This paper describes formalizations of Tait’s normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting. 1

ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The El-operator is omitted. For example the \Pi-type is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...

In this paper we present a simple argument for normalization of the fragment of Martin-Löf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types are generated simultaneously.

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. We introduce a simple translation from -calculus to combinatory logic (cl) such that: A is an sn -term iff the translation result of A is an sn term of cl (the reductions are fi-reduction in -calculus and weak reduction in cl). None of the conventional translations from -calculus to cl satisfy the above property. Our translation provides a simpler sn proof of Godel's -calculus by the ordinal number assignment method. By using our translation, we construct a homomorphism from a conditionally partial combinatory algebra which arises over sn -terms to a partial combinatory algebra which arises over sn cl-terms. 1 Introduction We often find some translations from -calculus to combinatory logic (cl) provide a pleasing viewpoint in the study of -calculus. The most typical example can be found in the study of the equational theories and the model theories of -calculus. The translations from -calculus to cl have been investigated comprehensively by Curry school [8], and we come to know ...