## Adapting Proofs-as-Programs: The Curry--Howard ProtocolThis book ?nds new things to do with an old idea. The proofs-as-programs paradigm constitutes a set of approaches to developing programs from proofs in constructive logic. It has been over thirty years since the paradigm was ?rst conceived. At that time, there was a belief that proofs-as-programs had the - tential for practical application to semi-automated software development. I- tial applications were mostly concerned with ?ne-grain, mathematical program synthesis. For various reasons, research interest in the area eventually tended toward more theoretic issues of constructive logic and type theory. However, in recent years, the situation has become more balanced, and there is increasingly active research in applying constructive techniques to industrial-scale, complex software engineering problems. Thismonographdetailsseveralimportantadvancesinthisdirectionofpr- tical proofs-as-programs. One of the central themes of the book is a general, abstract framework for developing new systems of program synthesis by adapting proofs-as-programs to new contexts. Framework-oriented approaches that facilitate analogous - proaches to building systems for solving particular problems have been popular and successful. Thesemethodsarehelpful asthey providea formal toolbox that enablesa“roll-your-own”approachtodevelopingsolutions.Itishopedthatour framework will have a similar impact. The framework is demonstrated by example. We will give two novel - plications of proofs-as-programs to large-scale, coarse-grain software engine- ing problems: contractual imperative program synthesis and structured p- gram synthesis. These applications constitute an exemplary justi?cation of the framework. Also, in and of themselves, these approaches to synthesis should be interesting for researchers working in the target problem domains. |

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### Contents

2 | 23 |

The CurryHoward Protocol | 77 |

Overview | 93 |

Properties of Intuitionistic Hoare Logic | 135 |

ProofsasImperativePrograms | 165 |

7 | 217 |

Prooftheoretic Properties of SSL | 253 |

9 | 296 |

12 | 391 |

415 | |

### Other editions - View all

Adapting Proofs-as-Programs: The Curry--Howard Protocol Iman Poernomo,John N. Crossley,Martin Wirsing No preview available - 2010 |

### Common terms and phrases

abstract account list application Assume assumption axioms basic specification boolean Chapter component constructive type theory corresponding counter Curry–Howard isomorphism Curry–Howard protocol define definition denote derive disjoint union etype(A example executable specification extended realizer extractint extraction map extractmod(p Fint function symbols given Harrop formula hide HInt Hoare logic immediate reduct imperative programs induction hypothesis Inl(a instantiation intuitionistic logic lambda calculus lambda terms Lemma length db logical type theory LTT(IHL LTT(SSL many-sorted models modified realizer modular realizer morphism natural deduction normalization obtain operational semantics password post-condition predicate program synthesis program/formula pairs programming language proof ends proof-term PwdCore PwdSys reduction rules refinement return value schema schemata Section side-effect side-effect-free signature Sk(A Skolem function SML programs sort specification expression structural rules structured programs structured specifications subterms theorem tologic true type inference rules variables