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| <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> | ||
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| <title>Goedel.PRrepresentable</title> | ||
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| <h1 class="libtitle">Library Goedel.PRrepresentable</h1> | ||
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| <div class="paragraph"> </div> | ||
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| Original version by Russel O'Connor | ||
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| This library is dedicated to the proof of the theorem: | ||
| "Every primitive recursive function is representable in <span class="inlinecode"><span class="id" title="var">NN</span></span>" | ||
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| <div class="paragraph"> </div> | ||
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| <ul class="doclist"> | ||
| <li> Let <span class="inlinecode"><span class="id" title="var">f</span></span> be any <span class="inlinecode"><span class="id" title="var">n</span></span>-ary arithmetic function, and <span class="inlinecode"><span class="id" title="var">p</span></span> a proof that this function is primitive recursive. | ||
| Out of <span class="inlinecode"><span class="id" title="var">p</span></span>, we can associate a <span class="inlinecode"><span class="id" title="var">NN</span></span>-formula which correctly expresses <span class="inlinecode"><span class="id" title="var">f</span></span>. | ||
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| <div class="paragraph"> </div> | ||
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| </li> | ||
| <li> The main difficulty is to associate a <span class="inlinecode"><span class="id" title="var">NN</span></span>-formula to the case where <span class="inlinecode"><span class="id" title="var">f</span></span> can be defined by a primitive | ||
| recursive scheme. | ||
| This is made possible with the help of Goedel's <span class="inlinecode"><span class="id" title="keyword">beta</span></span>-function, studied in the | ||
| <a=href="https://github.com/thery/coqprime/blob/master/src/Coqprime/N/ChineseRem.v">ChineseRem</a> | ||
| library. This function allows | ||
| to represent a sequence of computation steps through a first-order formula (theorem <span class="inlinecode"><span class="id" title="var">ChineseRem.BetaTheorem</span></span>). | ||
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| </li> | ||
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| <div class="paragraph"> </div> | ||
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| <div class="code"> | ||
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| <br/> | ||
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| <br/> | ||
| <span class="id" title="keyword">Theorem</span> <a id="primRecRepresentable" class="idref" href="#primRecRepresentable"><span class="id" title="lemma">primRecRepresentable</span></a> :<br/> | ||
| <span class="id" title="keyword">∀</span> (<a id="n:473" class="idref" href="#n:473"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:474" class="idref" href="#f:474"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a>) (<a id="p:475" class="idref" href="#p:475"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a>),<br/> | ||
| <a class="idref" href="Goedel.PRrepresentable.html#Representable"><span class="id" title="definition">Representable</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a> (<a class="idref" href="Goedel.PRrepresentable.html#primRecFormula"><span class="id" title="definition">primRecFormula</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#fun2PR"><span class="id" title="definition">fun2PR</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a>)).<br/> | ||
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| <hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> | ||
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