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The Haskell Road to Logic, Math and Programming Kees Doets and Jan van Eijck March 4, 2004 Contents Preface v 1 Getting Started 1.1 Starting up the Haskell Interpreter . . . . . . 1.2 Implementing a Prime Number Test . . . . . 1.3 Haskell Type Declarations . . . . . . . . . . 1.4 Identifiers in Haskell . . . . . . . . . . . . . 1.5 Playing the Haskell Game . . . . . . . . . . 1.6 Haskell Types . . . . . . . . . . . . . . . . . 1.7 The Prime Factorization Algorithm . . . . . . 1.8 The map and filter Functions . . . . . . . . 1.9 Haskell Equations and Equational Reasoning 1.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 8 11 12 17 19 20 24 26 2 Talking about Mathematical Objects 2.1 Logical Connectives and their Meanings . . 2.2 Logical Validity and Related Notions . . . . 2.3 Making Symbolic Form Explicit . . . . . . 2.4 Lambda Abstraction . . . . . . . . . . . . . 2.5 Definitions and Implementations . . . . . . 2.6 Abstract Formulas and Concrete Structures 2.7 Logical Handling of the Quantifiers . . . . 2.8 Quantifiers as Procedures . . . . . . . . . . 2.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 28 38 50 58 60 61 64 68 70 . . . . 71 72 75 78 90 3 The Use of Logic: Proof 3.1 Proof Style . . . . . . . 3.2 Proof Recipes . . . . . . 3.3 Rules for the Connectives 3.4 Rules for the Quantifiers . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS ii 3.5 3.6 3.7 3.8 Summary of the Proof Recipes . . . . . . Some Strategic Guidelines . . . . . . . . Reasoning and Computation with Primes . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 . 99 . 103 . 111 4 Sets, Types and Lists 4.1 Let’s Talk About Sets . . . . . . . . . . . 4.2 Paradoxes, Types and Type Classes . . . . 4.3 Special Sets . . . . . . . . . . . . . . . . 4.4 Algebra of Sets . . . . . . . . . . . . . . 4.5 Pairs and Products . . . . . . . . . . . . . 4.6 Lists and List Operations . . . . . . . . . 4.7 List Comprehension and Database Query 4.8 Using Lists to Represent Sets . . . . . . . 4.9 A Data Type for Sets . . . . . . . . . . . 4.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 114 121 125 127 136 139 145 149 153 158 5 Relations 5.1 The Notion of a Relation . . . . . . . . . . . . . . 5.2 Properties of Relations . . . . . . . .