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+ Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ A Bi-directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions
+
+
+ .pdf
+
+
+ Published in Logical Methods in Computer Science, volume 8, Pages 1-49, 2012. DOI:10.2168/LMCS-8(1:18)2012. ISSN: 1860-5974.
+
+
+ The paper describes the refinement algorithm for the Calculus of (Co)Inductive
+ Constructions (CIC) implemented in the interactive theorem prover Matita. The
+ refinement algorithm is in charge of giving a meaning to the terms, types and
+ proof terms directly written by the user or generated by using tactics,
+ decision procedures or general automation. The terms are written in an
+ âexternal syntaxâ meant to be user friendly that allows omission of
+ information, untyped binders and a certain liberal use of user defined
+ sub-typing. The refiner modifies the terms to obtain related well typed terms
+ in the internal syntax understood by the kernel of the ITP. In particular, it
+ acts as a type inference algorithm when all the binders are untyped. The
+ proposed algorithm is bi-directional: given a term in external syntax and a
+ type expected for the term, it propagates as much typing information as
+ possible towards the leaves of the term. Traditional mono-directional
+ algorithms, instead, proceed in a bottom- up way by inferring the type of a
+ sub-term and comparing (unifying) it with the type expected by its context only
+ at the end. We propose some novel bi-directional rules for CIC that are
+ particularly effective. Among the benefits of bi-directionality we have better
+ error message reporting and better inference of dependent types. Moreover,
+ thanks to bi-directionality, the coercion system for sub-typing is more
+ effective and type inference generates simpler unification problems that are
+ more likely to be solved by the inherently incomplete higher order unification
+ algorithms implemented. Finally we introduce in the external syntax the notion
+ of vector of placeholders that enables to omit at once an arbitrary number of
+ arguments. Vectors of placeholders allow a trivial implementation of implicit
+ arguments and greatly simplify the implementation of primitive and simple
+ tactics.
+
+
+
+ -
+
+ Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ The Matita Interactive Theorem Prover
+
+
+ .pdf
+
+
+ In Automated Deduction â CADE-23, LECTURE NOTES IN COMPUTER SCIENCE,
+ ISBN:978-3-642-22437-9, Pages 64-69, Volume 6803, Year 2011 64-69.
+
+
+
+ -
+
+ Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ Nonuniform Coercions via Unification Hints
+
+
+ .pdf
+
+
+ In Proceedings Types for Proofs and Programs, Revised Selected Papers,
+ ELECTRONIC PROCEEDINGS IN THEORETICAL COMPUTER SCIENCE (EPTCS),
+ ISSN:2075-2180, DOI:10.4204/EPTCS.53.2, vol. 53, 19-26, March 2011.
+
+
+ We introduce the notion of nonuniform coercion, which is the promotion of a
+ value of one type to an enriched value of a different type via a nonuniform
+ procedure. Nonuniform coercions are a generalization of the (uniform) coercions
+ known in the literature and they arise naturally when formalizing mathematics in
+ an higher order interactive theorem prover using convenient devices like
+ canonical structures, type classes or unification hints. We also show how
+ nonuniform coercions can be naturally implemented at the user level in an
+ interactive theorem prover that allows unification hints.
+
+
+
+ -
+
+ Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ Hints in Unification
+
+
+ .pdf
+
+
+ In Proceedings of Theorem Proving in Higher Order Logics (TPHOLs 2009),
+ Munich, August 17-20, 2009. LECTURE NOTES IN COMPUTER SCIENCE,
+ ISBN:978-3-642-03358-2, vol. 5674. 84-98.
+
+
+ Several mechanisms such as Canonical Structures, Type Classes, or
+ Pullbacks have been recently introduced with the aim to improve the power
+ and flexibility of the type inference algorithm for interactive theorem
+ provers. We claim that all these mechanisms are particular instances of a
+ simpler and more general technique, just consisting in providing suitable
+ hints to the unification procedure underlying type inference. This allows
+ a simple, modular and not intrusive implementation of all the above
+ mentioned techniques, opening at the same time innovative and unexpected
+ perspectives on its possible applications.
+
+
+
+ -
+
+ Andrea Asperti, Enrico Tassi
+
+
+ Smart Matching
+
+
+ .pdf
+
+
+ Published in the proceedings of MKM2010, LNCS, Volume 6167/2010, Year 2010,
+ Pages 263-277, DOI 10.1007/978-3-642-14128-7_23, ISBN 978-3-642-14127-0.
+
+
+ One of the most annoying aspects in the formalization of
+ mathematics is the need of transforming notions to match a given, existing
+ result. This kind of transformations, often based on a conspicuous background
+ knowledge in the given scientific domain (mostly expressed in the form of
+ equalities or isomorphisms), are usually implicit in the mathematical
+ discourse, and it would be highly desirable to obtain a similar behavior in
+ interactive provers. The paper describes the superposition-based implementation
+ of this feature inside the Matita interactive theorem prover, focusing in
+ particular on the so called smart application tactic, supporting smart matching
+ between a goal and a given result.
+
+
+
+ -
+
+ Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ A New Type For Tactics
+
+
+ .pdf
+
+
+ To appear in the proceedings of ACM SIGSAM PLMMS 2009, ISBN 978-1-60558-735-6.
+ Published as technical report UBLCS-2009-14.
+
+
+ The type of tactics in all (procedural) proof assistants is (a variant of)
+ the one introduced in LCF. We discuss why this is inconvenient and we
+ propose
+ a new type for tactics that 1) allows the implementation of more clever
+ tactics; 2) improves the implementation of declarative languages on top
+ of procedural ones; 3) allows for better proof structuring; 4) improves
+ proof automation; 5) allows tactics to rearrange and delay the goals to be
+ proved (e.g. in case of side conditions or PVS subtyping judgements).
+
+
+
+ -
+
+ Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi
+
+
+ A Compact Kernel for the Calculus of Inductive Constructions
+
+
+ .pdf
+
+
+ Published in the Journal Sadhana,
+
+
+ The paper describes the new kernel for the Calculus of Inductive
+ Constructions (CIC) implemented inside the Matita Interactive Theorem Prover.
+ The design of the new kernel has been completely revisited since
+ the first release, resulting in a remarkably compact implementation
+ of about 2300 lines of OCaml code. The work is meant for people
+ interested in implementation aspects of Interactive Provers, and
+ is not self contained. In particular, it requires good
+ acquaintance with Type Theory and functional programming languages.
+
+
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