# Derivation and Computation: Taking the Curry-Howard Correspondence Seriously

Cambridge University Press, May 18, 2000 - Computers - 384 pages
Mathematics is about proofs, that is the derivation of correct statements; and calculations, that is the production of results according to well-defined sets of rules. The two notions are intimately related. Proofs can involve calculations, and the algorithm underlying a calculation should be proved correct. The aim of the author is to explore this relationship. The book itself forms an introduction to simple type theory. Starting from the familiar propositional calculus the author develops the central idea of an applied lambda-calculus. This is illustrated by an account of Gödel's T, a system which codifies number-theoretic function hierarchies. Each of the book's 52 sections ends with a set of exercises, some 200 in total. These are designed to help the reader get to grips with the subject, and develop a further understanding. An appendix contains complete solutions of these exercises.

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### Contents

 DERIVATION SYSTEMS 3 COMPUTATION MECHANISMS 27 THE TYPED COMBINATOR CALCULUS 48 THE TYPED ACALCULUS 67 SUBSTITUTION ALGORITHMS 82 APPLIED ACALCULI 100 MULTIRECURSIVE ARITHMETIC 137 ORDINALS AND ORDINAL NOTATIONS 170
 THE TYPED COMBINATOR CALCULUS 249 THE TYPED ACALCULUS 264 E SUBSTITUTION ALGORITHMS 279 F APPLIED ACALCULI 290 G MULTIRECURSIVE ARITHMETIC 318 H ORDINALS AND ORDINAL NOTATIONS 341 HIGHER ORDER RECURSION 355 POSTVIEW 371

 HIGHER ORDER RECURSION 189 A DERIVATION SYSTEMS 215 B COMPUTATION MECHANISMS 234