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Program extraction from normalization proofs
 Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science
, 1993
"... This paper describes formalizations of Tait’s normalization proof for the simply typed λcalculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machineextracted that implement variants of the wellknown normalizationbyevaluation algorithm. The case stud ..."
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Cited by 60 (5 self)
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This paper describes formalizations of Tait’s normalization proof for the simply typed λcalculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machineextracted that implement variants of the wellknown normalizationbyevaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a nontrivial setting. 1
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... 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 ..."
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Cited by 40 (4 self)
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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 Eloperator is omitted. For example the \Pitype 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...
A realizability interpretation of MartinLöf's type theory
"... In this paper we present a simple argument for normalization of the fragment of MartinLöf'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 ge ..."
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Cited by 8 (1 self)
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In this paper we present a simple argument for normalization of the fragment of MartinLöf'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.
On the strength of proofirrelevant type theories
 of Lecture Notes in Computer Science
, 2006
"... Vol. 4 (3:13) 2008, pp. 1–20 ..."
A lambdatoCL Translation for Strong Normalization
, 1997
"... . 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 fireduction in calculus and weak reduction in cl). None of the conventional translations from calculus to cl satisfy t ..."
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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 fireduction 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 clterms. 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 ...
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric