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Universes for Generic Programs and Proofs in Dependent Type Theory
 Nordic Journal of Computing
, 2003
"... We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, el ..."
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Cited by 42 (2 self)
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We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductiverecursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.
Setoids in Type Theory
, 2000
"... Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we ..."
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Cited by 30 (4 self)
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Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we argue that a commonly advocated approach to partial setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
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...
A Type of Partial Recursive Functions
"... Abstract. Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. In both cases, we could obtain a type only by extending the theory with either an imp ..."
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Abstract. Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. In both cases, we could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second one evaluates them, and the last one computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor: a partial recursive function is created by giving the first and third of the above components. As in one of our previous methods, every partial recursive function is associated with an inductive domain predicate and the evaluation of the function requires a proof that the predicate holds on the input values. We give a constructive justification for the new construct by means of an interpretation from the extended type theory into the base one. This shows that the extended theory is consistent and constructive. 1
Formalisation of General Logics in the Calculus of Inductive Constructions: Towards an Abstract . . .
, 1999
"... Formal specifications of logics share many standard concepts and in order to avoid repetitious works, it seems desirable to express these specifications in a uniform framework. General logics à la J. Meseguer provide an uniform and modular way of encoding a logical language, its semantics and its in ..."
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Formal specifications of logics share many standard concepts and in order to avoid repetitious works, it seems desirable to express these specifications in a uniform framework. General logics à la J. Meseguer provide an uniform and modular way of encoding a logical language, its semantics and its inference system. Hence, we describe here a formalisation of general logics in the calculus of inductive constructions thus providing a generic and modular set of speci cations (with the proofs of s...