Preface
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Abstract:
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-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene's realizability interpretation---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. Outline Since most calculi found in type theory build on-calculus, the notes begin, in Chapter 1, with an introduction to type-free-calculus. The introduction derives the most rudimentary properties of fi-reduction including the Church-Rosser theorem. It also presents Kleene's theorem stating that all recursive functions are-definable and Church's theorem stating that fi-equality is undecidable. As explained above, an important part of the Curry-Howard isomorphism is the idea that a constructive proof of an implication is a certain procedure. This calls for some elaboration of what is meant by constructive proofs, and Chapter 2 therefore presents intuitionistic propositional logic. The chapter presents a natural deduction formulation of minimal and intuitionistic propositional logic. The usual semantics in terms of Heyting algebras and in terms of Kripke models are introduced---the former explained
