Derivation and Computation: Taking the Curry-Howard Correspondence Seriously
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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THE TYPED XCALCULUS
A DERIVATION SYSTEMS
THE TYPED COMBINATOR CALCULUS
E SUBSTITUTION ALGORITHMS
F APPLIED XCALCULI
ORDINALS AND ORDINAL NOTATIONS
HIGHER ORDER RECURSION
COMMONLY USED SYMBOLS
THE TYPED XCALCULUS
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1-placed 1-step reduction abbreviations arbitrary arboreal code calculate chapter component computation Consider construction context convenience let declaration DEFINITION derivation system example Exercise fºr formulas function f fundamental sequence given H-derivation hence hierarchy holds housing axiom induction hypothesis induction step iterator judgement Lemma limit ordinal multi-index mutation natural numbers nominated derivations notation obtain operator ordinal arithmetic ordinal notations pair parsing tree primitive recursive problem proceeds by recursion produce proof proof theory properties redex removal reduction axioms renaming identifier replacement required result rules snake f Solution spec clause standard subderivations substitution algorithm supremum T H 9 T H tº term H Theorem Tº H tº tº transition relation tree type erasure untyped verify X-terms XSig XX H