Read about our upcoming Code of Conduct on this issue

Commit 790e2692 authored by Lawrence Paulson's avatar Lawrence Paulson
Browse files

Hermite_Lindemann website

parent a9fb9e1730a7
......@@ -664,6 +664,39 @@ abstract =
developed in the AFP entry on the transcendence of
<em>e</em>.</p>
 
[Hermite_Lindemann]
title = The Hermite–Lindemann–Weierstraß Transcendence Theorem
author = Manuel Eberl <https://www21.in.tum.de/~eberlm>
topic = Mathematics/Number theory
date = 2021-03-03
notify = eberlm@in.tum.de
abstract =
<p>This article provides a formalisation of the
Hermite-Lindemann-Weierstraß Theorem (also known as simply
Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the
crowning achievements of 19th century number theory.</p>
<p>The theorem states that if $\alpha_1, \ldots,
\alpha_n\in\mathbb{C}$ are algebraic numbers that are linearly
independent over $\mathbb{Z}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$
are algebraically independent over $\mathbb{Q}$.</p>
<p>Like the <a
href="https://doi.org/10.1007/978-3-319-66107-0_5">previous
formalisation in Coq by Bernard</a>, I proceeded by formalising
<a
href="https://doi.org/10.1017/CBO9780511565977">Baker's
version of the theorem and proof</a> and then deriving the
original one from that. Baker's version states that for any
algebraic numbers $\beta_1, \ldots, \beta_n\in\mathbb{C}$ and distinct
algebraic numbers $\alpha_i, \ldots, \alpha_n\in\mathbb{C}$, we have
$\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0$ if and only
if all the $\beta_i$ are zero.</p> <p>This has a number of
direct corollaries, e.g.:</p> <ul> <li>$e$ and $\pi$
are transcendental</li> <li>$e^z$, $\sin z$, $\tan z$,
etc. are transcendental for algebraic
$z\in\mathbb{C}\setminus\{0\}$</li> <li>$\ln z$ is
transcendental for algebraic $z\in\mathbb{C}\setminus\{0,
1\}$</li> </ul>
[DFS_Framework]
title = A Framework for Verifying Depth-First Search Algorithms
author = Peter Lammich <http://www21.in.tum.de/~lammich>, René Neumann <mailto:neumannr@in.tum.de>
......
......@@ -156,7 +156,7 @@ To this end, we mechanized several results on resultants, which also required us
<td class="data"><a href="Berlekamp_Zassenhaus.html">Berlekamp_Zassenhaus</a>, <a href="Sturm_Sequences.html">Sturm_Sequences</a> </td></tr>
<tr><td class="datahead">Used by:</td>
<td class="data"><a href="LLL_Basis_Reduction.html">LLL_Basis_Reduction</a> </td></tr>
<td class="data"><a href="Hermite_Lindemann.html">Hermite_Lindemann</a>, <a href="LLL_Basis_Reduction.html">LLL_Basis_Reduction</a> </td></tr>
......
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>The Hermite–Lindemann–Weierstraß Transcendence Theorem - Archive of Formal Proofs
</title>
<link rel="stylesheet" type="text/css" href="../front.css">
<link rel="icon" href="../images/favicon.ico" type="image/icon">
<link rel="alternate" type="application/rss+xml" title="RSS" href="../rss.xml">
<!-- MathJax for LaTeX support in abstracts -->
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']]
},
processEscapes: true,
svg: {
fontCache: 'global'
}
};
</script>
<script id="MathJax-script" async src="../components/mathjax/es5/tex-mml-chtml.js"></script>
</head>
<body class="mathjax_ignore">
<table width="100%">
<tbody>
<tr>
<!-- Navigation -->
<td width="20%" align="center" valign="top">
<p>&nbsp;</p>
<a href="https://www.isa-afp.org/">
<img src="../images/isabelle.png" width="100" height="88" border=0>
</a>
<p>&nbsp;</p>
<p>&nbsp;</p>
<table class="nav" width="80%">
<tr>
<td class="nav" width="100%"><a href="../index.html">Home</a></td>
</tr>
<tr>
<td class="nav"><a href="../about.html">About</a></td>
</tr>
<tr>
<td class="nav"><a href="../submitting.html">Submission</a></td>
</tr>
<tr>
<td class="nav"><a href="../updating.html">Updating Entries</a></td>
</tr>
<tr>
<td class="nav"><a href="../using.html">Using Entries</a></td>
</tr>
<tr>
<td class="nav"><a href="../search.html">Search</a></td>
</tr>
<tr>
<td class="nav"><a href="../statistics.html">Statistics</a></td>
</tr>
<tr>
<td class="nav"><a href="../topics.html">Index</a></td>
</tr>
<tr>
<td class="nav"><a href="../download.html">Download</a></td>
</tr>
</table>
<p>&nbsp;</p>
<p>&nbsp;</p>
</td>
<!-- Content -->
<td width="80%" valign="top">
<div align="center">
<p>&nbsp;</p>
<h1> <font class="first">T</font>he
<font class="first">H</font>ermite–Lindemann–Weierstraß
<font class="first">T</font>ranscendence
<font class="first">T</font>heorem
</h1>
<p>&nbsp;</p>
<table width="80%" class="data">
<tbody>
<tr>
<td class="datahead" width="20%">Title:</td>
<td class="data" width="80%">The Hermite–Lindemann–Weierstraß Transcendence Theorem</td>
</tr>
<tr>
<td class="datahead">
Author:
</td>
<td class="data">
<a href="https://www21.in.tum.de/~eberlm">Manuel Eberl</a>
</td>
</tr>
<tr>
<td class="datahead">Submission date:</td>
<td class="data">2021-03-03</td>
</tr>
<tr>
<td class="datahead" valign="top">Abstract:</td>
<td class="abstract mathjax_process">
<p>This article provides a formalisation of the
Hermite-Lindemann-Weierstraß Theorem (also known as simply
Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the
crowning achievements of 19th century number theory.</p>
<p>The theorem states that if $\alpha_1, \ldots,
\alpha_n\in\mathbb{C}$ are algebraic numbers that are linearly
independent over $\mathbb{Z}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$
are algebraically independent over $\mathbb{Q}$.</p>
<p>Like the <a
href="https://doi.org/10.1007/978-3-319-66107-0_5">previous
formalisation in Coq by Bernard</a>, I proceeded by formalising
<a
href="https://doi.org/10.1017/CBO9780511565977">Baker's
version of the theorem and proof</a> and then deriving the
original one from that. Baker's version states that for any
algebraic numbers $\beta_1, \ldots, \beta_n\in\mathbb{C}$ and distinct
algebraic numbers $\alpha_i, \ldots, \alpha_n\in\mathbb{C}$, we have
$\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0$ if and only
if all the $\beta_i$ are zero.</p> <p>This has a number of
direct corollaries, e.g.:</p> <ul> <li>$e$ and $\pi$
are transcendental</li> <li>$e^z$, $\sin z$, $\tan z$,
etc. are transcendental for algebraic
$z\in\mathbb{C}\setminus\{0\}$</li> <li>$\ln z$ is
transcendental for algebraic $z\in\mathbb{C}\setminus\{0,
1\}$</li> </ul></td>
</tr>
<tr>
<td class="datahead" valign="top">BibTeX:</td>
<td class="formatted">
<pre>@article{Hermite_Lindemann-AFP,
author = {Manuel Eberl},
title = {The Hermite–Lindemann–Weierstraß Transcendence Theorem},
journal = {Archive of Formal Proofs},
month = mar,
year = 2021,
note = {\url{https://isa-afp.org/entries/Hermite_Lindemann.html},
Formal proof development},
ISSN = {2150-914x},
}</pre>
</td>
</tr>
<tr><td class="datahead">License:</td>
<td class="data"><a href="http://isa-afp.org/LICENSE">BSD License</a></td></tr>
<tr><td class="datahead">Depends on:</td>
<td class="data"><a href="Algebraic_Numbers.html">Algebraic_Numbers</a>, <a href="Pi_Transcendental.html">Pi_Transcendental</a>, <a href="Power_Sum_Polynomials.html">Power_Sum_Polynomials</a> </td></tr>
</tbody>
</table>
<p></p>
<table class="links">
<tbody>
<tr>
<td class="links">
<a href="../browser_info/current/AFP/Hermite_Lindemann/outline.pdf">Proof outline</a><br>
<a href="../browser_info/current/AFP/Hermite_Lindemann/document.pdf">Proof document</a>
</td>
</tr>
<tr>
<td class="links">
<a href="../browser_info/current/AFP/Hermite_Lindemann/index.html">Browse theories</a>
</td></tr>
<tr>
<td class="links">
<a href="../release/afp-Hermite_Lindemann-current.tar.gz">Download this entry</a>
</td>
</tr>
<tr><td class="links">Older releases:
None
</td></tr>
</tbody>
</table>
</div>
</td>
</tr>
</tbody>
</table>
<script src="../jquery.min.js"></script>
<script src="../script.js"></script>
</body>
</html>
\ No newline at end of file
......@@ -143,7 +143,9 @@ developed in the AFP entry on the transcendence of
<tr><td class="datahead">Depends on:</td>
<td class="data"><a href="E_Transcendental.html">E_Transcendental</a>, <a href="Symmetric_Polynomials.html">Symmetric_Polynomials</a> </td></tr>
<tr><td class="datahead">Used by:</td>
<td class="data"><a href="Hermite_Lindemann.html">Hermite_Lindemann</a> </td></tr>
</tbody>
......
......@@ -174,7 +174,9 @@ i&gt;k.</p></td>
<tr><td class="datahead">Depends on:</td>
<td class="data"><a href="Polynomial_Factorization.html">Polynomial_Factorization</a>, <a href="Symmetric_Polynomials.html">Symmetric_Polynomials</a> </td></tr>
<tr><td class="datahead">Used by:</td>
<td class="data"><a href="Hermite_Lindemann.html">Hermite_Lindemann</a> </td></tr>
</tbody>
......
......@@ -95,6 +95,14 @@ of a scientific journal, is indexed by <a href="http://dblp.uni-trier.de/db/jour
Author:
<a href="https://lig-membres.imag.fr/mechenim/">Mnacho Echenim</a>
</td>
</tr>
<tr>
<td class="entry">
2021-03-03: <a href="entries/Hermite_Lindemann.html">The Hermite–Lindemann–Weierstraß Transcendence Theorem</a>
<br>
Author:
<a href="https://www21.in.tum.de/~eberlm">Manuel Eberl</a>
</td>
</tr>
<tr>
<td class="entry">
......
......@@ -22,6 +22,39 @@ also known as von Neumann measurements, which are based on elements of
spectral theory. We also formalized the CHSH inequality, an inequality
involving expectations in a probability space that is violated by
quantum measurements, thus proving that quantum mechanics cannot be modeled with an underlying local hidden-variable theory.</description>
</item>
<item>
<title>The Hermite–Lindemann–Weierstraß Transcendence Theorem</title>
<link>https://www.isa-afp.org/entries/Hermite_Lindemann.html</link>
<guid>https://www.isa-afp.org/entries/Hermite_Lindemann.html</guid>
<dc:creator> Manuel Eberl </dc:creator>
<pubDate>03 Mar 2021 00:00:00 +0000</pubDate>
<description>
&lt;p&gt;This article provides a formalisation of the
Hermite-Lindemann-Weierstraß Theorem (also known as simply
Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the
crowning achievements of 19th century number theory.&lt;/p&gt;
&lt;p&gt;The theorem states that if $\alpha_1, \ldots,
\alpha_n\in\mathbb{C}$ are algebraic numbers that are linearly
independent over $\mathbb{Z}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$
are algebraically independent over $\mathbb{Q}$.&lt;/p&gt;
&lt;p&gt;Like the &lt;a
href=&#34;https://doi.org/10.1007/978-3-319-66107-0_5&#34;&gt;previous
formalisation in Coq by Bernard&lt;/a&gt;, I proceeded by formalising
&lt;a
href=&#34;https://doi.org/10.1017/CBO9780511565977&#34;&gt;Baker&#39;s
version of the theorem and proof&lt;/a&gt; and then deriving the
original one from that. Baker&#39;s version states that for any
algebraic numbers $\beta_1, \ldots, \beta_n\in\mathbb{C}$ and distinct
algebraic numbers $\alpha_i, \ldots, \alpha_n\in\mathbb{C}$, we have
$\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0$ if and only
if all the $\beta_i$ are zero.&lt;/p&gt; &lt;p&gt;This has a number of
direct corollaries, e.g.:&lt;/p&gt; &lt;ul&gt; &lt;li&gt;$e$ and $\pi$
are transcendental&lt;/li&gt; &lt;li&gt;$e^z$, $\sin z$, $\tan z$,
etc. are transcendental for algebraic
$z\in\mathbb{C}\setminus\{0\}$&lt;/li&gt; &lt;li&gt;$\ln z$ is
transcendental for algebraic $z\in\mathbb{C}\setminus\{0,
1\}$&lt;/li&gt; &lt;/ul&gt;</description>
</item>
<item>
<title>Mereology</title>
......@@ -562,30 +595,5 @@ href=&#34;https://www.isa-afp.org/entries/Goedel_HFSet_Semantic.html&#34;&gt;Goe
and &lt;a
href=&#34;https://www.isa-afp.org/entries/Goedel_HFSet_Semanticless.html&#34;&gt;Goedel_HFSet_Semanticless&lt;/a&gt;.</description>
</item>
<item>
<title>From Abstract to Concrete G&ouml;del's Incompleteness Theorems&mdash;Part II</title>
<link>https://www.isa-afp.org/entries/Goedel_HFSet_Semanticless.html</link>
<guid>https://www.isa-afp.org/entries/Goedel_HFSet_Semanticless.html</guid>
<dc:creator> Andrei Popescu, Dmitriy Traytel </dc:creator>
<pubDate>16 Sep 2020 00:00:00 +0000</pubDate>
<description>
We validate an abstract formulation of G&amp;ouml;del&#39;s Second
Incompleteness Theorem from a &lt;a
href=&#34;https://www.isa-afp.org/entries/Goedel_Incompleteness.html&#34;&gt;separate
AFP entry&lt;/a&gt; by instantiating it to the case of &lt;i&gt;finite
consistent extensions of the Hereditarily Finite (HF) Set
theory&lt;/i&gt;, i.e., consistent FOL theories extending the HF Set
theory with a finite set of axioms. The instantiation draws heavily
on infrastructure previously developed by Larry Paulson in his &lt;a
href=&#34;https://www.isa-afp.org/entries/Incompleteness.html&#34;&gt;direct
formalisation of the concrete result&lt;/a&gt;. It strengthens
Paulson&#39;s formalization of G&amp;ouml;del&#39;s Second from that
entry by &lt;i&gt;not&lt;/i&gt; assuming soundness, and in fact not
relying on any notion of model or semantic interpretation. The
strengthening was obtained by first replacing some of Paulson’s
semantic arguments with proofs within his HF calculus, and then
plugging in some of Paulson&#39;s (modified) lemmas to instantiate
our soundness-free G&amp;ouml;del&#39;s Second locale.</description>
</item>
</channel>
</rss>
This diff is collapsed.
......@@ -771,6 +771,7 @@
<a href="entries/Minkowskis_Theorem.html">Minkowskis_Theorem</a> &nbsp;
<a href="entries/E_Transcendental.html">E_Transcendental</a> &nbsp;
<a href="entries/Pi_Transcendental.html">Pi_Transcendental</a> &nbsp;
<a href="entries/Hermite_Lindemann.html">Hermite_Lindemann</a> &nbsp;
<a href="entries/Liouville_Numbers.html">Liouville_Numbers</a> &nbsp;
<a href="entries/Prime_Harmonic_Series.html">Prime_Harmonic_Series</a> &nbsp;
<a href="entries/Fermat3_4.html">Fermat3_4</a> &nbsp;
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment