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- <meta content="Ferruccio Guidi" name="author">
- <meta content="The formal system λδ" name="description">
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-<h1>The Formal System λδ (\lambda\delta)<br>
-</h1>
-<h2>Towards the unification of terms, types, environments and contexts</h2>
-<img style="width: 95%; height: 4px;" alt="[Separator]" title="Separator" src="images/rainbow.png"><br>
-<table style="text-align: left; width: 95%; margin-left: auto; margin-right: auto;" border="0" cellpadding="2" cellspacing="20">
- <tbody>
- <tr>
- <td style="vertical-align: top;">
- <ul>
- <li>Foreword</li>
- </ul>
- <ul>
- <li><a href="news.html">News</a></li>
- </ul>
- <ul>
- <li><a href="documentation.html">Papers</a></li>
- </ul>
- <ul>
- <li><a href="implementation.html">Resources</a><br>
- </li>
- </ul>
- </td>
- <td style="vertical-align: top; text-align: left;">
- <h3 style="text-align: right;">Foreword <img style="width: 37px; height: 37px;" alt="[Butterfly]" title="Butterfly" src="images/b9.png"></h3>
-The formal system λδ
-(\lambda\delta) is a typed lambda calculus that pursues the static and
-dynamic unification of terms, types, environments and contexts while
-enjoying a well-conceived theory, which includes the commonly
-desired properties.<br>
- <br>
-λδ takes some features from the calculi of the Automath family and
-some from the pure type systems, but it differs from both in that it
-does not include the Π construction while it provides for an
-abbreviation mechanism at the level of terms.<br>
- <br>
-λδ features explicit type annotations, which are borrowed from
-realistic type checker implementations and with which type checking is
-reduced to type inference.<br>
- <br>
-The reduction steps of λδ include β-contraction, δ-expansion,
-ζ-contraction and θ-swap. On the other hand,
-η-contraction is not included.<br>
- <br>
-The theory of λδ includes important properties such as the
-confluence of reduction, the correctness of types, the
-uniqueness of types up to conversion, the subject reduction of the type
-assignment, the strong normalization of the typed terms. The
-decidability of type inference and of type checking come as corollaries.<br>
- <br>
-λδ features uniformly dependent types and a predicative abstraction
-mechanism, so the calculus can serve as a formal specification language
-for the type theories that need a predicative foundation. λδ is
-expected to have the expressive power of λ→.<br>
- <br>
-λδ comes in several versions listed in the following table, which
-includes the major milestones:<br>
- <br>
- <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="0">
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+ <meta name="description" content="\lambda\delta home page" />
+ <title>\lambda\delta home page</title>
+ <link rel="stylesheet" type="text/css" href="http://lambdadelta.info/css/ld_web.css" />
+ <link rel="stylesheet" type="text/css" href="http://lambdadelta.info/css/lddl.css" />
+ <link rel="stylesheet" type="text/css" href="http://lambdadelta.info/css/xhtbl.css" />
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+ </head>
+ <body lang="en-US">
+ <div class="spacer">
+ <a href="http://lambdadelta.info/">
+ <img class="icon32" alt="[\lambda\delta home]" title="\lambda\delta home" src="http://lambdadelta.info/images/crux_32.png" />
+ </a>
+ </div>
+ <div class="head1">The Formal Systems of the λδ (\lambda\delta) Family</div>
+ <div class="spacer">
+ <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="spacer">
+ <br />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ <table cellpadding="4" cellspacing="0">
<tbody>
<tr>
- <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Version<br>
+ <td class="snns capitalize italic sky">
+ <a href="http://lambdadelta.info/index.html">home</a>
</td>
- <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Name<br>
+ <td class="snns capitalize italic magenta">
+ <a href="http://lambdadelta.info/news.html">news</a>
</td>
- <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Started<br>
+ <td class="snns capitalize italic white">
+ <a href="http://lambdadelta.info/specification.html">specification</a>
</td>
- <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Released<br>
+ <td class="snnn capitalize italic white">
+ <br />
</td>
- <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Dismissed<br>
+ <td class="snnn capitalize italic white">
+ <br />
+ </td>
+ <td class="snns capitalize italic orange">
+ <a href="http://lambdadelta.info/documentation.html">documentation</a>
+ </td>
+ <td class="snns capitalize italic green">
+ <a href="http://lambdadelta.info/implementation.html">implementation</a>
+ </td>
+ <td class="ssnn capitalize italic green">
+ <br />
</td>
</tr>
<tr>
- <td style="vertical-align: top; background-color: rgb(255, 223, 191);">2<br>
+ <td class="snns capitalize sky">
+ <a href="http://lambdadelta.info/index.html#foreword">foreword</a>
+ </td>
+ <td class="snns capitalize magenta">
+ <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
+ </td>
+ <td class="snns capitalize white">
+ <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 223, 191);">basic_2<br>
+ <td class="snnn capitalize white">(<a href="http://lambdadelta.info/ground_2.html">background</a> - <a href="http://lambdadelta.info/basic_2.html">core</a> - <a href="http://lambdadelta.info/apps_2.html">applications</a>)</td>
+ <td class="snnn capitalize white">
+ <br />
</td>
- <td style="vertical-align: top; background-color: rgb(255, 223, 191);">April
-2011<br>
+ <td class="snns capitalize orange">
+ <a href="http://lambdadelta.info/documentation.html#v2">version 2</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 223, 191);">planned
-in
-2014<br>
+ <td class="snns capitalize green">
+ <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 223, 191);">not
-planned yet<br>
+ <td class="ssnn capitalize green">
+ <br />
</td>
</tr>
<tr>
- <td style="vertical-align: top; background-color: rgb(255, 191, 191);">1<br>
+ <td class="snss capitalize sky">
+ <a href="http://lambdadelta.info/index.html#citations">citations</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 191, 191);">basic_1<br>
+ <td class="snss capitalize magenta">
+ <a href="http://lambdadelta.info/news.html#visibility">visibility</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 191, 191);">May
-2004<br>
+ <td class="snss capitalize white">
+ <a href="http://lambdadelta.info/specification.html#v1">version 1</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 191, 191);">November
-2006<br>
+ <td class="snsn capitalize white">(<a href="http://lambdadelta.info/ground_1.html">background</a> - <a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
+ <td class="snsn capitalize white">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
+ <td class="snss capitalize orange">
+ <a href="http://lambdadelta.info/documentation.html#v1">version 1</a>
</td>
- <td style="vertical-align: top; background-color: rgb(255, 191, 191);">May
-2008<br>
+ <td class="snss capitalize green">
+ <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
</td>
+ <td class="sssn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</td>
</tr>
</tbody>
</table>
- <br>
- <h3 style="text-align: right;">Notice
-for
-the
-Internet
-Explorer
-user <img style="width: 37px; height: 37px;" alt="[Butterfly]" title="Butterfly" src="images/b3.png"></h3>
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-<br>
-Last update 2014-02-24 by <a href="http://www.cs.unibo.it/%7Efguidi/">Ferruccio
-Guidi</a><br>
-</div>
-
-</body></html>
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="foreword">Foreword <img class="icon37" alt="[spacer]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b9.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ The formal systems of the λδ (\lambda\delta) family are typed λ-calculi aiming to support
+ the foundational frameworks for Mathematics that require an underlying specification language
+ (for example the <a href="http://www.math.unipd.it/~maietti/">Minimalist Foundation</a>
+ and its predecessors).
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ The λδ family is developed within the
+ <a href="http://helm.cs.unibo.it/">Hypertextual Electronic Library of Mathematics</a>
+ as a set of machine-checked digital specifications.
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ This is the family logo: <a href="http://lambdadelta.info/images/crux_177.png">crux_177.png</a>
+ (revised <span class="emph alpha">2012-09</span>).
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ <span class="emph alpha">Notice for the user of Internet Explorer.</span>
+ To view this site correctly, please select a font
+ with <a href="http://www.unicode.org/">Unicode</a> support.
+ For example "Lucida Sans Unicode" (it should be already installed on your system).
+ To change the current font follow:
+ "Tools" menu → "Internet Options" entry → "General" tab → "Fonts" button.
+ </div>
+ <!-- ===================================================================== -->
+ <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="citations">Citations <img class="icon37" alt="[spacer]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b9.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ This is a list of publications citing λδ documentation.
+ </div>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C7">
+ <li>
+ C. Dunchev, F. Guidi, C. Sacerdoti Coen, E. Tassi:
+ <span class="emph alpha">ELPI: Fast, Embeddable, λProlog Interpreter</span>
+ (2015). In proc. of LPAR 20. Lecture Notes in Computer Science, 9450, pp. 460-468. Springer.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C6">
+ <li>
+ A. Asperti, W. Ricciotti, C. Sacerdoti Coen, E. Tassi:
+ <span class="emph alpha">Formal metatheory of programming languages in the Matita interactive theorem prover</span>
+ (2012). In Journal of Automated Reasoning, 49(3), pp. 427-451. Springer.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C5">
+ <li>
+ M.E. Maietti:
+ <span class="emph alpha">Consistency of the minimalist foundation with Church thesis and Bar Induction</span>
+ (2012). Submitted article.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C4">
+ <li>
+ W. Ricciotti:
+ <span class="emph alpha">Theoretical and implementation aspects in the mechanization of the metatheory of programming languages</span>
+ (July 2011). Ph.D. Thesis in Computer Science, Technical Report UBLCS-2011-09, University of Bologna.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C3">
+ <li>
+ C.E. Brown:
+ <span class="emph alpha">Faithful Reproductions of the Automath Landau Formalization</span>
+ (2011). Technical report.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C2">
+ <li>
+ M.E. Maietti:
+ <span class="emph alpha">A minimalist two-level foundation for constructive mathematics</span>
+ (2009). In Annals of Pure and Applied Logic, 160(3), pp. 319-354. Elsevier.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="C1">
+ <li>
+ V. Rahili:
+ <span class="emph alpha">First Year Report: Realisability methods of proof and semantics with application to expansion</span>
+ (July 2007). Technical report.
+ </li>
+ </ul>
+ <!-- ===================================================================== -->
+ <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="disclaimer">Disclaimer <img class="icon37" alt="[spacer]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b9.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ The systens of the λδ family <span class="emph alpha">are not</span> related intentionally to
+ any other system having (variations of) the symbols λ and δ in its name or syntax.
+ Examples include (but are not limited to):
+ </div>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D1">
+ <li>
+ <span class="emph alpha">λ-δ</span> of
+ A. Church:
+ <span class="emph alpha">The calculi of lambda-conversion</span>
+ (1941).
+ Annals of Mathematics Studies 6.
+ Princeton University Press.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D2">
+ <li>
+ <span class="emph alpha">∆Λ</span> of
+ N.G. de Bruijn:
+ <span class="emph alpha">Generalizing Automath by means of a lambda-typed lambda calculus</span>
+ (1987).
+ In Lecture Notes in Pure and Applied Mathematics 106, pp. 71-92.
+ Marcel Dekker.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D3">
+ <li>
+ <span class="emph alpha">λ<sub>∆</sub>
+ </span> of
+ N.J. Rehof, M.H. Sørensen:
+ <span class="emph alpha">The λ<sub>∆</sub>-calculus</span>
+ (1994).
+ In Lecture Notes in Computer Science, 789, pp. 516–542.
+ Springer.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D4">
+ <li>
+ <span class="emph alpha">λ∆</span> of
+ S. Ronchi Della Rocca, L. Paolini:
+ <span class="emph alpha">The Parametric Lambda Calculus</span>
+ (2004).
+ Texts in Theoretical Computer Science, An EATCS Series.
+ Springer.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D5">
+ <li>
+ <span class="emph alpha">λD</span> of
+ R. Nederpelt, H. Geuvers:
+ <span class="emph alpha">Type Theory and Formal Proof</span>
+ (2014).
+ Cambridge University Press.
+ </li>
+ </ul>
+ <ul xmlns:ld="http://lambdadelta.info/" id="D6">
+ <li>
+ <span class="emph alpha">Cλξ</span> of
+ N.G. de Bruijn:
+ <span class="emph alpha">A namefree lambda calculus with facilities for internal definition of expressions and segments</span>
+ (1978).
+ TH-report 78-WSK-03.
+ Eindhoven University of Technology, Eindhoven.
+ </li>
+ </ul>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ <img class="icon32" alt="[Smiling face]" title="Smile!" src="http://lambdadelta.info/images/smile.png" />
+ Moreover, the systens of the λδ family <span class="emph alpha">are not</span> related intentionally to
+ <a href="http://umineko.wikia.com/wiki/Lambdadelta">Lady Lambdadelta</a>,
+ the Witch of Certainty of the sound novel
+ <a href="https://it.wikipedia.org/wiki/Umineko_no_naku_koro_ni">Umineko no Naku Koro ni</a>.
+ </div>
+ <div class="spacer">
+ <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="spacer">
+ <br />
+ </div>
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+ <img class="w3c" alt="[Generated from XML via XSL]" title="Generated from XML via XSL" src="http://lambdadelta.info/images/xml_xsl2.png" />
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+ <img class="w3c" alt="[PNG used here]" title="PNG used here" src="http://lambdadelta.info/images/PNGnow2.png" />
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+ <img class="w3c" alt="[Viewable with any browser]" title="Viewable with any browser" src="http://www.anybrowser.org/campaign/bvgraphics/abtfile.png" />
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+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="spacer">
+ <br />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Mon, 18 Apr 2016 20:35:45 +0200</div>
+ </body>
+</html>
projects / helm.git / blobdiff