Going Polymorphic - TH1 Reasoning for Leo-III

While interactive proof assistants for higher-order logic (HOL) commonly admit reasoning within rich type systems, current theorem provers for HOL are mainly based on simply typed λ-calculi and therefore do not allow such flexibility. In this paper, we present modifications to the higher-order automated theorem prover Leo-III for turning it into a reasoning system for rank-1 polymorphic HOL. To that end, a polymorphic version of HOL and a suitable paramodulation-based calculus are sketched. The implementation is evaluated using a set of polymorphic TPTP THF problems.

[1]  Andrei Popescu,et al.  Encoding Monomorphic and Polymorphic Types , 2013, TACAS.

[2]  Christoph Benzmüller,et al.  Higher-order semantics and extensionality , 2004, Journal of Symbolic Logic.

[3]  Harald Ganzinger,et al.  Rewrite-Based Equational Theorem Proving with Selection and Simplification , 1994, J. Log. Comput..

[4]  Norbert Völker.  HOL2P - A System of Classical Higher Order Logic with Second Order Polymorphism , 2007, TPHOLs.

[5]  Benjamin C. Pierce,et al.  Types and programming languages: the next generation , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[6]  Tanel Tammet,et al.  A Resolution Theorem Prover for Intuitonistic Logic , 1996, CADE.

[7]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[8]  Cezary Kaliszyk,et al.  HOL(y)Hammer: Online ATP Service for HOL Light , 2013, Math. Comput. Sci..

[9]  Gérard P. Huet,et al.  A Unification Algorithm for Typed lambda-Calculus , 1975, Theor. Comput. Sci..

[10]  Peter V. Homeier.  The HOL-Omega Logic , 2009, TPHOLs.

[11]  Thierry Coquand,et al.  A new paradox in type theory , 1995 .

[12]  Lawrence C. Paulson,et al.  The Higher-Order Prover Leo-II , 2015, Journal of Automated Reasoning.

[13]  Lawrence C. Paulson,et al.  Translating Higher-Order Clauses to First-Order Clauses , 2007, Journal of Automated Reasoning.

[14]  Christoph Benzmüller,et al.  Leo-III Version 1.1 (System description) , 2017, LPAR.

[15]  Jerzy Tiuryn,et al.  Alpha-Conversion and Typability , 1999, Inf. Comput..

[16]  Stephan Schulz,et al.  E - a brainiac theorem prover , 2002, AI Commun..

[17]  Lawrence Charles Paulson,et al.  Isabelle/HOL: A Proof Assistant for Higher-Order Logic , 2002 .

[18]  Martín Abadi,et al.  Explicit substitutions , 1989, POPL '90.

[19]  Daniel Gallin,et al.  Intensional and Higher-Order Modal Logic, With Applications to Montague Semantics , 1975 .

[20]  Christoph Benzmüller,et al.  Extensional Higher-Order Paramodulation and RUE-Resolution , 1999, CADE.

[21]  Albert Rubio,et al.  Paramodulation-Based Theorem Proving , 2001, Handbook of Automated Reasoning.

[22]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[23]  Geoff Sutcliffe,et al.  Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure , 2010, J. Formaliz. Reason..

[24]  Chad E. Brown,et al.  Satallax: An Automatic Higher-Order Prover , 2012, IJCAR.

[25]  de Ng Dick Bruijn,et al.  Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem , 1972 .

[26]  Hendrik Pieter Barendregt,et al.  Introduction to generalized type systems , 1991, Journal of Functional Programming.

[27]  Cezary Kaliszyk,et al.  TH1: The TPTP Typed Higher-Order Form with Rank-1 Polymorphism , 2016, PAAR@IJCAR.

[28]  M.J.C. Gordon,et al.  The HOL Logic and System , 1994 .

[29]  Frank Pfenning,et al.  A Linear Spine Calculus , 2003, J. Log. Comput..

[30]  John Harrison,et al.  HOL Light: An Overview , 2009, TPHOLs.

[31]  J. Roger Hindley,et al.  Lambda calculus with types (Perspectives in Logic) , 2014 .

[32]  Christoph Benzmüller,et al.  LeoPARD - A Generic Platform for the Implementation of Higher-Order Reasoners , 2015, CICM.

[33]  Dale Miller,et al.  Automation of Higher-Order Logic , 2014, Computational Logic.

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

[35]  Thomas F. Melham.  The HOL logic extended with quantification over type variables , 1993, Formal Methods Syst. Des..

[36]  Andrei Paskevich,et al.  TFF1: The TPTP Typed First-Order Form with Rank-1 Polymorphism , 2013, CADE.