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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.
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...
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