The Isabelle Cookbook A Gentle Tutorial for Programming Isabelle/ML (draft) by Christian Urban with contributions from: Stefan Berghofer Jasmin Blanchette Sascha Bohme¨ Lukas Bulwahn Jeremy Dawson Rafal Kolanski Alexander Krauss Tobias Nipkow Norbert Schirmer Andreas Schropp Christian Sternagel June 16, 2019 2 Contents Contentsi 1 Introduction1 1.1 Intended Audience and Prior Knowledge.................1 1.2 Isabelle Infrastructure...........................2 1.3 Existing Documentation..........................2 1.4 Typographic Conventions.........................3 1.5 How To Understand Isabelle Code.....................4 1.6 Aaaaargh! My Code Does not Work Anymore..............5 1.7 Serious Isabelle ML-Programming.....................6 1.8 Some Naming Conventions in the Isabelle Sources...........6 1.9 Acknowledgements.............................7 2 First Steps 11 2.1 Including ML-Code............................. 11 2.2 Printing and Debugging.......................... 12 2.3 Combinators................................ 16 2.4 ML-Antiquotations............................. 23 2.5 Storing Data in Isabelle.......................... 26 2.6 Summary.................................. 34 3 Isabelle Essentials 35 3.1 Terms and Types.............................. 35 3.2 Constructing Terms and Types Manually................. 40 3.3 Unification and Matching......................... 48 3.4 Sorts (TBD)................................. 57 3.5 Type-Checking............................... 58 3.6 Certified Terms and Certified Types.................... 60 3.7 Theorems.................................. 61 3.8 Theorem Attributes............................. 73 3.9 Pretty-Printing............................... 76 i ii CONTENTS 3.10 Summary.................................. 81 4 Advanced Isabelle 83 4.1 Theories................................... 83 4.2 Contexts................................... 85 4.3 Local Theories and Local Setups (TBD).................. 90 4.4 Morphisms (TBD).............................. 90 4.5 Misc (TBD)................................. 91 4.6 What Is In an Isabelle Name? (TBD)................... 91 4.7 Concurrency (TBD)............................. 92 4.8 Parse and Print Translations (TBD).................... 92 4.9 Summary.................................. 92 5 Tactical Reasoning 93 5.1 Basics of Reasoning with Tactics...................... 93 5.2 Simple Tactics................................ 97 5.3 Tactic Combinators............................. 106 5.4 Simplifier Tactics.............................. 112 5.5 Simprocs.................................. 119 5.6 Conversions................................. 123 5.7 Declarations (TBD)............................. 132 5.8 Structured Proofs (TBD).......................... 132 5.9 Summary.................................. 132 6 Parsing 133 6.1 Building Generic Parsers.......................... 133 6.2 Parsing Theory Syntax........................... 141 6.3 Parsers for ML-Code (TBD)........................ 145 6.4 Context Parser (TBD)............................ 145 6.5 Argument and Attribute Parsers (TBD).................. 145 6.6 Parsing Inner Syntax............................ 145 6.7 Parsing Specifications............................ 146 6.8 New Commands.............................. 149 6.9 Methods (TBD)............................... 152 7 How to Write a Definitional Package 155 7.1 Preliminaries................................ 156 7.2 Parsing and Typing the Specification................... 161 7.3 The Code in a Nutshell........................... 163 7.4 The Gory Details.............................. 166 CONTENTS iii 7.5 Extensions of the Package (TBD)..................... 183 7.6 Definitional Packages............................ 184 A Recipes 185 A.1 Useful Document Antiquotations..................... 185 A.2 Restricting the Runtime of a Function................... 188 A.3 Measuring Time............................... 188 A.4 Executing an External Application (TBD)................ 189 A.5 Writing an Oracle (TBD).......................... 190 A.6 SAT Solvers................................. 192 A.7 User Space Type-Systems (TBD)...................... 194 B Solutions to Most Exercises 195 C Comments for Authors 203 Bibliography 207 Index 208 iv CONTENTS Chapter 1 Introduction “My thesis is that programming is not at the bottom of the intellectual pyramid, but at the top. It’s creative design of the highest order. It isn’t monkey or donkey work; rather, as Edsger Dijkstra famously claimed, it’s amongst the hardest intellectual tasks ever attempted.” Richard Bornat, In Defence of Programming.[1] If your next project requires you to program Isabelle with ML, then this tutorial is for you. It will guide you through the first steps of Isabelle programming, and also explain “tricks of the trade”. We also hope the tutorial will encourage students and researchers to play with Isabelle and implement new ideas. The source code of Isabelle can look intimidating, but beginners can get by with knowledge of only a handful of concepts, a small number of functions and a few basic coding conventions. There is also a considerable amount of code written in Scala that allows Isabelle interface with the Jedit GUI. Explanation of this part is beyond this tutorial. The best way to get to know the Isabelle/ML is by experimenting with the many code examples included in the tutorial. The code is as far as possible checked against the Isabelle distribution. If something does not work, then please let us know. It is impossible for us to know every environment, operating system or editor in which Isabelle is used. If you have comments, criticism or like to add to the tutorial, please feel free—you are most welcome!! The tutorial is meant to be gentle and compre- hensive. To achieve this we need your help and feedback. 1.1 Intended Audience and Prior Knowledge This tutorial targets readers who already know how to use Isabelle for writing the- ories and proofs. We also assume that readers are familiar with the functional pro- gramming language ML, the language in which most of Isabelle is implemented. If you are unfamiliar with either of these two subjects, then you should first work through the Isabelle/HOL tutorial [4] or Paulson’s book on ML [5]. Recently, Isabelle has adopted a sizable amount of Scala code for a slick GUI based on jEdit. This part of the code is beyond the interest of this tutorial, since it mostly does not concern the regular Isabelle developer. 1 2 CHAPTER 1. INTRODUCTION 1.2 Isabelle Infrastructure There is a rich Isabelle infrastructure—it might be good to have the following rough mind map [7]: Logic Isabelle/Pure is the logical Framework and bootstrap environment. The Pure logic is used to represent rules for Higher-Order Natural Deduction declaratively. This allows the implementation and definition of object logics like HOL using the Pure logic and framework. Isabelle/HOL is the main library of theories and tools for applications that is used throughout this tutorial. Programming Isabelle/ML is the Isabelle tool implementation and extension language. It is based on Poly/ML1. Both Isabelle/Pure and Isabelle/ML emerge from the same boot- strap process: the result is a meta-language for programming the logic that is intertwined with it from a technological viewpoint, but logic and programming remain formally separated. Isabelle/Scala is the Isabelle system programming language. It connects the logical environment with the outside world. Most notably resulting in the Prover IDE Isabelle/jEdit and the command line tools. Proof Isabelle/Isar is the structured proof language of the Isabelle framework. Isar means, Intelligible semi-automated reasoning. Document language for HTML output and LATEXtype-setting of proof text. A proof document combines formal and informal text to describe what has been proven to a general audience. IDE Isabelle/jEdit is the IDE for proof and tool development. It provides a rich interac- tive frontend to the Isabelle framework in which logic and proof development, document creation as well as ML programming are seamlessly integrated. 1.3 Existing Documentation The following documentation about Isabelle programming already exists (and is part of the distribution of Isabelle): 1http://polyml.org 1.4. TYPOGRAPHIC CONVENTIONS 3 The Isabelle/Isar Reference Manual provides a top level view on the Isabelle sys- tem, explaining general concepts and specification material (like grammars, examples and so on) about Isabelle, Isar, Pure, HOL and the document lan- guage. The Isabelle/Isar Implementation Manual describes Isabelle implementation from a high-level perspective, documenting the major underlying concepts and in- terfaces. Isabelle/jEdit describes the IDE. The Old Introduction to Isabelle is an older document that used to be the main reference of Isabelle at a time when all proof scripts were written with ML. Many parts of this manual are outdated now, but some parts, particularly the chapters on tactics, are still useful. Then of course there are: The Isabelle sources. They are the ultimate reference for how things really work. Therefore you should not hesitate to look at the way things are actually imple- mented. While much of the Isabelle code is uncommented, some parts have very helpful comments—particularly the code about theorems and terms. De- spite the lack of comments in most parts, it is often good to look at code that does similar things as you want to do and learn from it. This tutorial contains frequently pointers to the Isabelle sources. The best way