This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. Ordered Geometry in Hilbert’s Grundlagen der Geometrie Phil Scott Doctor of Philosophy Centre for Intelligent Systems and their Applications School of Informatics University of Edinburgh 2015 Abstract The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean ge- ometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polyg- onal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem. i Acknowledgements Many thanks go to my supervisor Jacques Fleuriot for his support and encourage- ment. Thanks to Laura Meikle for spotting a simplification of Theorem 3.11, and especially to Steven Obua for helping me prove transitivity of polygonal rotations de- scribed in §11.5.1. ii Declaration I declare that this thesis was composed by myself, that the work contained herein is my own except for work included which forms part of jointly-authored publications. Our contribution and that of the other authors to this work is explicitly indicated below. We confirm that appropriate credit has been given within the thesis where reference has been made to the work of others, and that this work has not been submitted for any other degree or professional qualification except as specified. In Chapters 4 and 5, we expand on a combinator language which evolved between Composable Discovery Engines for Interactive Theorem Proving and A Combinator Language for Theorem Discovery, published respectively in Interactive Theorem Prov- ing in 2011 and Intelligent Computer Mathematics in 2012. The intended application of this language was first described in An Investigation of Hilbert’s Implicit Reasoning through Proof Discovery in Idle-Time, published in Automated Deduction in Geometry in 2010, and our analysis from this paper has been updated and can be found in §5.2.3 and §5.3.1.1. The work was co-authored with Jacques Fleuriot. (Phil Scott) January 15, 2015 iii Table of Contents 1 Introduction 1 1.1 The Grundlagen der Geometrie .....................2 1.2 Ordered Geometry . .2 1.3 Verification . .3 1.3.1 Computer Assistance . .4 1.3.2 Readable Verifications . .5 1.4 Contributions and Organisation . .6 2 Background 8 2.1 Object Logic . .8 2.1.1 Definitions . .9 2.1.2 Higher-order Logic . 12 2.2 Proof Assistant . 13 2.2.1 Edinburgh LCF . 13 2.2.2 Additional Functionality . 14 2.3 Classical Logic . 15 2.3.1 Axiom of Infinity . 17 2.4 Verification Tools . 17 2.4.1 Tactics . 17 2.4.2 Fully Automated Procedures . 18 2.5 Declarative Proof . 18 2.5.1 Mizar Light . 19 2.5.2 Extending Mizar Light for Interactivity . 21 2.5.3 Concluding Remarks . 23 2.6 Conventions . 24 iv 3 Axiomatics 27 3.1 Primitives . 27 3.2 Group I . 28 3.2.1 Incidence Relations . 28 3.2.2 Axioms and Formalisation . 30 3.2.3 Related Axiomatisations . 33 3.2.4 Elementary Consequences . 33 3.2.5 Absent Arguments . 36 3.2.6 Point sets . 38 3.3 Group II . 42 3.3.1 Axioms and Primitive Notions . 42 3.3.2 Pasch and Incidence Reasoning . 44 3.4 Conclusion . 45 4 Automation 47 4.1 Background . 47 4.1.1 Wu’s Method . 48 4.1.2 Other Methods . 49 4.2 Basis for an Algorithm . 49 4.2.1 Inference Rules . 50 4.3 Forward Chaining . 52 4.3.1 Concurrency . 52 4.3.2 Discovery . 53 4.4 An Implementation in Combinators . 54 4.4.1 Related Work . 55 4.4.2 Streams . 55 4.4.3 A Monad for Breadth-First Search . 57 4.5 Case-analysis . 59 4.5.1 Trees . 60 4.6 Additional Primitives and Derived Discoverers . 64 4.6.1 Case-splitting . 64 4.6.2 Delaying . 65 4.6.3 Filtering . 65 4.6.4 Accumulating . 67 4.6.5 Deduction . 67 v 4.7 Integration . 69 4.7.1 Concurrency . 70 4.7.2 Dependency Tracking . 71 4.8 Implementation Details . 72 4.8.1 Implementation Issues . 72 4.9 Applicative Functors . 73 4.10 The Problem Revisited . 75 4.11 Conclusion and Further Work . 75 5 Elementary Consequences in Group II 78 5.1 THEOREM 3 . 78 5.1.1 Verification . 79 5.1.2 The Outer and Inner Pasch Axioms . 81 5.2 THEOREM 4 . 83 5.2.1 Discovering Applications of Pasch . 84 5.2.2 Verifying Hilbert’s Proof . 85 5.2.3 Alternative Proof . 88 5.3 THEOREM 5 . 90 5.3.1 Part 1 of THEOREM 5 . 90 5.3.2 Discovery at work . 95 5.3.3 Part 2 of THEOREM 5 . 102 5.4 Conclusion . 104 6 Infinity and Linear Ordering 105 6.1 THEOREM 6 at the Meta-level . 105 6.1.1 Representation . 107 6.1.2 Enumerating Possible Orderings . 109 6.2 THEOREM 6 at the Object Level . 110 6.3 Natural Numbers . 110 6.3.1 The Axiom of Infinity . 111 6.3.2 Models and a Finite Interpretation . 112 6.4 Infinity . 115 6.5 A Geometric Successor . 115 6.5.1 Lemmas . 118 6.6 Theorem of Infinity . 119 6.7 THEOREM 6 Revisited . 120 vi 6.7.1 At Least One Ordering . 120 6.7.2 Exactly Two Orderings . 123 6.8 An Ordering Tactic . 123 6.8.1 Example . 124 6.9 Conclusion . 125 7 Ordering in the Plane 127 7.1 Definitions and Formalisation . 127 7.1.1 Rays . 128 7.1.2 Quotienting . 129 7.1.3 Automatic Lifting . 130 7.2 Theory of Half-Planes . 132 7.2.1 Transitivity . 133 7.2.2 Covering . 135 7.3 THEOREM 8 . 137 7.4 Conclusion . 140 8 Background to the Jordan Curve Theorem for Polygons 141 8.1 Relationship with the Full Jordan Curve Theorem . 141 8.2 Generality of the Polygonal Case . 143 8.3 Polygonal Case: Formulation . 145 8.4 Veblen’s Proof . 146 8.4.1 Veblen’s Lemma . ..