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-<h1>The Formal System λδ (lambdadelta)<br>
-</h1>
-<h2>Towards the unification of terms, types, environments and contexts</h2>
-<img style="width: 95%; height: 4px;" alt="[Separator]"
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- 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
-(lambdadelta) is a typed lambda calculus that pursues the static and
-dynamic unification of terms, types, environments and contexts while
-enjoying a well-conceived meta-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 meta-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"
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+ <div class="head1">The Formal System λδ (\lambda\delta)</div>
+ <div class="spacer">
+ <img class="rule" alt="[Spacer]" title="lambdadelta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
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+ <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 orange">
+ <a href="http://lambdadelta.info/documentation.html">documentation</a>
</td>
- <td
- style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Released<br>
+ <td class="snns capitalize italic green">
+ <a href="http://lambdadelta.info/specification.html">specification</a>
</td>
- <td
- style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Dismissed<br>
+ <td class="snnn capitalize italic green">
+ <br />
+ </td>
+ <td class="snnn capitalize italic green">
+ <br />
+ </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
- style="vertical-align: top; background-color: rgb(255, 223, 191);">basic_2<br>
+ <td class="snns capitalize magenta">
+ <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
</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
-2013<br>
+ <td class="snns capitalize green">
+ <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
</td>
- <td
- style="vertical-align: top; background-color: rgb(255, 223, 191);">not
-planned yet<br>
+ <td class="snnn capitalize green">(<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 green">
+ <br />
</td>
+ <td class="snns capitalize green">
+ <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
+ </td>
+ <td class="ssnn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</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 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);">basic_1<br>
+ <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
-2004<br>
+ <td class="snss capitalize green">
+ <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 green">(<a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
+ <td class="snsn capitalize green">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
+ <td class="snss capitalize green">
+ <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
</td>
- <td
- style="vertical-align: top; background-color: rgb(255, 191, 191);">May
-2008<br>
+ <td class="sssn capitalize green">
+ <br />
</td>
</tr>
</tbody>
</table>
- <br>
- <h3 style="text-align: right;">Notice
-for
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-Guidi</a><br>
-</div>
-</body>
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="foreword">Foreword <img class="icon37" alt="[spacer]" title="lambdadelta butterfly" src="http://lambdadelta.info/images/b9.png" />
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ The formal system λδ (\lambda\delta) is a typed λ-calculus aiming to support
+ the foundations of Mathematics that require an underlying specification language
+ (for example the <a href="http://www.math.unipd.it/~maietti/">Minimal Type Theory</a>
+ and its predecessors).
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ λδ is developed in the context of the
+ <a href="http://helm.cs.unibo.it/">Hypertextual Electronic Library of Mathematics</a>
+ as a machine-checked digital specification
+ that is not the formal counterpart of previous informal material.
+ </div>
+ <div xmlns:ld="http://lambdadelta.info/" class="text">
+ This is the System 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
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+ </div>
+ <!-- ===================================================================== -->
+ <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="citations">Citations <img class="icon37" alt="[spacer]" title="lambdadelta 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 (not including our own).
+ </div>
+ <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 JAR 49(3), pp. 427-451.
+ </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). Typescript note.
+ </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 APAL 160(3), pp. 319-354.
+ </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). Typescript note.
+ </li>
+ </ul>
+ <div class="spacer">
+ <img class="rule" alt="[Spacer]" title="lambdadelta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
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+ <div xmlns:ld="http://lambdadelta.info/" class="spacer">Last update: Mon, 19 Jan 2015 23:52:51 +0100</div>
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