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<h1 class="libtitle">Library Flocq.Prop.Fprop_Sterbenz</h1>
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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
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Copyright (C) 2010-2013 Guillaume Melquiond
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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.
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<a name="lab22"></a><h1 class="section">Sterbenz conditions for exact subtraction</h1>
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<br/>
<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/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#"><span class="id" title="library">Fcore_defs</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#"><span class="id" title="library">Fcore_generic_fmt</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Calc.Fcalc_ops.html#"><span class="id" title="library">Fcalc_ops</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fprop_Sterbenz"><span class="id" title="section">Fprop_Sterbenz</span></a>.<br/>
<br/>
<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/>
<span class="id" title="keyword">Notation</span> <a name="bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span> := (<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> <span class="id" title="var">e</span>).<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fprop_Sterbenz.fexp"><span class="id" title="variable">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> <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.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">valid_exp</span> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_exp"><span class="id" title="class">Valid_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#Fprop_Sterbenz.fexp"><span class="id" title="variable">fexp</span></a> }.<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">monotone_exp</span> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Monotone_exp"><span class="id" title="class">Monotone_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#Fprop_Sterbenz.fexp"><span class="id" title="variable">fexp</span></a> }.<br/>
<span class="id" title="keyword">Notation</span> <a name="format"><span class="id" title="abbreviation">format</span></a> := (<a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#Fprop_Sterbenz.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="generic_format_plus"><span class="id" title="lemma">generic_format_plus</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</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.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</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><br/>
(<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.Prop.Fprop_Sterbenz.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#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#Zmin"><span class="id" title="abbreviation">Zmin</span></a> (<a class="idref" href="Flocq.Core.Fcore_Raux.html#ln_beta"><span class="id" title="definition">ln_beta</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="Flocq.Core.Fcore_Raux.html#ln_beta"><span class="id" title="definition">ln_beta</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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><br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="generic_format_plus_weak"><span class="id" title="lemma">generic_format_plus_weak</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</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.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</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><br/>
(<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.Prop.Fprop_Sterbenz.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#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</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#Rmin"><span class="id" title="definition">Rmin</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.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</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.Prop.Fprop_Sterbenz.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><br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sterbenz_aux"><span class="id" title="lemma">sterbenz_aux</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</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.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</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><br/>
(<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#777fa820f32591f17d50731841113fa3"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#777fa820f32591f17d50731841113fa3"><span class="id" title="notation">≤</span></a> 2 <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.Prop.Fprop_Sterbenz.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><br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="sterbenz"><span class="id" title="lemma">sterbenz</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#x"><span class="id" title="variable">x</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.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</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><br/>
(<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0c0207ff801f7c795ce431f130ae670a"><span class="id" title="notation">/</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#777fa820f32591f17d50731841113fa3"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#777fa820f32591f17d50731841113fa3"><span class="id" title="notation">≤</span></a> 2 <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.Prop.Fprop_Sterbenz.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><br/>
<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_Sterbenz.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#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Prop.Fprop_Sterbenz.html#Fprop_Sterbenz"><span class="id" title="section">Fprop_Sterbenz</span></a>.<br/>
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