Abstract

We consider Martin-Lof's well-ordering type constructor in the context of an impredicative type theory. We show that the well-ordering types can represent various inductive types faithfully in the presence of the filling-up equality rules or j-rules. We also discuss various properties of the filling-up rules. 1 Introduction Type theory is on the edge of two disciplines, constructive logic and computer science. Logicians see type theory as interesting because it offers a foundation for constructive mathematics and its formalization. For computer scientists, type theory promises to provide a uniform framework for programs, proofs, specifications, and their development. From each perspective, incorporating a general mechanism for inductively defined data types into type theory is an important next step. Various type-theoretic approaches to inductive data types have been considered in the literature, both in Martin-Lof's predicative type theories (e.g., [ML84, Acz86, Dyb88, Dyb91, B...

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