Recursive Definitions in Type Theory
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The type theories we consider are adequate for the foundations of mathematics and computer science. Recursive type definitions are important practical ways to organize data, and they express powerful axioms about the termination of procedures. In the theory examined here, the demands of practicality arising from our implemented system, Nuprl, suggest an approach to recursive types that significantly increases the proof theoretic power of the theory and leads to insights into computational semantics. We offer a new account of recursive definitions for both types and partial functions. The computational requirements of the theory restrict recursive type definitions involving the total function-space constructor (