Importing SMT and Connection proofs as expansion trees

Different automated theorem provers reason in various deductive systems and, thus, produce proof objects which are in general not compatible. To understand and analyze these objects, one needs to study the corresponding proof theory, and then study the language used to represent proofs, on a prover by prover basis. In this work we present an implementation that takes SMT and Connection proof objects from two different provers and imports them both as expansion trees. By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression , visualization, sequent calculus proof construction, proof checking, etc.) can be employed uniformly. The expansion proofs can also be used as a validation tool for the proof objects produced.

[1]  Geoff Sutcliffe The TPTP Problem Library and Associated Infrastructure , 2009, Journal of Automated Reasoning.

[2]  Gopalan Nadathur,et al.  Nominal abstraction , 2011, Inf. Comput..

[3]  Tobias Nipkow,et al.  The Isabelle Framework , 2008, TPHOLs.

[4]  Christoph Benzmüller Automating Quantified Conditional Logics in HOL , 2013, IJCAI.

[5]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[6]  M. Gordon,et al.  Introduction to HOL: a theorem proving environment for higher order logic , 1993 .

[7]  Cezary Kaliszyk,et al.  Certified Connection Tableaux Proofs for HOL Light and TPTP , 2014, CPP.

[8]  Dale Miller A Proposal for Broad Spectrum Proof Certificates , 2011, CPP.

[9]  Benjamin Grégoire,et al.  A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses , 2011, CPP.

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

[11]  Damien Doligez,et al.  Zenon : An Extensible Automated Theorem Prover Producing Checkable Proofs , 2007, LPAR.

[12]  Tobias Nipkow,et al.  A Revision of the Proof of the Kepler Conjecture , 2009, Discret. Comput. Geom..

[13]  Stephan Schulz,et al.  System Description: E 1.8 , 2013, LPAR.

[14]  Michael D. Ernst,et al.  Computer Aided Verification , 2016, Lecture Notes in Computer Science.

[15]  Frank Pfenning,et al.  System Description: Twelf - A Meta-Logical Framework for Deductive Systems , 1999, CADE.

[16]  Lawrence C. Paulson,et al.  Quantified Multimodal Logics in Simple Type Theory , 2009, Logica Universalis.

[17]  Dale Miller PROOFCERT – Broad Spectrum Proof Certificates – ERC , 2017 .

[18]  Damien Doligez,et al.  Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps , 2013 .

[19]  Jean-Christophe Filliâtre,et al.  Why3 - Where Programs Meet Provers , 2013, ESOP.

[20]  Claude Kirchner,et al.  Theorem Proving Modulo , 2003, Journal of Automated Reasoning.

[21]  Hendrik Pieter Barendregt,et al.  Autarkic Computations in Formal Proofs , 2002, Journal of Automated Reasoning.

[22]  Michael Norrish,et al.  seL4: formal verification of an OS kernel , 2009, SOSP '09.

[23]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[24]  Gopalan Nadathur,et al.  Abella: A System for Reasoning about Relational Specifications , 2014, J. Formaliz. Reason..

[25]  Florian Rabe,et al.  Representing Model Theory in a Type-Theoretical Logical Framework , 2009, LSFA.

[26]  Dale Miller,et al.  Proof and refutation in MALL as a game , 2010, Ann. Pure Appl. Log..

[27]  Dale Miller,et al.  Unification of Simply Typed Lamda-Terms as Logic Programming , 1991, ICLP.

[28]  Bruno Woltzenlogel Paleo,et al.  Gödel's God in Isabelle/HOL , 2013, Arch. Formal Proofs.

[29]  J. Hurd First-Order Proof Tactics in Higher-Order Logic Theorem Provers In Proc , 2003 .

[30]  Alexander Leitsch,et al.  CERES for first-order schemata , 2013, J. Log. Comput..

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

[32]  Geoff Sutcliffe The TPTP Problem Library and Associated Infrastructure , 2017, Journal of Automated Reasoning.

[33]  Rob Arthan HOL Constant Definition Done Right , 2014, ITP.

[34]  Steven Obua,et al.  Importing HOL into Isabelle/HOL , 2006, IJCAR.

[35]  Herman Geuvers,et al.  Some logical and syntactical observations concerning the first-order dependent type system lambda-P , 1999, Mathematical Structures in Computer Science.

[36]  Program FOUNDATIONS OF CONSTRUCTIVE MATHEMATICS , 2014 .

[37]  Joe Hurd,et al.  The OpenTheory Standard Theory Library , 2011, NASA Formal Methods.

[38]  Christoph Weidenbach,et al.  SPASS Version 3.5 , 2009, CADE.

[39]  Andrei Voronkov,et al.  First-Order Theorem Proving and Vampire , 2013, CAV.

[40]  Philip B. Clayton,et al.  CLawZ: cost-effective formal verification for control systems , 2005, 24th Digital Avionics Systems Conference.

[41]  Ulrich Furbach,et al.  Proceedings of the Third international joint conference on Automated Reasoning , 2006 .

[42]  Davide Sangiorgi,et al.  Enhancements of the bisimulation proof method , 2012, Advanced Topics in Bisimulation and Coinduction.

[43]  Jean-Raymond Abrial,et al.  The B-book - assigning programs to meanings , 1996 .

[44]  Ronan Saillard Towards explicit rewrite rules in the λΠ-calculus modulo , 2013 .

[45]  Olivier Hermant,et al.  The λΠ-calculus Modulo as a Universal Proof Language , 2012, PxTP.

[46]  G. Gentzen Untersuchungen über das logische Schließen. I , 1935 .

[47]  Guillaume Burel Experimenting with Deduction Modulo , 2011, CADE.

[48]  Cezary Kaliszyk,et al.  Scalable LCF-Style Proof Translation , 2013, ITP.

[49]  Dale A. Miller,et al.  A compact representation of proofs , 1987, Stud Logica.

[50]  Lawrence C. Paulson,et al.  Logic and computation - interactive proof with Cambridge LCF , 1987, Cambridge tracts in theoretical computer science.

[51]  Jasmin Christian Blanchette,et al.  Three years of experience with Sledgehammer, a Practical Link Between Automatic and Interactive Theorem Provers , 2012, IWIL@LPAR.

[52]  Florian Rabe Representing Isabelle in LF , 2010, LFMTP.

[53]  Thomas C. Hales,et al.  The Jordan Curve Theorem, Formally and Informally , 2007, Am. Math. Mon..

[54]  Benjamin Werner,et al.  Importing HOL Light into Coq , 2010, ITP.

[55]  Alexander Leitsch,et al.  PROOFTOOL: a GUI for the GAPT Framework , 2012, UITP.

[56]  Peter B. Andrews An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[57]  Claude Marché,et al.  The BWare Project: Building a Proof Platform for the Automated Verification of B Proof Obligations , 2014, ABZ.

[58]  Nikolai Sultana Higher-order proof translation , 2015 .

[59]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

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

[61]  Mark-Oliver Stehr,et al.  An Executable Formalization of the HOL/Nuprl Connection in the Metalogical Framework Twelf , 2006, LPAR.

[62]  Michael Kohlhase,et al.  A scalable module system , 2011, Inf. Comput..

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

[64]  Dale Miller,et al.  Cut-elimination for a logic with definitions and induction , 2000, Theor. Comput. Sci..

[65]  Gopalan Nadathur,et al.  The Bedwyr System for Model Checking over Syntactic Expressions , 2007, CADE.

[66]  Michael Norrish,et al.  A Brief Overview of HOL4 , 2008, TPHOLs.

[67]  Dale A. Miller,et al.  Proofs in Higher-Order Logic , 1983 .

[68]  Gopalan Nadathur,et al.  Uniform Proofs as a Foundation for Logic Programming , 1991, Ann. Pure Appl. Log..

[69]  Anna Philippou,et al.  Tools and Algorithms for the Construction and Analysis of Systems , 2018, Lecture Notes in Computer Science.

[70]  JEAN-MARC ANDREOLI,et al.  Logic Programming with Focusing Proofs in Linear Logic , 1992, J. Log. Comput..

[71]  Koen Claessen,et al.  Using the TPTP Language for Writing Derivations and Finite Interpretations , 2006, IJCAR.

[72]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[73]  Christoph Benzmüller,et al.  Verifying the Modal Logic Cube Is an Easy Task (For Higher-Order Automated Reasoners) , 2010, Verification, Induction, Termination Analysis.

[74]  Tobias Nipkow,et al.  A FORMAL PROOF OF THE KEPLER CONJECTURE , 2015, Forum of Mathematics, Pi.

[75]  Alexander Leitsch,et al.  Cut-elimination and Redundancy-elimination by Resolution , 2000, J. Symb. Comput..

[76]  Michael Rathjen,et al.  Lambda Calculus with Types , 2014 .

[77]  Guillaume Burel,et al.  CoqInE: Translating the Calculus of Inductive Constructions into the λΠ-calculus Modulo , 2012, PxTP.

[78]  Damien Doligez,et al.  Zenon Modulo: When Achilles Outruns the Tortoise Using Deduction Modulo , 2013, LPAR.

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

[80]  Andrew W. Appel,et al.  Foundational proof-carrying code , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[81]  Bor-Yuh Evan Chang,et al.  Boogie: A Modular Reusable Verifier for Object-Oriented Programs , 2005, FMCO.

[82]  Pierre Halmagrand,et al.  Checking Zenon Modulo Proofs in Dedukti , 2015, PxTP@CADE.

[83]  David Baelde,et al.  Least and Greatest Fixed Points in Linear Logic , 2007, TOCL.

[84]  Alwen Tiu,et al.  Programming in Higher-Order Logic , 2009 .

[85]  J. H. Geuvers Logics and type systems , 1993 .

[86]  Geoff Sutcliffe,et al.  Solving the $100 modal logic challenge , 2009, J. Appl. Log..

[87]  Michael J. C. Gordon,et al.  Edinburgh LCF: A mechanised logic of computation , 1979 .

[88]  Robin Milner,et al.  Algebraic laws for nondeterminism and concurrency , 1985, JACM.

[89]  Raphaël Cauderlier,et al.  Mixing HOL and Coq in Dedukti (Rough Diamond) , 2015 .

[90]  Peter Schroeder-Heister,et al.  Rules of definitional reflection , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[91]  Guillaume Burel,et al.  A Shallow Embedding of Resolution and Superposition Proofs into the λΠ-Calculus Modulo , 2013, PxTP@CADE.

[92]  Bruno Woltzenlogel Paleo,et al.  Automating Gödel's Ontological Proof of God's Existence with Higher-order Automated Theorem Provers , 2014, ECAI.

[93]  Jens Otten Restricting backtracking in connection calculi , 2010, AI Commun..

[94]  Gopalan Nadathur,et al.  Mixing Finite Success and Finite Failure in an Automated Prover , 2005 .

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

[96]  Ali Assaf,et al.  A Calculus of Constructions with Explicit Subtyping , 2014, TYPES.

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

[98]  Chad E. Brown,et al.  Analytic Tableaux for Higher-Order Logic with Choice , 2010, Journal of Automated Reasoning.

[99]  Tobias Nipkow,et al.  Nitpick: A Counterexample Generator for Higher-Order Logic Based on a Relational Model Finder , 2010, ITP.

[100]  Zakaria Chihani,et al.  Foundational Proof Certificates in First-Order Logic , 2013, CADE.

[101]  Dale Miller,et al.  A proof theory for generic judgments , 2005, TOCL.

[102]  Konrad Slind An Implementation of higher order logic , 1990 .

[103]  Geoff Sutcliffe,et al.  An Interactive Derivation Viewer , 2007, UITP@FLoC.

[104]  Ali Assaf Conservativity of Embeddings in the lambda Pi Calculus Modulo Rewriting , 2015, TLCA.

[105]  Alexander Leitsch,et al.  Introducing Quantified Cuts in Logic with Equality , 2014, IJCAR.