Assertion Level Proof Planning with Compiled Strategies Dominik Dietrich Dissertation zur Erlangung des Grades des Doktors der Ingenieurwissenschaften der Natur- wissenschaftlich-Technischen Fakult¨aten der Universit¨at des Saarlandes. Saarbrucken,¨ 2011. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibli- ografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. Dietrich, Dominik: Assertion Level Proof Planning with Compiled Strategies ISBN 978-3-86376-015-1 Dekan: Prof. Dr. Holger Hermanns, Universit¨at des Saarlandes Vorsitzender: Prof. Dr. Gerd Smolka, Universit¨at des Saarlandes Gutachter: Prof. Dr. J¨org Siekmann, Universit¨at des Saarlandes Prof. Dr. Alan Bundy, University of Edinburgh Prof. Dr. Fairouz Kamareddine, Heriot-Watt University Beisitzer: Dr. Helmut Horacek Tag des Kolloquiums: 27. September 2011 All Rights Reserved New Edition 2012, G¨ottingen c Optimus Verlag URL: www.optimus-verlag.de Printed in Germany Paper is FSC certified (wood-free, chlorine free and acid-free, and resistant to aging ANSI 3948 and ISO 9706) No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted in any form or by any means, electronic, mechanical, photocopying, scanning, or otherwise without the prior written permission of the Publisher. Request to the Publisher for permission should be addressed to [email protected]. Kurzzusammenfassung Die vorliegende Arbeit besch¨aftigt sich damit, das Formalisieren von Beweisen zu verein- fachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu k¨onnen. Dazu wird ein abstrakter Kalkul¨ entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen gefuhrten¨ Beweisen relativ nahe kommt. An- hand einer Fallstudie wird gezeigt, dass die abstrakte Beweisfuhrung¨ auf der Fakteneben vorteilhaft fur¨ automatische Suchverfahren ist. Zus¨atzlich wird eine Strategiesprache ent- wickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden k¨onnen. Zusammen bilden die einander erg¨anzenden Methoden den Rahmen zur Automa- tisierung von deklarativen Beweisen auf der Faktenebene, die bisher uberwiegend¨ manuell entwickelt werden mussten. iii Abstract The objective of this thesis is to ease the formalization of proofs by being able to ver- ify as well as to automatically construct abstract human-style proofs. This is achieved by lifting the logical basis to the abstract assertion level, which has been identified as a style of reasoning that can be found in textbooks. A case study shows that automatic reasoning procedures benefit from the abstract assertion level reasoning. In addition, a strategy language is developed that allows the specification of abstract underspecified declarative proof patterns within the proof document and supports their refinement. Case studies show that complex reasoning patterns can concisely be specified within the devel- oped language. Together, the complementary methods provide a framework to automate declarative proofs at the assertion level. v Acknowledgements First of all, I would like to thank Prof. Dr. J¨org Siekmann who accepted me as his Ph.D. student, and who has, over the past years, given me all the encouragement and conditions necessary to carry out this thesis. He has given me a lot of freedom to work on my thesis and his knowledge, interest in the research, and guidance has helped me to complete this thesis. My sincere gratitude goes to Prof. Dr. Alan Bundy, who with great experience in the field and his thorough understanding of my work engaged me in valuable discussions. I am grateful that he agreed to serve as an examiner of the thesis. I would also like to thank Prof. Dr. Fairouz Kamareddine for agreeing to become an examiner of this thesis. Moreover, I want to express my deeply-felt thanks to my thesis advisor Serge Autexier for his warm encouragement and thoughtful guidance during the entire period. This thesis greatly benefited from his scientific advice, including the more technical parts of this thesis. My research visit at the Carnegie Mellon University was one of the most wonderful experiences during the time of my Ph.D. study. Therefore, my special thanks go to Prof. Wilfried Sieg for giving me this great opportunity to be a guest at his laboratory. I sincerely thank him for his time, discussions and hospitality, from which this thesis benefited enormously. Moreover, I want to thank the DAAD for the financial support. The members of the Ωmega group in Saarbr¨ucken and FormalSafe group at DFKI Bremen have contributed immensely to my personal and professional time. The groups have been a source of friendships as well as good advice and collaboration. In particular, I wish to mention Christoph Benzm¨uller, Mark Buckley, Dieter Hutter, Christian Maeder, Till Mossakowski, Martin Pollet, Marvin Schiller, Ewaryst Schulz, Lutz Schr¨oder, Holger T¨aubig, Marc Wagner, Dennis Walter, and Claus-Peter Wirth. I wish to thank all anonymous and known reviewers of my papers for providing thoughtful comments, from which this thesis greatly benefited, and all people who sup- ported me during the time of writing this thesis. In particular I want to mention my colleague and friend Ewaryst Schulz for patiently proof-reading almost the complete the- sis and for discussions about this work, as well as Christoph Benzm¨uller, who also care- fully read many parts of this thesis. Moreover, I want to express my appreciation to Till Mossakowski, Lutz Schr¨oder, and Dennis Walter for reading parts of the thesis. Finally I would like to thank my parents for the financial support over the years, as well as my girlfriend Sandra for her love and patience. vii Contents Kurzzusammenfassung ii Abstract iv Acknowledgements vii Zusammenfassung xvii Extended Abstract xix I Introduction 1 1 Introduction 3 1.1 Contributions .................................. 5 1.2 OutlineoftheThesis .............................. 8 2 HistoricalOverviewandStateoftheArt 9 2.1 ClassicalAutomatedTheoremProving . ... 10 2.1.1 Rewriting ................................ 10 2.2 Interactive Theorem Proving and Proof Style. ....... 11 2.2.1 Proceduralvs. DeclarativeProof . .. 11 2.2.2 TacticLanguages ............................ 14 2.2.3 Deduction Modulo, Supernatural Deduction, and Superdeduction . 15 2.2.4 Proof Transformation and Presentation . .... 16 2.3 ProofPlanningandProofRefinement. ... 17 2.4 Practical Applications of Theorem Proving . ...... 19 2.4.1 Verification of Software, Hardware, and Mathematics . ...... 20 2.4.2 TutoringSystemsforMathematics . 20 2.5 Summary .................................... 22 II Assertion Level Proofs 23 3 Assertion Level Proofs 25 3.1 ExamplesofAssertionApplications . ... 27 3.2 DeepApplication ................................ 31 3.3 Summary .................................... 36 ix CONTENTS 4 Foundations 37 4.1 Syntax,SemanticsandUniformNotation . ... 37 4.1.1 Syntax.................................. 37 4.1.2 Type Inference – Algorithm .................... 40 W 4.1.3 Semantics ................................ 41 4.1.4 UniformnotationandPolarities . 43 4.2 Higher-OrderUnification. 46 4.3 Summary .................................... 47 5 Core Proof Theory 49 5.1 IndexedFormulaTrees . .. .. .. .. ... .. .. .. .. .. ... .. .. 49 5.1.1 Instantiations .............................. 53 5.1.2 Core ExpansionRules......................... 53 5.1.3 IncreasingMultiplicities . .. 55 5.2 FreeVariableIndexedFormulaTrees . .... 56 5.2.1 ReplacementRules . 59 5.2.2 Contraction,WeakeningandCut . 62 5.2.3 Simplification .............................. 63 5.2.4 ExtensionalityRules . 64 5.2.5 Instantiation .............................. 65 5.2.6 IncreaseofMultiplicities . .. 65 5.2.7 Sch¨utte’sRule.............................. 65 5.3 TwoExampleProofs .............................. 66 5.3.1 SimpleSetTheory ........................... 66 5.3.2 EquationalReasoning . 72 5.4 Summary .................................... 73 6 The Core calculusandtheAssertionLevel 75 6.1 WindowsandInferenceRepresentation . ..... 76 6.2 RepresentingAssertions . 82 6.2.1 Preprocessing .............................. 86 6.3 AssertionApplication.............................. 87 6.4 Assertions: BackwardApplication . ... 92 6.4.1 Generation of New Premises and Task Splitting . 100 6.5 Assertions: ForwardApplication. 101 6.6 ApplicationofRewriteRules. 108 6.7 RelatedWork ..................................109 6.7.1 LeanTAP .................................109 6.7.2 Focusing.................................110 6.7.3 Prawitz, Supernatural Deduction, Superdeduction . ........111 6.7.4 DeductionModulo . .111 6.7.5 Relationship to Hyperresolution and SLD Resolution . ......111 6.7.6 Imps...................................112 6.7.7 Muscadet ................................112 6.7.8 Theorema ................................113 6.8 Summary ....................................113 x CONTENTS 7 Proof Theory 115 7.1 Formal Characterization of Assertion Applications . .........115 7.2 SoundnessandCompleteness. 116 7.2.1 Sequent Calculus and Block Tableau Systems . 117 7.2.2 SystematicBlockTableau . .117 7.2.3 SystematicAssertionLevelTableau . 122 7.3 Summary ....................................128 III ProofPlansandProofStrategies 129 8 Proof