## Derivation and Computation: Taking the Curry-Howard Correspondence SeriouslyMathematics 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 |

COMMONLY USED SYMBOLS | 377 |

### Other editions - View all

Derivation and Computation: Taking the Curry-Howard Correspondence Seriously H. Simmons No preview available - 2000 |

Derivation and Computation: Taking the Curry-Howard Correspondence Seriously H. Simmons No preview available - 2000 |

### Common terms and phrases

1-placed 1-step reduction A-calculus A-terms abbreviate algorithm proceeds alphabetic variants arbitrary arboreal code arithmetic ASig calculate chapter combinator terms component computation mechanism Consider constants construction context F Curry-Howard correspondence declared in F DEFINITION derivation system example Exercise F h t F is legal finite formulas function F fundamental sequence given H-derivation hence hierarchy holds housing axiom indicated induction hypothesis induction step iterator judgement leaf leap Lemma limit ordinal multi-index mutation natural numbers nominated derivations obtain ordinal arithmetic ordinal notations pair parsing tree possible primitive recursive problem proceeds by recursion produce proof proof theory properties propositional calculus propositional logic redex removal reduction axioms reduction relation renaming identifier replacement required result return a derivation root rules shown snake Solution spec clause standard substitution algorithm Suppose supremum Theorem transition relation tree type erasure untyped verify write