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<h1 class="libtitle">Library Flocq.Core.Fcore_defs</h1>
<div class="code">
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<div class="doc">
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
<div class="paragraph"> </div>
Copyright (C) 2010-2013 Sylvie Boldo
<br />
Copyright (C) 2010-2013 Guillaume Melquiond
<div class="paragraph"> </div>
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
<div class="paragraph"> </div>
<a name="lab2"></a><h1 class="section">Basic definitions: float and rounding property</h1>
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<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_Raux.html#"><span class="id" title="library">Fcore_Raux</span></a>.<br/>
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<span class="id" title="keyword">Section</span> <a name="Def"><span class="id" title="section">Def</span></a>.<br/>
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<div class="doc">
Definition of a floating-point number
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<div class="code">
<span class="id" title="keyword">Record</span> <a name="float"><span class="id" title="record">float</span></a> (<span class="id" title="keyword">beta</span> : <a class="idref" href="Flocq.Core.Fcore_Zaux.html#radix"><span class="id" title="record">radix</span></a>) := <a name="Float"><span class="id" title="constructor">Float</span></a> { <a name="Fnum"><span class="id" title="projection">Fnum</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ; <a name="Fexp"><span class="id" title="projection">Fexp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> }.<br/>
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<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> [[<span class="id" title="keyword">beta</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> [[<span class="id" title="keyword">beta</span>]].<br/>
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<span class="id" title="keyword">Variable</span> <span class="id" title="keyword">beta</span> : <a class="idref" href="Flocq.Core.Fcore_Zaux.html#radix"><span class="id" title="record">radix</span></a>.<br/>
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<span class="id" title="keyword">Definition</span> <a name="F2R"><span class="id" title="definition">F2R</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="Flocq.Core.Fcore_defs.html#float"><span class="id" title="record">float</span></a> <span class="id" title="keyword">beta</span>) :=<br/>
(<a class="idref" href="Flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#361afbb9b173fae6251865f2da226003"><span class="id" title="notation">×</span></a> <a class="idref" href="Flocq.Core.Fcore_Raux.html#bpow"><span class="id" title="definition">bpow</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="Flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>))%<span class="id" title="var">R</span>.<br/>
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<div class="doc">
Requirements on a rounding mode
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<div class="code">
<span class="id" title="keyword">Definition</span> <a name="round_pred_total"><span class="id" title="definition">round_pred_total</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">f</span><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_pred_monotone"><span class="id" title="definition">round_pred_monotone</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span>, <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_pred"><span class="id" title="definition">round_pred</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#round_pred_total"><span class="id" title="definition">round_pred_total</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#round_pred_monotone"><span class="id" title="definition">round_pred_monotone</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#Def"><span class="id" title="section">Def</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> [[<span class="id" title="keyword">beta</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> [[<span class="id" title="keyword">beta</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> [[<span class="id" title="keyword">beta</span>]].<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="RND"><span class="id" title="section">RND</span></a>.<br/>
<br/>
</div>
<div class="doc">
property of being a round toward -inf
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Rnd_DN_pt"><span class="id" title="definition">Rnd_DN_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">g</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_DN"><span class="id" title="definition">Rnd_DN</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_DN_pt"><span class="id" title="definition">Rnd_DN_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
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</div>
<div class="doc">
property of being a round toward +inf
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Rnd_UP_pt"><span class="id" title="definition">Rnd_UP_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">g</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_UP"><span class="id" title="definition">Rnd_UP</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_UP_pt"><span class="id" title="definition">Rnd_UP_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
</div>
<div class="doc">
property of being a round toward zero
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Rnd_ZR_pt"><span class="id" title="definition">Rnd_ZR_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_DN_pt"><span class="id" title="definition">Rnd_DN_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_UP_pt"><span class="id" title="definition">Rnd_UP_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_ZR"><span class="id" title="definition">Rnd_ZR</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_ZR_pt"><span class="id" title="definition">Rnd_ZR_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
</div>
<div class="doc">
property of being a round to nearest
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">g</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>))%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_N"><span class="id" title="definition">Rnd_N</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_NG_pt"><span class="id" title="definition">Rnd_NG_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">f2</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_NG"><span class="id" title="definition">Rnd_NG</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_NG_pt"><span class="id" title="definition">Rnd_NG_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_NA_pt"><span class="id" title="definition">Rnd_NA_pt</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">x</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">f2</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_N_pt"><span class="id" title="definition">Rnd_N_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#f"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_NA"><span class="id" title="definition">Rnd_NA</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">rnd</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>, <a class="idref" href="Flocq.Core.Fcore_defs.html#Rnd_NA_pt"><span class="id" title="definition">Rnd_NA_pt</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#RND"><span class="id" title="section">RND</span></a>.<br/>
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