Typed Lambda Calculi and Applications
Volume 902 of the series Lecture Notes in Computer Science pp 124138
Higherorder abstract syntax in Coq
 Joëlle DespeyrouxAffiliated withSophiaAntipolis, INRIA
 , Amy FeltyAffiliated withAT&T Bell Laboratories
 , André HirschowitzAffiliated withCNRS URA 168, University of Nice
Abstract
The terms of the simplytyped λcalculus can be used to express the higherorder abstract syntax of objects such as logical formulas, proofs, and programs. Support for the manipulation of such objects is provided in several programming languages (e.g. λProlog, Elf). Such languages also provide embedded implication, a tool which is widely used for expressing hypothetical judgments in natural deduction. In this paper, we show how a restricted form of secondorder syntax and embedded implication can be used together with induction in the Coq Proof Development system. We specify typing rules and evaluation for a simple functional language containing only function abstraction and application, and we fully formalize a proof of type soundness in the system. One difficulty we encountered is that expressing the higherorder syntax of an objectlanguage as an inductive type in Coq generates a class of terms that contains more than just those that directly represent objects in the language. We overcome this difficulty by defining a predicate in Coq that holds only for those terms that correspond to programs. We use this predicate to express and prove the adequacy for our syntax.
 Title
 Higherorder abstract syntax in Coq
 Book Title
 Typed Lambda Calculi and Applications
 Book Subtitle
 Second International Conference on Typed Lambda Calculi and Applications, TLCA '95 Edinburgh, United Kingdom, April 10–12, 1995 Proceedings
 Pages
 pp 124138
 Copyright
 1995
 DOI
 10.1007/BFb0014049
 Print ISBN
 9783540590484
 Online ISBN
 9783540491781
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 902
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Joëlle Despeyroux ^{(1)}
 Amy Felty ^{(2)}
 André Hirschowitz ^{(3)}
 Author Affiliations

 1. SophiaAntipolis, INRIA, 2004 Route des Lucioles, F06565, Valbonne Cedex, France
 2. AT&T Bell Laboratories, 600 Mountain Ave., 07974, Murray Hill, NJ, USA
 3. CNRS URA 168, University of Nice, 06108, Nice Cedex 2, France
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