Induction and Recursion on the Partial Real Line via Biquotients of Bifree Algebras

The partial real line is the continuous domain of compact real intervals ordered by reverse inclusion. The idea is that singleton intervals represent total real numbers, and that the remaining intervals represent partial real numbers. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. The theory is based on a domain-equation-like presentation of the partial unit interval, which we refer to as a biquotient of a bifree algebra.