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Proof theory --- Lambda calculus --- Type Theory --- Curry-Howard isomorphism

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The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.The isomorphism has many aspects, even at the syntactic level:formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation,

Curry-Howard isomorphism. --- Curry-Howard isomorphism. Lambda calculus. Proof theory. --- Lambda calculus. --- Proof theory. --- Curry-Howard isomorphism --- Lambda calculus --- Proof theory --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Calculus, Lambda --- Correspondence, Curry-Howard --- Curry-Howard correspondence --- Formulae-as-types principle --- Logic, Symbolic and mathematical --- Isomorphisms (Mathematics)

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This 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.

Curry-Howard isomorphism. --- Proof theory. --- Logic, Symbolic and mathematical. --- Functional programming (Computer science) --- Lambda calculus. --- Abstract data types (Computer science) --- Data types (Computer science) --- Computer programming --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Logic, Symbolic and mathematical --- Correspondence, Curry-Howard --- Curry-Howard correspondence --- Formulae-as-types principle --- Isomorphisms (Mathematics) --- Calculus, Lambda --- Logic design. --- Computer science. --- Software engineering. --- Logics and Meanings of Programs. --- Mathematical Logic and Foundations. --- Logic Design. --- Programming Techniques. --- Software Engineering/Programming and Operating Systems. --- Models and Principles. --- Computer software engineering --- Engineering --- Informatics --- Science --- Design, Logic --- Design of logic systems --- Digital electronics --- Electronic circuit design --- Logic circuits --- Machine theory --- Switching theory --- Computer logic. --- Mathematical logic. --- Computer programming. --- Computers. --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Cybernetics --- Calculators --- Cyberspace --- Computers --- Electronic computer programming --- Electronic data processing --- Electronic digital computers --- Programming (Electronic computers) --- Coding theory --- Computer science logic --- Programming