Termination Casts: A Flexible Approach to Termination With General Recursion (Technical Appendix)

This paper proposes a type-and-effect system called Teq↓, which distinguishes terminating terms and total functions from possibly diverging terms and partial functions, for a lambda calculus with general recursion and equality types. The central idea is to include a primitive type-form "Terminates t", expressing that term t is terminating; and then allow terms t to be coerced from possibly diverging to total, using a proof of Terminates t. We call such coercions termination casts, and show how to implement terminating recursion using them. For the meta-theory of the system, we describe a translation from Teq↓ to a logical theory of termination for general recursive, simply typed functions. Every typing judgment of Teq↓ is translated to a theorem expressing the appropriate termination property of the computational part of the Teq↓ term.