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14 <h1>The Formal System λδ (\lambda\delta)<br>
16 <h2>Towards the unification of terms, types, environments and contexts</h2>
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37 <h3 style="text-align: right;">Foreword <img style="width: 37px; height: 37px;" alt="[Butterfly]" title="Butterfly" src="images/b9.png"></h3>
39 (\lambda\delta) is a typed lambda calculus that pursues the static and
40 dynamic unification of terms, types, environments and contexts while
41 enjoying a well-conceived theory, which includes the commonly
42 desired properties.<br>
44 λδ takes some features from the calculi of the Automath family and
45 some from the pure type systems, but it differs from both in that it
46 does not include the Π construction while it provides for an
47 abbreviation mechanism at the level of terms.<br>
49 λδ features explicit type annotations, which are borrowed from
50 realistic type checker implementations and with which type checking is
51 reduced to type inference.<br>
53 The reduction steps of λδ include β-contraction, δ-expansion,
54 ζ-contraction and θ-swap. On the other hand,
55 η-contraction is not included.<br>
57 The theory of λδ includes important properties such as the
58 confluence of reduction, the correctness of types, the
59 uniqueness of types up to conversion, the subject reduction of the type
60 assignment, the strong normalization of the typed terms. The
61 decidability of type inference and of type checking come as corollaries.<br>
63 λδ features uniformly dependent types and a predicative abstraction
64 mechanism, so the calculus can serve as a formal specification language
65 for the type theories that need a predicative foundation. λδ is
66 expected to have the expressive power of λ→.<br>
68 λδ comes in several versions listed in the following table, which
69 includes the major milestones:<br>
71 <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="0">
74 <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Version<br>
76 <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Name<br>
78 <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Started<br>
80 <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Released<br>
82 <td style="vertical-align: top; background-color: rgb(223, 223, 223); text-align: center;">Dismissed<br>
86 <td style="vertical-align: top; background-color: rgb(255, 223, 191);">2<br>
88 <td style="vertical-align: top; background-color: rgb(255, 223, 191);">basic_2<br>
90 <td style="vertical-align: top; background-color: rgb(255, 223, 191);">April
93 <td style="vertical-align: top; background-color: rgb(255, 223, 191);">planned
97 <td style="vertical-align: top; background-color: rgb(255, 223, 191);">not
102 <td style="vertical-align: top; background-color: rgb(255, 191, 191);">1<br>
104 <td style="vertical-align: top; background-color: rgb(255, 191, 191);">basic_1<br>
106 <td style="vertical-align: top; background-color: rgb(255, 191, 191);">May
109 <td style="vertical-align: top; background-color: rgb(255, 191, 191);">November
112 <td style="vertical-align: top; background-color: rgb(255, 191, 191);">May
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141 Last update 2012-12-02 by <a href="http://www.cs.unibo.it/%7Efguidi/">Ferruccio