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Nested General Recursion and Partiality in Type Theory
 Theorem Proving in Higher Order Logics: 14th International Conference, TPHOLs 2001, volume 2152 of Lecture Notes in Computer Science
, 2000
"... We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. The method consists in defining an inductive specialpurpose accessibility predicate, that characterises the inputs on which the algorithm terminates. As a resul ..."
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Cited by 24 (10 self)
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We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. The method consists in defining an inductive specialpurpose accessibility predicate, that characterises the inputs on which the algorithm terminates. As a result, the typetheoretic version of the algorithm can be defined by structural recursion on the proof that the input values satisfy this predicate. This technique results in definitions in which the computational and logical parts are clearly separated; hence, the typetheoretic version of the algorithm is given by its purely functional content, similarly to the corresponding program in a functional programming language. In the case of nested recursion, the special predicate and the typetheoretic algorithm must be defined simultaneously, because they depend on each other. This kind of definitions is not allowed in ordinary type theory, but it is provided in type theories extended wit...
Recursive Families of Inductive Types
, 2000
"... Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong p ..."
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Cited by 1 (1 self)
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Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong positivity, that characterizes these families. Then we investigate its solutions. First, we construct a model using wellorderings. Second, we use an extension...
Parigot's Second Order λμCalculus and Inductive Types
, 2001
"... . A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associat ..."
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Cited by 1 (0 self)
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. A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot's reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (:: ! ) to that for atomic formulae. Therefore, it even extends to noninterleaving positive xedpoint types. As a nontrivial application, strong normalization of calculus, extended by primitive recursion on monotone inductive types, is established. 1 Introduction calculus [12] essentially is the extension of nat...