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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, introductio ..."
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Cited by 52 (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 44 (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 ..."
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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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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...
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...
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
Noname manuscript No. (will be inserted by the editor) Modelling Algebraic Structures and Morphisms in ACL2
"... Abstract In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structures — as a result, an algebraic hierarchy ranging ..."
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Abstract In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structures — as a result, an algebraic hierarchy ranging from setoids to vector spaces has been developed. The resultant tools can be used to simplify the development of generic theories about algebraic structures. In particular, the benefits of using the tools presented in this paper, compared to a fromscratch approach, are especially relevant when working with complex mathematical structures; for example, the structures employed in Algebraic Topology. This work shows that ACL2 can be a suitable tool for formalising algebraic concepts coming, for instance, from computer algebra systems.
www.lmcsonline.org GENERAL RECURSION VIA COINDUCTIVE TYPES
, 2004
"... Abstract. A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of dom ..."
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Abstract. A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of domain theory, and inductive definition of domain predicates. Here, a different solution is proposed: exploiting coinductive types to model infinite computations. To every type A we associate a type of partial elements A ν, coinductively generated by two constructors: the first, �a � just returns an element a: A; the second, ⊲ x, adds a computation step to a recursive element x: A ν. We show how this simple device is sufficient to formalize all recursive functions between two given types. It allows the definition of fixed points of finitary, that is, continuous, operators. We will compare this approach to different ones from the literature. Finally, we mention that the formalization, with appropriate structural maps, defines a strong monad. 1.
Equational Reasoning in Type Theory
, 2000
"... We dene the notions of equational theory and equational logic in Type Theory using the development of Universal Algebra presented in a previous paper. The main result is the formal proof of Birkho's validity and completeness theorem, that gives a theoretical basis to the two level approach ..."
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We dene the notions of equational theory and equational logic in Type Theory using the development of Universal Algebra presented in a previous paper. The main result is the formal proof of Birkho's validity and completeness theorem, that gives a theoretical basis to the two level approach to interactive theorem proving. The whole development has been implemented using the proof assistant Coq. Keywords: Type Theory, Universal Algebra, Equational Logic, Interactive Theorem Proving. 1