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Well-founded Relations in Type Theory

We want to have general recursion in type theory without giving up the requirement on termination. To this end, we offer an account of well-founded relations definition. We present schemas that allow the construction of well-founded relations from simpler ones. These schemas when properly instantiated give particular proofs of well-foundedness useful in program construction. We present the definitions in the context of Martin-Löf’s logical framework extended with pattern matching. The formalizations were verified using the ALF system.1