-family: MathJax_Main; padding-left: 0.278em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.308em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-38">\overline{Ts}=2</script></span>, <code>X</code> is <code>variant&lt;</code><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo>,</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-259" style="width: 3.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.64em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1002.64em, 2.834em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-260"><span class="msubsup" id="MathJax-Span-261"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-262" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-263" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-264" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-265" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.288em; border-left: 0px solid; width: 0px; height: 1.249em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup><mo>,</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-39">T^-,U</script></span><code>&gt;</code>.</p><ul>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-266" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-267"><span class="mi" id="MathJax-Span-268" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-40">T</script></span> is not convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-269" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-270"><span class="mi" id="MathJax-Span-271" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-41">U</script></span>, the effects are the same as the case <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-272" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.94em, 2.596em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-273"><span class="munderover" id="MathJax-Span-274"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-275"><span class="mi" id="MathJax-Span-276" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-277" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-278" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-279" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-280" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.308em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-42">\overline{Ts}=1</script></span>.</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-281" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-282"><span class="mi" id="MathJax-Span-283" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-284" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-285" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-43">T\rightharpoonup U</script></span> is an Exact Match or a Promotion, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-286" style="width: 4.259em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.664em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1003.66em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-287"><span class="mi" id="MathJax-Span-288" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-289" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="msubsup" id="MathJax-Span-290" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-291" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-292" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-44">T\rightharpoonup T^-</script></span> has a worse conversion sequence, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-293" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-294"><span class="mi" id="MathJax-Span-295" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-45">U</script></span> is constructed in <code>X v = r</code> under the existing rule.  The proposed resolution constructs <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-296" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-297"><span class="mi" id="MathJax-Span-298" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-46">U</script></span> as well because <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-299" style="width: 4.259em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.664em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1003.66em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-300"><span class="mi" id="MathJax-Span-301" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-302" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="msubsup" id="MathJax-Span-303" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-304" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-305" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-47">T\rightharpoonup T^-</script></span> is not viable.</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-306" style="width: 0.811em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1000.7em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-307"><span class="mi" id="MathJax-Span-308" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.91em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-48">T</script></span> is potentially unrepresentable converted to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-309" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-310"><span class="mi" id="MathJax-Span-311" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-49">U</script></span>, according to Lemma 2, both <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-312" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-313"><span class="mi" id="MathJax-Span-314" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-315" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-316" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-50">T\rightharpoonup U</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-317" style="width: 4.259em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.664em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1003.66em, 2.64em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-318"><span class="mi" id="MathJax-Span-319" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-320" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="msubsup" id="MathJax-Span-321" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-322" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-323" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-51">T\rightharpoonup T^-</script></span> have Conversion rank, so <code>X v = r</code> is ill-formed due to ambiguity under the existing rule.  Meanwhile, it’s also ill-formed given the proposed resolution, because both conversions are potentially unrepresentable.</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-324" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-325"><span class="mi" id="MathJax-Span-326" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-327" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-328" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-52">T\rightharpoonup U</script></span> has Conversion rank and is not potentially unrepresentable, <code>X v = r</code> is still ill-formed under the existing rule.  However, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-329" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-330"><span class="mi" id="MathJax-Span-331" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-53">U</script></span> is constructed given the proposed resolution.</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-332" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-333"><span class="mi" id="MathJax-Span-334" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-335" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-336" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-54">T\rightharpoonup U</script></span> is a user-defined conversion, <code>X v = r</code> constructs <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-337" style="width: 1.619em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.401em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.488em, 1001.4em, 2.586em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-338"><span class="msubsup" id="MathJax-Span-339"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-340" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-341" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.024em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-55">T^-</script></span> under the existing rule.  With the proposed resolution, it constructs <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-342" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-343"><span class="mi" id="MathJax-Span-344" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-56">U</script></span> instead.</li>
</ul><p>The effects are summarized in the order of the bullets:</p><table>
<thead>
<tr>
<th><code>X v = r;</code></th>
<th>before</th>
<th>after</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>variant&lt;float, vector&lt;int&gt;&gt; v = 0;</code></td>
<td>holds <code>float</code></td>
<td>ill-formed</td>
</tr>
<tr>
<td><code>variant&lt;float, int&gt; v = 'a';</code></td>
<td>holds <code>int('a')</code></td>
<td>holds <code>int('a')</code></td>
</tr>
<tr>
<td><code>variant&lt;float, char&gt; v = 0;</code></td>
<td>ill-formed</td>
<td>ill-formed</td>
</tr>
<tr>
<td><code>variant&lt;float, long&gt; v = 0;</code></td>
<td>ill-formed</td>
<td>holds <code>long</code></td>
</tr>
<tr>
<td><code>variant&lt;float, big_int&lt;256&gt;&gt; v = 0;</code></td>
<td>holds <code>float</code></td>
<td>holds <code>big_int</code></td>
</tr>
</tbody>
</table><p>When <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>&amp;gt;</mo><mn>2</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-345" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.97em, 2.626em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-346"><span class="munderover" id="MathJax-Span-347"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-348"><span class="mi" id="MathJax-Span-349" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-350" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-351" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-352" style="font-family: MathJax_Main; padding-left: 0.278em;">&gt;</span><span class="mn" id="MathJax-Span-353" style="font-family: MathJax_Main; padding-left: 0.278em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.109em; border-left: 0px solid; width: 0px; height: 1.342em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>&gt;</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-57">\overline{Ts}>2</script></span>, let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><msub><mi>s</mi><mn>1</mn></msub><mo>=</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-354" style="width: 5.067em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.364em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1004.36em, 2.79em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-355"><span class="mi" id="MathJax-Span-356" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="msubsup" id="MathJax-Span-357"><span style="display: inline-block; position: relative; width: 0.898em; height: 0px;"><span style="position: absolute; clip: rect(3.383em, 1000.42em, 4.159em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-358" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.837em; left: 0.469em;"><span class="mn" id="MathJax-Span-359" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-360" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-361" style="padding-left: 0.278em;"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-362" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-363" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.236em; border-left: 0px solid; width: 0px; height: 1.198em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><msub><mi>s</mi><mn>1</mn></msub><mo>=</mo><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-58">Ts_1=T^-</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>S</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-364" style="width: 0.757em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.647em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.612em, 1000.65em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-365"><span class="mi" id="MathJax-Span-366" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.968em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-59">S</script></span> be an overload set <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><mi>&amp;#x03C4;</mi><mo stretchy=&quot;false&quot;>)</mo><mo>&amp;#x2223;</mo><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-367" style="width: 7.654em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.573em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1006.51em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-368"><span class="mo" id="MathJax-Span-369" style="font-family: MathJax_Main;">{</span><span class="mi" id="MathJax-Span-370" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-371" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-372" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span><span class="mo" id="MathJax-Span-373" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-374" style="font-family: MathJax_Main; padding-left: 0.278em;">∣</span><span class="mi" id="MathJax-Span-375" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span><span class="mo" id="MathJax-Span-376" style="font-family: MathJax_Main; padding-left: 0.278em;">∈</span><span class="mi" id="MathJax-Span-377" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-378" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-379" style="font-family: MathJax_Main;">}</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>∣</mo><mi>τ</mi><mo>∈</mo><mi>T</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-60">\{f(\tau) \mid \tau \in Ts\}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo>=</mo><mi>S</mi><mo>&amp;#x2212;</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo stretchy=&quot;false&quot;>)</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-380" style="width: 9.216em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.92em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1007.85em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-381"><span class="msup" id="MathJax-Span-382"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.12em, 1000.65em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-383" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.705em;"><span class="mo" id="MathJax-Span-384" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-385" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-386" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span class="mo" id="MathJax-Span-387" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mo" id="MathJax-Span-388" style="font-family: MathJax_Main; padding-left: 0.222em;">{</span><span class="mi" id="MathJax-Span-389" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-390" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-391"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-392" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-393" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-394" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-395" style="font-family: MathJax_Main;">}</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.314em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>′</mo></msup><mo>=</mo><mi>S</mi><mo>−</mo><mo fence="false" stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-61">S'=S-\{f(T^-)\}</script></span>.</p><ul>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>S</mi><mo stretchy=&quot;false&quot;>(</mo><mi>r</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-396" style="width: 2.212em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.886em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1001.79em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-397"><span class="mi" id="MathJax-Span-398" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span class="mo" id="MathJax-Span-399" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-400" style="font-family: MathJax_Math; font-style: italic;">r</span><span class="mo" id="MathJax-Span-401" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-62">S(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-402" style="width: 3.182em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.748em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.542em, 1002.65em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-403"><span class="mi" id="MathJax-Span-404" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-405" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-406"><span style="display: inline-block; position: relative; width: 1.415em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1000.7em, 4.149em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-407" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.79em;"><span class="mo" id="MathJax-Span-408" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-409" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.314em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>−</mo></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-63">f(T^-)</script></span>, this is equivalent to the case <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-410" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.94em, 2.596em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-411"><span class="munderover" id="MathJax-Span-412"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-413"><span class="mi" id="MathJax-Span-414" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-415" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-416" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-417" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-418" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.308em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-64">\overline{Ts}=1</script></span>; or</li>
<li>if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-65-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mi>r</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-419" style="width: 2.589em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.209em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1002.11em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-420"><span class="msup" id="MathJax-Span-421"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.12em, 1000.65em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-422" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.705em;"><span class="mo" id="MathJax-Span-423" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-424" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-425" style="font-family: MathJax_Math; font-style: italic;">r</span><span class="mo" id="MathJax-Span-426" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.295em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-65">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><mi>U</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-427" style="width: 2.481em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.101em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1002.01em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-428"><span class="mi" id="MathJax-Span-429" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-430" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-431" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span class="mo" id="MathJax-Span-432" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-66">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-67-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-433" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-434"><span class="mi" id="MathJax-Span-435" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-436" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-437" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-67">T\rightharpoonup U</script></span> has Conversion rank, <code>X v = r</code> is ill-formed without the proposed resolution; or</li>
<li>if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-68-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mi>r</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-438" style="width: 2.589em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.209em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1002.11em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-439"><span class="msup" id="MathJax-Span-440"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.12em, 1000.65em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-441" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.705em;"><span class="mo" id="MathJax-Span-442" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-443" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-444" style="font-family: MathJax_Math; font-style: italic;">r</span><span class="mo" id="MathJax-Span-445" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.295em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-68">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-69-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><mi>U</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-446" style="width: 2.481em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.101em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1002.01em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-447"><span class="mi" id="MathJax-Span-448" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-449" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-450" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span class="mo" id="MathJax-Span-451" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-69">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-70-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-452" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-453"><span class="mi" id="MathJax-Span-454" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-455" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-456" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-70">T\rightharpoonup U</script></span> is a user-defined conversion, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-71-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo>&amp;#x2212;</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><mi>U</mi><mo stretchy=&quot;false&quot;>)</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-457" style="width: 6.145em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.28em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1005.21em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-458"><span class="msup" id="MathJax-Span-459"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.12em, 1000.65em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-460" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.705em;"><span class="mo" id="MathJax-Span-461" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-462" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mo" id="MathJax-Span-463" style="font-family: MathJax_Main; padding-left: 0.222em;">{</span><span class="mi" id="MathJax-Span-464" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-465" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-466" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span class="mo" id="MathJax-Span-467" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-468" style="font-family: MathJax_Main;">}</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.295em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>′</mo></msup><mo>−</mo><mo fence="false" stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-71">S'-\{f(U)\}</script></span> has no viable function against  <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-72-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>r</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-469" style="width: 0.541em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.431em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.875em, 1000.41em, 2.651em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-470"><span class="mi" id="MathJax-Span-471" style="font-family: MathJax_Math; font-style: italic;">r</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.075em; border-left: 0px solid; width: 0px; height: 0.65em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-72">r</script></span>, and the situation is equivalent to the last bullet in the case <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-73-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent=&quot;false&quot;>&amp;#x00AF;</mo></mover><mo>=</mo><mn>2</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-472" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.253em, 1002.97em, 2.596em, -1000em); top: -2.425em; left: 0em;"><span class="mrow" id="MathJax-Span-473"><span class="munderover" id="MathJax-Span-474"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(3.148em, 1001.12em, 4.159em, -1000em); top: -3.987em; left: 0.013em;"><span class="mrow" id="MathJax-Span-475"><span class="mi" id="MathJax-Span-476" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-477" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.308em, 1001.2em, 3.764em, -1000em); top: -4.479em; left: 0em;"><span class="mo" id="MathJax-Span-478" style=""><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; top: -3.987em; left: -0.049em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.894em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.12em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.507em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -3.987em; left: 0.7em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-479" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-480" style="font-family: MathJax_Main; padding-left: 0.278em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.425em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.308em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-73">\overline{Ts}=2</script></span>.</li>
<li>Otherwise, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-74-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mi>r</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-481" style="width: 2.589em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.209em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.558em, 1002.11em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-482"><span class="msup" id="MathJax-Span-483"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.12em, 1000.65em, 4.171em, -1000em); top: -3.987em; left: 0em;"><span class="mi" id="MathJax-Span-484" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; top: -4.35em; left: 0.705em;"><span class="mo" id="MathJax-Span-485" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span class="mo" id="MathJax-Span-486" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-487" style="font-family: MathJax_Math; font-style: italic;">r</span><span class="mo" id="MathJax-Span-488" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.295em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-74">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-75-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>f</mi><mo stretchy=&quot;false&quot;>(</mo><mi>U</mi><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-489" style="width: 2.481em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.101em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1002.01em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-490"><span class="mi" id="MathJax-Span-491" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-492" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-493" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span class="mo" id="MathJax-Span-494" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-75">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-76-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-495" style="width: 3.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.017em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.02em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-496"><span class="mi" id="MathJax-Span-497" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mo" id="MathJax-Span-498" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-499" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-76">T\rightharpoonup U</script></span> is an Exact Match or a Promotion, <code>X v = r</code> constructs <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-77-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-500" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-501"><span class="mi" id="MathJax-Span-502" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-77">U</script></span> with or without the proposed resolution.</li>
</ul><p>The effects are summarized in the order of the bullets:</p><table>
<thead>
<tr>
<th><code>X v = r;</code></th>
<th>before</th>
<th>after</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>variant&lt;float, big_int&lt;256&gt;, big_int&lt;128&gt;&gt; v = 0;</code></td>
<td>holds <code>float</code></td>
<td>ill-formed</td>
</tr>
<tr>
<td><code>variant&lt;float, long, double&gt; v = 0;</code></td>
<td>ill-formed</td>
<td>holds <code>long</code></td>
</tr>
<tr>
<td><code>variant&lt;float, vector&lt;int&gt;, big_int&lt;256&gt;&gt; v = 0;</code></td>
<td>holds <code>float</code></td>
<td>holds <code>big_int</code></td>
</tr>
<tr>
<td><code>variant&lt;float, int, big_int&lt;256&gt;&gt; v = 'a';</code></td>
<td>holds <code>int</code></td>
<td>holds <code>int</code></td>
</tr>
</tbody>
</table><p><strong>Theorem 2.</strong> For <code>variant&lt;</code><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-78-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-503" style="width: 2.697em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.317em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1002.24em, 2.65em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-504"><span class="mi" id="MathJax-Span-505" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-506" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-507" style="font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-508" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span><span class="mo" id="MathJax-Span-509" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.922em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-78">Ts...</script></span><code>&gt; v = r</code>, where <code>r</code> is a value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-79-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>R</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-510" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.661em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-511"><span class="mi" id="MathJax-Span-512" style="font-family: MathJax_Math; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.087em; border-left: 0px solid; width: 0px; height: 0.942em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-79">R</script></span>, when there exists one and only one <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-80-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-513" style="width: 3.397em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.909em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.64em, 1002.86em, 2.68em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-514"><span class="mi" id="MathJax-Span-515" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span><span class="mo" id="MathJax-Span-516" style="font-family: MathJax_Main; padding-left: 0.278em;">∈</span><span class="mi" id="MathJax-Span-517" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-518" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.109em; border-left: 0px solid; width: 0px; height: 0.957em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo>∈</mo><mi>T</mi><mi>s</mi></math></span></span><script type="math/tex" id="MathJax-Element-80">\tau \in Ts</script></span> rendering <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-81-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>R</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-519" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.661em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-520"><span class="mi" id="MathJax-Span-521" style="font-family: MathJax_Math; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.087em; border-left: 0px solid; width: 0px; height: 0.942em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-81">R</script></span> to be potentially unrepresentable converted to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-82-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>&amp;#x03C4;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-522" style="width: 0.649em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.886em, 1000.54em, 2.653em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-523"><span class="mi" id="MathJax-Span-524" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.078em; border-left: 0px solid; width: 0px; height: 0.64em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math></span></span><script type="math/tex" id="MathJax-Element-82">\tau</script></span>, the proposed resolution may cause breaking changes</p><ul>
<li>by preventing the initialization if it used to construct <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-83-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>&amp;#x03C4;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-525" style="width: 0.649em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.886em, 1000.54em, 2.653em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-526"><span class="mi" id="MathJax-Span-527" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.078em; border-left: 0px solid; width: 0px; height: 0.64em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math></span></span><script type="math/tex" id="MathJax-Element-83">\tau</script></span>, or</li>
<li>by preferring a different and user-defined conversion <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-84-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>R</mi><mo stretchy=&quot;false&quot;>&amp;#x21C0;</mo><mi>U</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-528" style="width: 3.559em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.071em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1003.07em, 2.662em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-529"><span class="mi" id="MathJax-Span-530" style="font-family: MathJax_Math; font-style: italic;">R</span><span class="mo" id="MathJax-Span-531" style="font-family: MathJax_Main; padding-left: 0.278em;">⇀</span><span class="mi" id="MathJax-Span-532" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.088em; border-left: 0px solid; width: 0px; height: 0.943em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">⇀</mo><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-84">R\rightharpoonup U</script></span> when there is exactly one <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-85-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>U</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi><mo>&amp;#x2212;</mo><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>{</mo><mi>&amp;#x03C4;</mi><mo fence=&quot;false&quot; stretchy=&quot;false&quot;>}</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-533" style="width: 6.899em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.927em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.567em, 1005.86em, 2.89em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-534"><span class="mi" id="MathJax-Span-535" style="font-family: MathJax_Math; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span class="mo" id="MathJax-Span-536" style="font-family: MathJax_Main; padding-left: 0.278em;">∈</span><span class="mi" id="MathJax-Span-537" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span class="mi" id="MathJax-Span-538" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-539" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mo" id="MathJax-Span-540" style="font-family: MathJax_Main; padding-left: 0.222em;">{</span><span class="mi" id="MathJax-Span-541" style="font-family: MathJax_Math; font-style: italic;">τ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.08em;"></span></span><span class="mo" id="MathJax-Span-542" style="font-family: MathJax_Main;">}</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.352em; border-left: 0px solid; width: 0px; height: 1.285em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>∈</mo><mi>T</mi><mi>s</mi><mo>−</mo><mo fence="false" stretchy="false">{</mo><mi>τ</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-85">U\in Ts-\{\tau\}</script></span> that is convertible from <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-86-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>R</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-543" style="width: 0.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.634em, 1000.75em, 2.661em, -1000em); top: -2.478em; left: 0em;"><span class="mrow" id="MathJax-Span-544"><span class="mi" id="MathJax-Span-545" style="font-family: MathJax_Math; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.087em; border-left: 0px solid; width: 0px; height: 0.942em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-86">R</script></span>.</li>
</ul><p>The first case is easy to fix while the second gives the desired outcome for this paper.  Both behaviors to be changed are bugs, not features, as shown in <a href="#problems-to-solve">Section 1</a>.</p><p>However, the definition of <em>potentially unrepresentable</em> conversions here makes it unable to be fully detected in the library, because while we understand type conversions as mappings, C++ type conversions happen in conversion sequences.  A boolean conversion in a standard conversion sequence that follows a user-defined conversion sequence cannot be detected.  In short, we cannot distinguish</p><pre><code class="c++ hljs"><span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Bad</span> {</span> <span class="hljs-keyword">operator</span> <span class="hljs-keyword">char</span> <span class="hljs-keyword">const</span>*(); };
</code></pre><p>from</p><pre><code class="c++ hljs"><span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">LessBad</span> {</span> <span class="hljs-function"><span class="hljs-keyword">operator</span> <span class="hljs-title">bool</span><span class="hljs-params">()</span></span>; };
</code></pre><p>The proposed resolution considers both to be bad.  Throughout the library, only two types are affected: <code>true_type</code> and <code>false_type</code>.  The most commonly seen cases, types that are contextually converted to <code>bool</code>, are not affected by this compromise, because <code>std::variant</code> is not contextual.  The compromise also does not break our proof, because user-defined conversions’ rank is lower than that of narrowing conversions.</p><h2 id="Alternative-designs" style=""><a class="anchor hidden-xs" href="#Alternative-designs" title="Alternative-designs"><span class="octicon octicon-link"></span></a>Alternative designs</h2><p>The author came up with and experimented a few other designs, here we list two basic ideas.</p><ol>
<li>
<p>Use the alternatives’ order information in determining which one to construct.  The idea defeats the purpose of the converting constructor because if the construction is sensitive to the order of the alternatives declared in the <code>variant</code> template argument list, <code>in_place_index</code> would be a better choice.  The converting constructor and assignment operator assume unordered alternatives.</p>
</li>
<li>
<p>Distinguish implicit and explicit conversions.  First, the idea doesn’t work well with the converting assignment operator; applying the implicit policy seems to be the only choice to maintain a consistent behavior, but this may be overkill.  Second, it is counterintuitive to have an explicit constructor accepting fewer types comparing to an implicit one because of <a href="#proposed-resolution">Theorem 1</a>.</p>
</li>
</ol><p>An additional suggestion is to allow narrowing but unambiguous conversions.  In the following example, if <code>double_double</code> is a user-defined type that does not convert from an integer type,</p><pre><code class="c++ hljs">variant&lt;<span class="hljs-keyword">char</span>, double_double&gt; v = <span class="hljs-number">42</span>;
</code></pre><p>, it might be not so problematic to construct the <code>char</code> alternative.  However, special casing this makes the construction less stable.  If <code>double_double</code> is upgraded to support converting from <code>int</code>, the code silently constructs the <code>double_double</code> alternative.</p><h2 id="Wording" style=""><a class="anchor hidden-xs" href="#Wording" title="Wording"><span class="octicon octicon-link"></span></a>Wording</h2><p>This wording is relative to N4762.</p><p>Modify 19.7.3.1 [variant.ctor]/12 as indicated:</p><pre><code>template&lt;class T&gt; constexpr variant(T&amp;&amp; t) noexcept(see below );
</code></pre><blockquote>
<p>Let <code>T</code><i><sub>j</sub></i> be a type that is determined as follows: build an
imaginary function <em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code> for each alternative
type <code>T</code><i><sub>i</sub></i><ins> for which
<code>T</code><i><sub>i</sub></i> <code>x[] = {std::forward&lt;T&gt;(t)};</code> is well-formed
for some invented variable <code>x</code> and,
if <code>T</code><i><sub>i</sub></i> is <em>cv</em> <code>bool</code>,
<code>remove_cvref_t&lt;T&gt;</code> is <code>bool</code></ins>. The overload
<em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code> selected by overload resolution for the
expression <em><code>FUN</code></em><code>(std::forward&lt;T&gt;(t))</code> defines the
alternative <code>T</code><i><sub>j</sub></i> which is the type of the contained value
after construction.</p>
</blockquote><blockquote>
<p>[…]</p>
</blockquote><div class="alert alert-warning">
<p><em>[Drafting note:</em>
The above uses <code>T</code><i><sub>i</sub></i> <code>x[] = {std::forward&lt;T&gt;(t)};</code> rather than <code>T</code><i><sub>i</sub></i><code>{std::forward&lt;T&gt;(t)}</code> to test whether the conversion sequence from <code>T</code> to <code>T</code><i><sub>i</sub></i> contains a narrowing conversion because whether <code>T</code><i><sub>i</sub></i><code>{std::forward&lt;T&gt;(t)}</code> is well-formed may subject to whether <code>T</code><i><sub>i</sub></i> has an <code>initializer_list</code> constructor or whether <code>T</code><i><sub>i</sub></i> is an aggregate.  Here is an example: <a href="https://godbolt.org/z/Ck5w-L" target="_blank">https://godbolt.org/z/Ck5w-L</a>
<em>–end note]</em></p>
</div><p>Modify 19.7.3.3 [variant.assign]/10 as indicated:</p><pre><code>template&lt;class T&gt; variant&amp; operator=(T&amp;&amp; t) noexcept(see below );
</code></pre><blockquote>
<p>Let <code>T</code><i><sub>j</sub></i> be a type that is determined as follows: build an
imaginary function <em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code> for each alternative
type <code>T</code><i><sub>i</sub></i><ins> for which
<code>T</code><i><sub>i</sub></i> <code>x[] = {std::forward&lt;T&gt;(t)};</code> is well-formed
for some invented variable <code>x</code> and,
if <code>T</code><i><sub>i</sub></i> is <em>cv</em> <code>bool</code>,
<code>remove_cvref_t&lt;T&gt;</code> is <code>bool</code></ins>. The overload
<em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code> selected by overload resolution for the
expression <em><code>FUN</code></em><code>(std::forward&lt;T&gt;(t))</code> defines the
alternative <code>T</code><i><sub>j</sub></i> which is the type of the contained value
after assignment.</p>
</blockquote><blockquote>
<p>[…]</p>
</blockquote></div>
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