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        A sane variant converting constructor - HackMD
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<tr><th>Doc. no.:</th>    <td>P0608R1</td></tr>
<tr><th>Date:</th>    <td>2017-12-30</td></tr>
<tr><th>Audience:</th>    <td>LEWG, 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"><a title="A-sane-variant-converting-constructor" class="anchor hidden-xs" href="#A-sane-variant-converting-constructor"><span class="octicon octicon-link"></span></a>A sane variant converting constructor</h1><h2 id="Problems-to-solve"><a title="Problems-to-solve" class="anchor hidden-xs" href="#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>
variant&lt;<span class="hljs-keyword">bool</span>, <span class="hljs-built_in">unique_ptr</span>&lt;<span class="hljs-keyword">int</span>&gt;&gt; x = <span class="hljs-literal">nullptr</span>;  <span class="hljs-comment">// holds bool</span>
</code></pre><p>The above holds <code>string</code> and <code>unique_ptr</code>, respectively, 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"><a title="Proposed-resolution" class="anchor hidden-xs" href="#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 boolean conversions.  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 tabindex="0" class="MathJax" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 1.02em; display: inline-block;"><span style="width: 0.86em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.86em, 2.53em, -1000em);"><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="width: 0.02em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-1" type="math/tex">X</script></span> be a sum type <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>+</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4" style="width: 3.12em; display: inline-block;"><span style="width: 2.69em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1002.69em, 2.61em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-7" style="padding-left: 0.22em; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-8" style="padding-left: 0.22em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.01em; overflow: hidden; vertical-align: -0.15em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-2" type="math/tex">T+U</script></span>.  Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" data-mathml='<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>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-9" style="width: 10.13em; display: inline-block;"><span style="width: 8.72em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1008.66em, 2.78em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mo" id="MathJax-Span-15" style="padding-left: 0.27em; font-family: MathJax_Main;">{</span><span class="msubsup" id="MathJax-Span-16"><span style="width: 1.01em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-17" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.58em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-18" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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: 0.16em;"><span style="width: 1.01em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-21" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.58em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-22" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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: 0.16em;"><span style="width: 1.01em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-25" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.58em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-26" style="font-family: MathJax_Main; font-size: 70.7%;">3</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-29" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-30" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-31" style="padding-left: 0.16em; font-family: MathJax_Main;">}</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 stretchy="false" fence="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 stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-3" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-32" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-4" type="math/tex">T</script></span>, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" data-mathml='<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>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-35" style="width: 10.4em; display: inline-block;"><span style="width: 8.94em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1008.87em, 2.78em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mo" id="MathJax-Span-41" style="padding-left: 0.27em; font-family: MathJax_Main;">{</span><span class="msubsup" id="MathJax-Span-42"><span style="width: 1.11em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.76em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-43" style="font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.68em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-44" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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: 0.16em;"><span style="width: 1.11em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.76em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.68em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-48" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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: 0.16em;"><span style="width: 1.11em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.76em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.68em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-52" style="font-family: MathJax_Main; font-size: 70.7%;">3</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-55" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-56" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-57" style="padding-left: 0.16em; font-family: MathJax_Main;">}</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 stretchy="false" fence="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 stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-5" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-58" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-6" type="math/tex">U</script></span>.  Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Y</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-61" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.74em, 2.53em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-7" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-66" style="width: 1.02em; display: inline-block;"><span style="width: 0.86em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.86em, 2.53em, -1000em);"><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="width: 0.02em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-8" type="math/tex">X</script></span>.  <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-9-Frame" role="presentation" style="position: relative;" data-mathml='<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>&amp;#x2296;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-69" style="width: 5.49em; display: inline-block;"><span style="width: 4.74em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.49em, 1004.68em, 2.61em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="texatom" id="MathJax-Span-75" style="padding-left: 0.27em;"><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="padding-left: 0.22em; font-family: MathJax_Main;">⊖</span><span class="texatom" id="MathJax-Span-79" style="padding-left: 0.22em;"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.05em; overflow: hidden; vertical-align: -0.15em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-9" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-10-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>&amp;#x222A;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-82" style="width: 3.02em; display: inline-block;"><span style="width: 2.58em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.49em, 1002.53em, 2.55em, -1000em);"><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="padding-left: 0.22em; font-family: MathJax_Main;">∪</span><span class="texatom" id="MathJax-Span-88" style="padding-left: 0.22em;"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.98em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-10" type="math/tex">{\rm A}\cup {\rm B}</script></span>, because each <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-11-Frame" role="presentation" style="position: relative;" data-mathml='<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>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-91" style="width: 3.77em; display: inline-block;"><span style="width: 3.23em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.23em, 2.82em, -1000em);"><span class="mrow" id="MathJax-Span-92"><span class="msubsup" id="MathJax-Span-93"><span style="width: 0.9em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-94" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.58em; top: -3.83em; position: absolute;"><span class="mi" id="MathJax-Span-95" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">i</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-96" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="msubsup" id="MathJax-Span-97" style="padding-left: 0.27em;"><span style="width: 1.04em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.76em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-98" style="font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.68em; top: -3.83em; position: absolute;"><span class="mi" id="MathJax-Span-99" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">j</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.25em; overflow: hidden; vertical-align: -0.4em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-11" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-12-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-100" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.49em, 1000.72em, 2.53em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.95em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-12" type="math/tex">{\rm A}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-13-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-105" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.64em, 2.53em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-13" type="math/tex">{\rm B}</script></span> are extended to include a type <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-14-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi></mrow><mo>&amp;#x2229;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">B</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-110" style="width: 4.96em; display: inline-block;"><span style="width: 4.25em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.49em, 1004.2em, 2.57em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-113" style="padding-left: 0.27em; font-family: MathJax_Main;">∈</span><span class="texatom" id="MathJax-Span-114" style="padding-left: 0.27em;"><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="padding-left: 0.22em; font-family: MathJax_Main;">∩</span><span class="texatom" id="MathJax-Span-118" style="padding-left: 0.22em;"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1em; overflow: hidden; vertical-align: -0.1em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-14" type="math/tex">\tau \in {\rm A}\cap {\rm B}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-15-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Y</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-121" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.74em, 2.53em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-15" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-16-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-126" style="width: 2.69em; display: inline-block;"><span style="width: 2.31em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1002.23em, 2.54em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-132" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.92em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-16" type="math/tex">Ts...</script></span><code>&gt;</code> (with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-17-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>&amp;gt;</mo><mn>1</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-133" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.94em, 2.57em, -1000em);"><span class="mrow" id="MathJax-Span-134"><span class="munderover" id="MathJax-Span-135"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-138" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-139"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-140" style="padding-left: 0.27em; font-family: MathJax_Main;">&gt;</span><span class="mn" id="MathJax-Span-141" style="padding-left: 0.27em; font-family: MathJax_Main;">1</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.34em; overflow: hidden; vertical-align: -0.1em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-17" type="math/tex">\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 tabindex="0" class="MathJax" id="MathJax-Element-18-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-142" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-18" type="math/tex">T</script></span> that is convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-19-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-145" style="width: 1.18em; display: inline-block;"><span style="width: 1.02em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1001.02em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-146"><span class="msup" id="MathJax-Span-147"><span style="width: 1.05em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-148" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-149" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-19" type="math/tex">T'</script></span>, let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-20-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-150" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.64em, 2.53em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-20" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-21-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-155" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-21" type="math/tex">T</script></span>, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-22-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-158" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.5em, 1000.7em, 2.72em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.16em; overflow: hidden; vertical-align: -0.28em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-22" type="math/tex">{\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 tabindex="0" class="MathJax" id="MathJax-Element-23-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-163" style="width: 1.18em; display: inline-block;"><span style="width: 1.02em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1001.02em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-164"><span class="msup" id="MathJax-Span-165"><span style="width: 1.05em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-166" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-167" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-23" type="math/tex">T'</script></span>.  If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-24-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow><mo>&amp;#x2286;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-168" style="width: 3.18em; display: inline-block;"><span style="width: 2.74em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.5em, 1002.69em, 2.72em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_Main;">⊆</span><span class="texatom" id="MathJax-Span-174" style="padding-left: 0.27em;"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.16em; overflow: hidden; vertical-align: -0.28em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-24" type="math/tex">{\rm P}\subseteq{\rm Q}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-25-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-177" style="width: 1.18em; display: inline-block;"><span style="width: 1.02em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1001.02em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-178"><span class="msup" id="MathJax-Span-179"><span style="width: 1.05em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-180" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-181" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-25" type="math/tex">T'</script></span> is denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-26-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-182" style="width: 1.61em; display: inline-block;"><span style="width: 1.4em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1001.4em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-183"><span class="msubsup" id="MathJax-Span-184"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-185" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-186" style="font-family: MathJax_Main; font-size: 70.7%;">+</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-26" type="math/tex">T^+</script></span>.  Otherwise, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-27-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow><mo>&amp;#x2288;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Q</mi></mrow></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-187" style="width: 3.18em; display: inline-block;"><span style="width: 2.74em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.58em; position: absolute; clip: rect(1.62em, 1002.69em, 3.05em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_AMS;">⊈</span><span class="texatom" id="MathJax-Span-193" style="padding-left: 0.27em;"><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="width: 0px; height: 2.58em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.4em; overflow: hidden; vertical-align: -0.41em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-27" type="math/tex">{\rm P}\nsubseteq{\rm Q}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-28-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2032;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-196" style="width: 1.18em; display: inline-block;"><span style="width: 1.02em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1001.02em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-197"><span class="msup" id="MathJax-Span-198"><span style="width: 1.05em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-199" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-200" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-28" type="math/tex">T'</script></span> is denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-29-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-201" style="width: 1.61em; display: inline-block;"><span style="width: 1.4em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1001.4em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-202"><span class="msubsup" id="MathJax-Span-203"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-204" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-205" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-29" type="math/tex">T^-</script></span>.  The conversion from <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-30-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-206" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-30" type="math/tex">T</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-31-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-209" style="width: 1.61em; display: inline-block;"><span style="width: 1.4em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1001.4em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-210"><span class="msubsup" id="MathJax-Span-211"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-212" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-213" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-31" type="math/tex">T^-</script></span> (denoted as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-32-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-214" style="width: 4.25em; display: inline-block;"><span style="width: 3.66em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1003.66em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-217" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="msubsup" id="MathJax-Span-218" style="padding-left: 0.27em;"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-219" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-220" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-32" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-33-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-221" style="width: 2.69em; display: inline-block;"><span style="width: 2.31em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1002.23em, 2.54em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-227" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.92em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-33" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-34-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-228" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-34" type="math/tex">T</script></span>.</p><p>When <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-35-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-231" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.94em, 2.54em, -1000em);"><span class="mrow" id="MathJax-Span-232"><span class="munderover" id="MathJax-Span-233"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-236" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-237"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-238" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-239" style="padding-left: 0.27em; font-family: MathJax_Main;">1</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.3em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-35" type="math/tex">\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 tabindex="0" class="MathJax" id="MathJax-Element-36-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-240" style="width: 1.61em; display: inline-block;"><span style="width: 1.4em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1001.4em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-241"><span class="msubsup" id="MathJax-Span-242"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-243" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-244" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-36" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-37-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>+</mo><mi mathvariant="normal">&amp;#x22A5;</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-245" style="width: 3.18em; display: inline-block;"><span style="width: 2.74em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1002.69em, 2.61em, -1000em);"><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="width: 0.18em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-248" style="padding-left: 0.22em; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-249" style="padding-left: 0.22em; font-family: MathJax_Main;">⊥</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.01em; overflow: hidden; vertical-align: -0.15em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-37" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-38-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>=</mo><mn>2</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-250" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.96em, 2.54em, -1000em);"><span class="mrow" id="MathJax-Span-251"><span class="munderover" id="MathJax-Span-252"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-255" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-256"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-257" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-258" style="padding-left: 0.27em; font-family: MathJax_Main;">2</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.3em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-38" type="math/tex">\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 tabindex="0" class="MathJax" id="MathJax-Element-39-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo>,</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-259" style="width: 3.02em; display: inline-block;"><span style="width: 2.58em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1002.58em, 2.72em, -1000em);"><span class="mrow" id="MathJax-Span-260"><span class="msubsup" id="MathJax-Span-261"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-262" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-263" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.24em; overflow: hidden; vertical-align: -0.28em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-39" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-40-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-266" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-40" type="math/tex">T</script></span> is not convertible to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-41-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-269" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-41" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-42-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-272" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.94em, 2.54em, -1000em);"><span class="mrow" id="MathJax-Span-273"><span class="munderover" id="MathJax-Span-274"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-277" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-278"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-279" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-280" style="padding-left: 0.27em; font-family: MathJax_Main;">1</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.3em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-42" type="math/tex">\overline{Ts}=1</script></span>.</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-43-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-281" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-284" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-285" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-43" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-44-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-286" style="width: 4.25em; display: inline-block;"><span style="width: 3.66em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1003.66em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-289" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="msubsup" id="MathJax-Span-290" style="padding-left: 0.27em;"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-291" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-292" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-44" type="math/tex">T\rightharpoonup T^-</script></span> has a worse conversion sequence, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-45-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-293" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-45" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-46-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-296" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-46" type="math/tex">U</script></span> as well because <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-47-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-299" style="width: 4.25em; display: inline-block;"><span style="width: 3.66em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1003.66em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-302" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="msubsup" id="MathJax-Span-303" style="padding-left: 0.27em;"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-304" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-305" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-47" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-48-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-306" style="width: 0.81em; display: inline-block;"><span style="width: 0.7em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1000.7em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.91em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-48" type="math/tex">T</script></span> is potentially unrepresentable converted to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-49-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-309" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-49" type="math/tex">U</script></span>, according to Lemma 2, both <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-50-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-312" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-315" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-316" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-50" type="math/tex">T\rightharpoonup U</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-51-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-317" style="width: 4.25em; display: inline-block;"><span style="width: 3.66em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1003.66em, 2.53em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-320" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="msubsup" id="MathJax-Span-321" style="padding-left: 0.27em;"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-322" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-323" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-51" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-52-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-324" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-327" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-328" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-52" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-53-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-329" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-53" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-54-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-332" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-335" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-336" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-54" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-55-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-337" style="width: 1.61em; display: inline-block;"><span style="width: 1.4em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1001.4em, 2.53em, -1000em);"><span class="mrow" id="MathJax-Span-338"><span class="msubsup" id="MathJax-Span-339"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-340" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-341" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.02em; overflow: hidden; vertical-align: -0.06em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-55" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-56-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-342" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-56" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-57-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>&amp;gt;</mo><mn>2</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-345" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.96em, 2.57em, -1000em);"><span class="mrow" id="MathJax-Span-346"><span class="munderover" id="MathJax-Span-347"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-350" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-351"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-352" style="padding-left: 0.27em; font-family: MathJax_Main;">&gt;</span><span class="mn" id="MathJax-Span-353" style="padding-left: 0.27em; font-family: MathJax_Main;">2</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.34em; overflow: hidden; vertical-align: -0.1em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-57" type="math/tex">\overline{Ts}>2</script></span>, let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-58-Frame" role="presentation" style="position: relative;" data-mathml='<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>&amp;#x2212;</mo></msup></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-354" style="width: 5.01em; display: inline-block;"><span style="width: 4.31em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1004.31em, 2.68em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="msubsup" id="MathJax-Span-357"><span style="width: 0.89em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.38em, 1000.42em, 4.15em, -1000em);"><span class="mi" id="MathJax-Span-358" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.46em; top: -3.83em; position: absolute;"><span class="mn" id="MathJax-Span-359" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-360" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="msubsup" id="MathJax-Span-361" style="padding-left: 0.27em;"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-362" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-363" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.19em; overflow: hidden; vertical-align: -0.23em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-58" type="math/tex">Ts_1=T^-</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-59-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-364" style="width: 0.75em; display: inline-block;"><span style="width: 0.64em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.5em, 1000.64em, 2.55em, -1000em);"><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="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.96em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-59" type="math/tex">S</script></span> be an overload set <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-60-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&amp;#x03C4;</mi><mo stretchy="false">)</mo><mo>&amp;#x2223;</mo><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-367" style="width: 7.65em; display: inline-block;"><span style="width: 6.57em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1006.5em, 2.78em, -1000em);"><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="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-373" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-374" style="padding-left: 0.27em; font-family: MathJax_Main;">∣</span><span class="mi" id="MathJax-Span-375" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">τ<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-376" style="padding-left: 0.27em; font-family: MathJax_Main;">∈</span><span class="mi" id="MathJax-Span-377" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" fence="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 stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-60" type="math/tex">\{f(\tau) \mid \tau \in Ts\}</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-61-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo>=</mo><mi>S</mi><mo>&amp;#x2212;</mo><mo fence="false" stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-380" style="width: 9.16em; display: inline-block;"><span style="width: 7.86em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1007.8em, 2.78em, -1000em);"><span class="mrow" id="MathJax-Span-381"><span class="msup" id="MathJax-Span-382"><span style="width: 0.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.12em, 1000.64em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-383" style="font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-384" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-385" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mi" id="MathJax-Span-386" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-387" style="padding-left: 0.22em; font-family: MathJax_Main;">−</span><span class="mo" id="MathJax-Span-388" style="padding-left: 0.22em; font-family: MathJax_Main;">{</span><span class="mi" id="MathJax-Span-389" style="font-family: MathJax_Math; font-style: italic;">f<span style="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-390" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-391"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-392" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-393" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.31em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 stretchy="false" fence="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-61" type="math/tex">S'=S-\{f(T^-)\}</script></span>.</p><ul>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-62-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-396" style="width: 2.21em; display: inline-block;"><span style="width: 1.88em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1001.79em, 2.78em, -1000em);"><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="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-62" type="math/tex">S(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-63-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>&amp;#x2212;</mo></msup><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-402" style="width: 3.18em; display: inline-block;"><span style="width: 2.74em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.43em, 1002.65em, 2.78em, -1000em);"><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="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-405" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-406"><span style="width: 1.41em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.14em, 1000.7em, 4.14em, -1000em);"><span class="mi" id="MathJax-Span-407" style="font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.79em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-408" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-409" style="font-family: MathJax_Main;">)</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.31em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-63" type="math/tex">f(T^-)</script></span>, this is equivalent to the case <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-64-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>=</mo><mn>1</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-410" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.94em, 2.54em, -1000em);"><span class="mrow" id="MathJax-Span-411"><span class="munderover" id="MathJax-Span-412"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-415" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-416"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-417" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-418" style="padding-left: 0.27em; font-family: MathJax_Main;">1</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.3em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-64" type="math/tex">\overline{Ts}=1</script></span>; or</li>
<li>if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-65-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-419" style="width: 2.58em; display: inline-block;"><span style="width: 2.2em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002.11em, 2.78em, -1000em);"><span class="mrow" id="MathJax-Span-420"><span class="msup" id="MathJax-Span-421"><span style="width: 0.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.12em, 1000.64em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-422" style="font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-423" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.29em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-65" type="math/tex">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-66-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-427" style="width: 2.48em; display: inline-block;"><span style="width: 2.1em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002em, 2.78em, -1000em);"><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="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-432" style="font-family: MathJax_Main;">)</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-66" type="math/tex">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-67-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-433" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-436" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-437" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-67" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-68-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-438" style="width: 2.58em; display: inline-block;"><span style="width: 2.2em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002.11em, 2.78em, -1000em);"><span class="mrow" id="MathJax-Span-439"><span class="msup" id="MathJax-Span-440"><span style="width: 0.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.12em, 1000.64em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-441" style="font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-442" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.29em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-68" type="math/tex">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-69-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-446" style="width: 2.48em; display: inline-block;"><span style="width: 2.1em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002em, 2.78em, -1000em);"><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="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-451" style="font-family: MathJax_Main;">)</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-69" type="math/tex">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-70-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-452" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-455" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-456" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-70" type="math/tex">T\rightharpoonup U</script></span> is a user-defined conversion, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-71-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo>&amp;#x2212;</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>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-457" style="width: 6.14em; display: inline-block;"><span style="width: 5.28em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1005.21em, 2.78em, -1000em);"><span class="mrow" id="MathJax-Span-458"><span class="msup" id="MathJax-Span-459"><span style="width: 0.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.12em, 1000.64em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-460" style="font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-461" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-462" style="padding-left: 0.22em; font-family: MathJax_Main;">−</span><span class="mo" id="MathJax-Span-463" style="padding-left: 0.22em; font-family: MathJax_Main;">{</span><span class="mi" id="MathJax-Span-464" style="font-family: MathJax_Math; font-style: italic;">f<span style="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.29em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 stretchy="false" fence="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-71" type="math/tex">S'-\{f(U)\}</script></span> has no viable function against  <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-72-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-469" style="width: 0.54em; display: inline-block;"><span style="width: 0.43em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.76em, 1000.41em, 2.54em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.65em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-72" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-73-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>T</mi><mi>s</mi></mrow><mo accent="false">&amp;#x00AF;</mo></mover><mo>=</mo><mn>2</mn></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-472" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.2em, 1002.96em, 2.54em, -1000em);"><span class="mrow" id="MathJax-Span-473"><span class="munderover" id="MathJax-Span-474"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: 0.01em; top: -3.98em; position: absolute; clip: rect(3.14em, 1001.12em, 4.15em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-477" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0em; top: -4.47em; position: absolute; clip: rect(3.3em, 1001.19em, 3.76em, -1000em);"><span class="mo" id="MathJax-Span-478"><span style="width: 1.19em; height: 0px; display: inline-block; position: relative;"><span style="left: -0.04em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.89em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.12em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.31em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.5em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -3.98em; position: absolute;"><span style="font-family: MathJax_Main; font-size: 70.7%;">¯</span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span></span></span><span class="mo" id="MathJax-Span-479" style="padding-left: 0.27em; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-480" style="padding-left: 0.27em; font-family: MathJax_Main;">2</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.3em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-73" type="math/tex">\overline{Ts}=2</script></span>.</li>
<li>Otherwise, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-74-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>&amp;#x2032;</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-481" style="width: 2.58em; display: inline-block;"><span style="width: 2.2em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002.11em, 2.78em, -1000em);"><span class="mrow" id="MathJax-Span-482"><span class="msup" id="MathJax-Span-483"><span style="width: 0.97em; height: 0px; display: inline-block; position: relative;"><span style="left: 0em; top: -3.98em; position: absolute; clip: rect(3.12em, 1000.64em, 4.17em, -1000em);"><span class="mi" id="MathJax-Span-484" style="font-family: MathJax_Math; font-style: italic;">S<span style="width: 0.03em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span style="width: 0px; height: 3.98em; display: inline-block;"></span></span><span style="left: 0.7em; top: -4.35em; position: absolute;"><span class="mo" id="MathJax-Span-485" style="font-family: MathJax_Main; font-size: 70.7%;">′</span><span style="width: 0px; height: 3.98em; display: inline-block;"></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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.29em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-74" type="math/tex">S'(r)</script></span> resolves to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-75-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-489" style="width: 2.48em; display: inline-block;"><span style="width: 2.1em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1002em, 2.78em, -1000em);"><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="width: 0.06em; height: 1px; overflow: hidden; display: inline-block;"></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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-494" style="font-family: MathJax_Main;">)</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-75" type="math/tex">f(U)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-76-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-495" style="width: 3.5em; display: inline-block;"><span style="width: 3.01em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.01em, 2.55em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-498" style="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-499" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-76" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-77-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-500" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-77" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-78-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>s</mi><mo>.</mo><mo>.</mo><mo>.</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-503" style="width: 2.69em; display: inline-block;"><span style="width: 2.31em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1002.23em, 2.54em, -1000em);"><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="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></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="padding-left: 0.16em; font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-509" style="padding-left: 0.16em; font-family: MathJax_Main;">.</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.92em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-78" type="math/tex">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 tabindex="0" class="MathJax" id="MathJax-Element-79-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-510" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-79" type="math/tex">R</script></span>, when there exists one and only one <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-80-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&amp;#x03C4;</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-513" style="width: 3.39em; display: inline-block;"><span style="width: 2.9em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.53em, 1002.86em, 2.57em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-516" style="padding-left: 0.27em; font-family: MathJax_Main;">∈</span><span class="mi" id="MathJax-Span-517" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mi" id="MathJax-Span-518" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.95em; overflow: hidden; vertical-align: -0.1em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-80" type="math/tex">\tau \in Ts</script></span> rendering <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-81-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-519" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-81" type="math/tex">R</script></span> to be potentially unrepresentable converted to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-82-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&amp;#x03C4;</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-522" style="width: 0.64em; display: inline-block;"><span style="width: 0.53em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.77em, 1000.54em, 2.54em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.64em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-82" type="math/tex">\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 tabindex="0" class="MathJax" id="MathJax-Element-83-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&amp;#x03C4;</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-525" style="width: 0.64em; display: inline-block;"><span style="width: 0.53em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.77em, 1000.54em, 2.54em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.64em; overflow: hidden; vertical-align: -0.07em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-83" type="math/tex">\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 tabindex="0" class="MathJax" id="MathJax-Element-84-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">&amp;#x21C0;</mo><mi>U</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-528" style="width: 3.55em; display: inline-block;"><span style="width: 3.07em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1003.07em, 2.55em, -1000em);"><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="padding-left: 0.27em; font-family: MathJax_Main;">⇀</span><span class="mi" id="MathJax-Span-532" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">U<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-84" type="math/tex">R\rightharpoonup U</script></span> when there is exactly one <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-85-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>&amp;#x2208;</mo><mi>T</mi><mi>s</mi><mo>&amp;#x2212;</mo><mo fence="false" stretchy="false">{</mo><mi>&amp;#x03C4;</mi><mo fence="false" stretchy="false">}</mo></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-533" style="width: 6.84em; display: inline-block;"><span style="width: 5.87em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.45em, 1005.8em, 2.78em, -1000em);"><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="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-536" style="padding-left: 0.27em; font-family: MathJax_Main;">∈</span><span class="mi" id="MathJax-Span-537" style="padding-left: 0.27em; font-family: MathJax_Math; font-style: italic;">T<span style="width: 0.12em; height: 1px; overflow: hidden; display: inline-block;"></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="padding-left: 0.22em; font-family: MathJax_Main;">−</span><span class="mo" id="MathJax-Span-540" style="padding-left: 0.22em; font-family: MathJax_Main;">{</span><span class="mi" id="MathJax-Span-541" style="font-family: MathJax_Math; font-style: italic;">τ<span style="width: 0.08em; height: 1px; overflow: hidden; display: inline-block;"></span></span><span class="mo" id="MathJax-Span-542" style="font-family: MathJax_Main;">}</span></span><span style="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 1.28em; overflow: hidden; vertical-align: -0.35em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 stretchy="false" fence="false">{</mo><mi>τ</mi><mo stretchy="false" fence="false">}</mo></math></span></span><script id="MathJax-Element-85" type="math/tex">U\in Ts-\{\tau\}</script></span> that is convertible from <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span tabindex="0" class="MathJax" id="MathJax-Element-86-Frame" role="presentation" style="position: relative;" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>'><nobr aria-hidden="true"><span class="math" id="MathJax-Span-543" style="width: 0.91em; display: inline-block;"><span style="width: 0.75em; height: 0px; font-size: 116%; display: inline-block; position: relative;"><span style="left: 0em; top: -2.37em; position: absolute; clip: rect(1.52em, 1000.75em, 2.55em, -1000em);"><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="width: 0px; height: 2.37em; display: inline-block;"></span></span></span><span style="width: 0px; height: 0.94em; overflow: hidden; vertical-align: -0.08em; border-left-color: currentColor; border-left-width: 0px; border-left-style: solid; display: inline-block;"></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 id="MathJax-Element-86" type="math/tex">R</script></span>.</li>
</ul><p>The first case is easy to fix while the second gives 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><h2 id="Alternative-designs"><a title="Alternative-designs" class="anchor hidden-xs" href="#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><h2 id="Wording"><a title="Wording" class="anchor hidden-xs" href="#Wording"><span class="octicon octicon-link"></span></a>Wording</h2><p>This wording is relative to N4713.</p><p>Modify 23.7.3.1 [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>, where <em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code>
shall not participate in overload resolution unless
<code>T</code><i><sub>i</sub></i><code>{t}</code> is well-formed and
is not a boolean conversion (7.14)</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><tt>(std::forward&lt;T&gt;(t))</tt> 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><p>Modify 23.7.3.3 [variant.assign]/8 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>, where <em><code>FUN</code></em><code>(T</code><i><sub>i</sub></i><code>)</code>
shall not participate in overload resolution unless
<code>T</code><i><sub>i</sub></i><code>{t}</code> is well-formed and
is not a boolean conversion (7.14)</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><tt>(std::forward&lt;T&gt;(t))</tt> 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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