We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's "Introduction to Functional Programming" can be expressed using these operators. 1 Introduction Among the many styles and methodologies for the construction of computer programs the Squiggol style in our opinion deserves attention from the functional programming community. The overall goal of Squiggol is to calculate programs from their specification in the way a mathematician calculates solutions to differential equations, or uses arithmetic to solve numerical problems. It is not hard to state, prove and use laws for well-known operations such as addition, multiplication and ---at the function level--- composition. It is, however, quite hard to state, prove and use laws for arbitrarily recursively ...

Defining data types as initial algebras, or dually as final co-algebras, is beneficial, if not indispensible, for an algebraic calculus for program construction, in view of the nice equational properties that then become available. It is not hard to render finite lists as an initial algebra and, dually, infinite lists as a final co-algebra. However, this would mean that there are two distinct data types for lists, and then a program that is applicable to both finite and infinite lists is not possible, and arbitrary recursive definitions are not allowed. We prove the existence of algebras that are both initial in one category of algebras and final in the closely related category of co-algebras, and for which arbitrary (continuous) fixed point definitions ("recursion") do have a solution. Thus there is a single data type that comprises both the finite and the infinite lists. The price to be paid, however, is that partiality (of functions and values) is unavoidable.