Superdeduction at Work

Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalisation of a proof term language that models appropriately superdeduction. We finaly examplify on several examples, including equality and noetherian induction, the usefulness of this approach which is implemented in the lemuridae system, written in TOM.

[1]  Christine Paulin-Mohring,et al.  The coq proof assistant reference manual , 2000 .

[2]  Claude Kirchner,et al.  External Rewriting for Skeptical Proof Assistants , 2003, Journal of Automated Reasoning.

[3]  Philip Wadler,et al.  Call-by-value is dual to call-by-name , 2003, ACM SIGPLAN International Conference on Functional Programming.

[4]  Alexander Simpson.  Workshop on Types for Proofs and Programs , 1993 .

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

[6]  Maribel Fernández,et al.  Curry-Style Types for Nominal Terms , 2006, TYPES.

[7]  Emmanuel Beffara,et al.  ICFP '03 Proceedings of the eighth ACM SIGPLAN international conference on Functional programming , 2003 .

[8]  Claude Kirchner,et al.  Principles of Superdeduction , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[9]  Gilles Dowek,et al.  Normalization in Supernatural deduction and in Deduction modulo , 2007 .

[10]  Natarajan Shankar,et al.  PVS: A Prototype Verification System , 1992, CADE.

[11]  Jean-Pierre Jouannaud,et al.  Higher-Order Rewriting: Framework, Confluence and Termination , 2005, Processes, Terms and Cycles.

[12]  Gilles Dowek,et al.  Proof normalization modulo , 1998, Journal of Symbolic Logic.

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

[14]  Hélène Kirchner,et al.  Automatic Combinability of Rewriting-Based Satisfiability Procedures , 2006, LPAR.

[15]  Toby Walsh,et al.  Automated Deduction—CADE-11 , 1992, Lecture Notes in Computer Science.

[16]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[17]  David Aspinall,et al.  Formalising Java's Data Race Free Guarantee , 2007, TPHOLs.

[18]  Benjamin Wack.  Typage et déduction dans le calcul de réécriture. (Type systems and deduction in the rewriting calculus) , 2005 .

[19]  W. V. Quine,et al.  Natural deduction , 2021, An Introduction to Proof Theory.

[20]  G. D. Liveing,et al.  The University of Cambridge , 1897, British medical journal.

[21]  Lawrence Charles Paulson,et al.  Isabelle: A Generic Theorem Prover , 1994 .

[22]  Christian Urban,et al.  Strong Normalisation of Cut-Elimination in Classical Logic , 1999, Fundam. Informaticae.

[23]  Christian Urban.  Strong Normalisation for a Gentzen-like Cut-Elimination Procedure , 2001, TLCA.

[24]  Hans de Nivelle,et al.  Automated Proof Construction in Type Theory Using Resolution , 2000, Journal of Automated Reasoning.

[25]  Gilles Dowek,et al.  Proof Normalization for a First-Order Formulation of Higher-Order Logic , 1997, TPHOLs.

[26]  Frank Wolter,et al.  Monodic fragments of first-order temporal logics: 2000-2001 A.D , 2001, LPAR.

[27]  Hugo Herbelin,et al.  The duality of computation , 2000, ICFP '00.

[28]  Jean-Marc Andreoli,et al.  Fucusing and Proof-Nets in Linear and Non-commutative Logic , 1999, LPAR.

[29]  Luigi Liquori,et al.  Rewriting Calculus with Fixpoints: Untyped and First-Order Systems , 2003, TYPES.

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

[31]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[32]  Hugo Herbelin,et al.  The Coq proof assistant : reference manual, version 6.1 , 1997 .

[33]  Eelco Visser,et al.  A core language for rewriting , 1998, WRLA.

[34]  Lawrence C. Paulson,et al.  Automation for interactive proof: First prototype , 2006, Inf. Comput..

[35]  Sam Lindley,et al.  Extensional Rewriting with Sums , 2007, TLCA.

[36]  Jean-Pierre Jouannaud,et al.  Inductive-data-type systems , 2002, Theor. Comput. Sci..

[37]  Florent Kirchner.  A Finite First-Order Theory of Classes , 2006, TYPES.

[38]  Stéphane Lengrand.  Call-by-value, call-by-name, and strong normalization for the classical sequent calculus , 2003, Electron. Notes Theor. Comput. Sci..

[39]  Virgile Prevosto,et al.  Certified mathematical hierarchies: the FoCal system , 2006, Mathematics, Algorithms, Proofs.

[40]  Christian Urban.  Classical Logic and Computation , 2000 .

[41]  Gilles Dowek,et al.  Arithmetic as a Theory Modulo , 2005, RTA.

[42]  Jean-Marc Andreoli.  Focussing and proof construction , 2001, Ann. Pure Appl. Log..

[43]  Guillaume Burel,et al.  Unbounded Proof-Length Speed-Up in Deduction Modulo , 2007, CSL.

[44]  Philip Wadler.  Call-by-Value Is Dual to Call-by-Name - Reloaded , 2005, RTA.

[45]  Frank Pfenning,et al.  Logic Programming and Automated Reasoning , 1994, Lecture Notes in Computer Science.

[46]  Paul Brauner.  Un calcul des séquents extensible , 2006 .

[47]  Steffen van Bakel,et al.  The Language chi: Circuits, Computations and Classical Logic , 2005, ICTCS.

[48]  Gilles Dowek,et al.  Cut elimination for Zermelo set theory , 2023, ArXiv.