Polymorphic Higher-order Termination

We generalise the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism. Instead of using weakly monotonic functionals, we interpret terms in a suitable extension of System F-omega. This enables a direct interpretation of rewrite rules which make essential use of impredicative polymorphism. In addition, our generalisation eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types. As an illustration of the potential of our method, we prove termination of a substantial fragment of full intuitionistic second-order propositional logic with permutative conversions.

[1]  Cynthia Kop,et al.  Dynamic Dependency Pairs for Algebraic Functional Systems , 2012, Log. Methods Comput. Sci..

[2]  Helmut Schwichtenberg,et al.  Strict Functionals for Termination Proofs , 1995, TLCA.

[3]  Jaco van de Pol,et al.  Termination Proofs for Higher-order Rewrite Systems , 1993, HOA.

[4]  Makoto Hamana,et al.  Polymorphic Rewrite Rules: Confluence, Type Inference, and Instance Validation , 2018, FLOPS.

[5]  Pawel Urzyczyn,et al.  A Syntactic Embedding of Predicate Logic into Second-Order Propositional Logic , 2010, Notre Dame J. Formal Log..

[6]  Makoto Tatsuta.  Simple Saturated Sets for Disjunction and Second-Order Existential Quantification , 2007, TLCA.

[7]  Enno Ohlebusch,et al.  Term Rewriting Systems , 2002 .

[8]  Terese.  Term rewriting systems , 2003, Cambridge tracts in theoretical computer science.

[9]  Albert Rubio,et al.  Polymorphic higher-order recursive path orderings , 2007, JACM.

[10]  Gilles Dowek.  Models and termination of proof-reduction in the $λ$$Π$-calculus modulo theory , 2015, ArXiv.

[11]  Frédéric Blanqui,et al.  Definitions by rewriting in the calculus of constructions , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[12]  J. Girard.  Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des Types , 1971 .

[13]  Cynthia Kop.  Higher Order Termination: Automatable Techniques for Proving Termination of Higher-Order Term Rewriting Systems , 2012 .

[14]  David Wahlstedt.  Type Theory with First-Order Data Types and Size-Change Termination , 2004 .

[15]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[16]  Aleksander Wojdyga.  Short Proofs of Strong Normalization , 2008, MFCS.

[17]  Denis Cousineau,et al.  Embedding Pure Type Systems in the Lambda-Pi-Calculus Modulo , 2007, TLCA.

[18]  Daria Walukiewicz-Chrzaszcz,et al.  Termination of rewriting in the Calculus of Constructions , 2003, Journal of Functional Programming.

[19]  Carsten Fuhs,et al.  Polynomial Interpretations for Higher-Order Rewriting , 2012, RTA.

[20]  Alexandra Silva,et al.  Practical coinduction , 2016, Mathematical Structures in Computer Science.

[21]  Nishida Naoki,et al.  Argument Filterings and Usable Rules in Higher-Order Rewrite Systems , 2009 .

[22]  D. Sangiorgi.  Introduction to Bisimulation and Coinduction , 2011 .

[23]  J. Girard,et al.  Proofs and types , 1989 .

[24]  M. Sørensen,et al.  Lectures on the Curry-Howard Isomorphism , 2013 .

[25]  Bart Jacobs,et al.  An introduction to (co)algebra and (co)induction , 2011, Advanced Topics in Bisimulation and Coinduction.

[26]  Makoto Hamana,et al.  Multiversal Polymorphic Algebraic Theories , 2013 .

[27]  J. C. van dePol.  Termination of higher-order rewrite systems , 1996 .