-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.
-
-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 Î construction while it provides for an
-abbreviation mechanism at the level of terms.
-
-λδ features explicit type annotations, which are borrowed from
-realistic type checker implementations and with which type checking is
-reduced to type inference.
-
-The reduction steps of λδ include β-contraction, δ-expansion,
-ζ-contraction and θ-swap. On the other hand,
-η-contraction is not included.
-
-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.
-
-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 λâ.
-
-λδ comes in several versions listed in the following table, which
-includes the major milestones:
-
-
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-
-
+ 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 Minimalist Foundation
+ and its predecessors).
+
+ This is the family logo: crux_177.png
+ (revised 2012-09).
+
+
+ Notice for the user of Internet Explorer.
+ To view this site correctly, please select a font
+ with Unicode support.
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+
+
+
Citations
+
+
+ This is a list of publications citing λδ documentation.
+
+
+
+ C. Dunchev, F. Guidi, C. Sacerdoti Coen, E. Tassi:
+ ELPI: Fast, Embeddable, λProlog Interpreter
+ (2015). In proc. of LPAR 20. Lecture Notes in Computer Science, 9450, pp. 460-468. Springer.
+
+
+
+
+ A. Asperti, W. Ricciotti, C. Sacerdoti Coen, E. Tassi:
+ Formal metatheory of programming languages in the Matita interactive theorem prover
+ (2012). In Journal of Automated Reasoning, 49(3), pp. 427-451. Springer.
+
+
+
+
+ M.E. Maietti:
+ Consistency of the minimalist foundation with Church thesis and Bar Induction
+ (2012). Submitted article.
+
+
+
+
+ W. Ricciotti:
+ Theoretical and implementation aspects in the mechanization of the metatheory of programming languages
+ (July 2011). Ph.D. Thesis in Computer Science, Technical Report UBLCS-2011-09, University of Bologna.
+
+
+
+
+ C.E. Brown:
+ Faithful Reproductions of the Automath Landau Formalization
+ (2011). Technical report.
+
+
+
+
+ M.E. Maietti:
+ A minimalist two-level foundation for constructive mathematics
+ (2009). In Annals of Pure and Applied Logic, 160(3), pp. 319-354. Elsevier.
+
+
+
+
+ V. Rahili:
+ First Year Report: Realisability methods of proof and semantics with application to expansion
+ (July 2007). Technical report.
+
+
+
+
Disclaimer
+
+
+ The systens of the λδ family are not 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):
+
+
+
+ λ-δ of
+ A. Church:
+ The calculi of lambda-conversion
+ (1941).
+ Annals of Mathematics Studies 6.
+ Princeton University Press.
+
+
+
+
+ âÎ of
+ N.G. de Bruijn:
+ Generalizing Automath by means of a lambda-typed lambda calculus
+ (1987).
+ In Lecture Notes in Pure and Applied Mathematics 106, pp. 71-92.
+ Marcel Dekker.
+
+
+
+
+ λâ
+ of
+ N.J. Rehof, M.H. Sørensen:
+ The λâ-calculus
+ (1994).
+ In Lecture Notes in Computer Science, 789, pp. 516â542.
+ Springer.
+
+
+
+
+ λâ of
+ S. Ronchi Della Rocca, L. Paolini:
+ The Parametric Lambda Calculus
+ (2004).
+ Texts in Theoretical Computer Science, An EATCS Series.
+ Springer.
+
+
+
+
+ λD of
+ R. Nederpelt, H. Geuvers:
+ Type Theory and Formal Proof
+ (2014).
+ Cambridge University Press.
+
+
+
+
+ Cλξ of
+ N.G. de Bruijn:
+ A namefree lambda calculus with facilities for internal definition of expressions and segments
+ (1978).
+ TH-report 78-WSK-03.
+ Eindhoven University of Technology, Eindhoven.
+
+
+
+
+ Moreover, the systens of the λδ family are not related intentionally to
+ Lady Lambdadelta,
+ the Witch of Certainty of the sound novel
+ Umineko no Naku Koro ni.
+
![[Crux Logo]](images/crux_32.png "The Crux")
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![[\lambda\delta home]](http://lambdadelta.info/images/crux_32.png "\lambda\delta home")
+ ![[Spacer]](http://lambdadelta.info/images/rainbow.png "\lambda\delta rainbow rule")
+ ![[Butterfly]](images/b3.png "Butterfly")
![[Use Any Browser Here]](images/globe_trans.png "Use Any Browser Here")
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![[spacer]](http://lambdadelta.info/images/b9.png "\lambda\delta butterfly")
+ ![[Smiling face]](http://lambdadelta.info/images/smile.png "Smile!")
+ Moreover, the systens of the λδ family ![[Generated from XML via XSL]](http://lambdadelta.info/images/xml_xsl2.png "Generated from XML via XSL")
+ ![[PNG used here]](http://lambdadelta.info/images/PNGnow2.png "PNG used here")
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+ 