ABSTRACT
We describe the theoretical underpinnings to support the construction of an extension to the Isabelle/HOL theorem prover to support the creation of datatypes for weak higher-order abstract syntax, and give an example of its application. This theoretical basis is centered around the concept of variable types (i.e. types whose elements are variables), and the concept of two terms in a given type having the "same structure" up to a given set of substitutions (the difference set) of one variable for another as allowed by that set. We provide an axiomatization of types for which the notion of having the same structure is well-behaved with the axiomatic class of same_struct_thy. We show that being a same_struct_thy is preserved by products, sums and certain function spaces.
Within a same_struct_thy, not all terms necessarily have the same structure as anything, including themselves. Those terms having the same structure as themselves relative to the empty difference set are said to be proper. A proper function from variables to terms corresponds to an abstraction of a variable in a term and also corresponds to substitution of variables for that variable in the term. Proper functions form the basis for a formalization of weak higher-order abstract syntax.
- Simon Ambler, Roy L. Crole, and Alberto Momigliano. Combining higher order abstract syntax with tactical theorem proving and (co)induction. In TPHOLs '02: Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics, pages 13--30, 2002. Google ScholarDigital Library
- Brian E. Aydemir, Aaron Bohannon, Matthew Fairbairn, J. Nathan Foster, Benjamin C. Pierce, Peter Sewell, Dimitrios Vytiniotis, Geoffrey Washburn, Stephanie Weirich, and Steve Zdancewic. Mechanized metatheory for the masses: The PoplMark Challenge. In Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science, pages 50--65. Springer, August 2005. Google ScholarDigital Library
- Anna Bucalo, Furio Honsell, Marino Miculan, Ivan Scagnetto, and Martin Hofmann. Consistency of the theory of contexts. J. Funct. Program., 16(3):327--372, 2006. Google ScholarDigital Library
- Joëlle Despeyroux, Amy Felty, and André Hirschowitz. Higher-order abstract syntax in Coq. In M. Dezani-Ciancaglini and G. Plotkin, editors, Proceedings of the International Conference on Typed Lambda Calculi and Applications, volume 902 of LNCS, pages 124--138, Edinburgh, Scotland, 1995. Springer-Verlag. Google ScholarDigital Library
- Marcelo Fiore, Gordon Plotkin, and Daniele Turi. Abstract syntax and variable binding (extended abstract). In Proc. 14th LICS Conf., pages 193--202. IEEE, Computer Society Press, 1999. Google ScholarDigital Library
- Murdoch James Gabbay and Martin Hofmann. Nominal renaming sets. In LPAR, pages 158--173, 2008. Google ScholarDigital Library
- Andrew Gacek. The Abella interactive theorem prover (system description). In A. Armando, P. Baumgartner, and G. Dowek, editors, Proceedings of IJCAR 2008, volume 5195 of Lecture Notes in Artificial Intelligence, pages 154--161. Springer, August 2008. Google ScholarDigital Library
- Martin Hofmann. Semantical analysis of higher-order abstract syntax. In LICS '99: Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, page 204, 1999. Google ScholarDigital Library
- Furio Honsell, Marino Miculan, and Ivan Scagnetto. An axiomatic approach to metareasoning on nominal algebras in HOAS. In Leeuwen (Eds.), 28th International Colloquium on Automata, Languages and Programming, ICALP 2001, pages 963--978, 2001. Google ScholarDigital Library
- Alberto Momigliano, Alan J. Martin, and Amy P. Felty. Two-level hybrid: A system for reasoning using higher-order abstract syntax. Electron. Notes Theor. Comput. Sci., 196:85--93, 2008. Google ScholarDigital Library
- T. Nipkow, L. C. Paulson, and M. Wenzel. Isabelle/HOL - A Proof Assistant for Higher-Order Logic. Springer-Verlag, 2002. Google ScholarDigital Library
- Frank Pfenning and Carsten Schürmann. System description: Twelf - a meta-logical framework for deductive systems. In Proceedings of the 16th International Conference on Automated Deduction (CADE-16, pages 202--206. Springer-Verlag LNAI, 1999. Google ScholarDigital Library
- Andrew M. Pitts. Alpha-structural recursion and induction. Journal of the ACM, 53:459--506, 2006. Google ScholarDigital Library
- Christine Röckl and Daniel Hirschkoff. A fully adequate shallow embedding of the {pi}-calculus in Isabelle/HOL with mechanized syntax analysis. J. Funct. Program., 13(2):415--451, 2003. Google ScholarDigital Library
- Yong Sun. An algebraic generalization of Frege structures - binding algebras. Theor. Comput. Sci., 211(1--2):189--232, 1999. Google ScholarDigital Library
- Christian Urban, Julien Narboux, and Stefan Berghofer. The Nominal Datatype Package, 2007.Google Scholar
- M Wenzel. Type classes and overloading in higher-order logic. In Elsa L. Gunter and Amy Felty, editors, Theorem Proving in Higher Order Logics, TPHOLs'97, volume 1275 of LNCS, pages 307--322, Murray Hill, NJ, USA, August 1997. Springer. Google ScholarDigital Library
- Theory support for weak higher order abstract syntax in Isabelle/HOL
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