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A General Formulation of Simultaneous InductiveRecursive Definitions in Type Theory
 Journal of Symbolic Logic
, 1998
"... The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by re ..."
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The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in MartinLöf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous inductionrecursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...
Intuitionistic and Classical Natural Deduction Systems with the Catch and the Throw Rules
 Theoretical Computer Science
, 1995
"... this paper, we introduce two natural deduction systems NJ c=t and NK c=t . NJ c=t is an extension of the intuitionistic natural deduction system NJ with the catch and the throw rules, and NK c=t is an extension of the classical natural deduction system ..."
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Cited by 11 (1 self)
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this paper, we introduce two natural deduction systems NJ c=t and NK c=t . NJ c=t is an extension of the intuitionistic natural deduction system NJ with the catch and the throw rules, and NK c=t is an extension of the classical natural deduction system
OpenEndedness of Objects and Types in MartinL\"of’s Type Theory
"... This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formu ..."
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This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formulation of computational openendedness. Suppose that a language underlying the theory is specified as a method system, which consists of a preobject system as the computational part and a pretype system as the structural part. Then types and their objects are unifoImly and inductively constructed as a type system that is built from the method system and that can provide a semantics of the theory. The main theorem shows that the original inference rules concerning objects or types remain valid in any type system built from a deterministic and regular extension of the original method system. This result includes a prescription for the class of types that can be introduced into the theory, which prescription is useful for checking whether specific new types can be introduced. 1