Modelling general recursion in type theory

Constructive type theory is a very expressive programming language. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that do not satisfy any syntactic condition that guarantees termination. We present a method to formalise general recursive algorithms in type theory that uses an inductive predicate to characterise termination and that separates the computational and logical parts of the definitions. As a consequence, the resulting typetheoretic algorithms are clear, compact and easy to understand. They are as simple as their Haskell-like versions, where there is no restriction on the recursive calls. Given a general recursive algorithm, our method consists in defining an inductive special-purpose accessibility predicate that characterises the inputs on which the algorithm terminates. The typetheoretic version of the algorithm is then defined by structural recursion on the proof that the input values satisfy this predicate. We give a formal definition of the method and discuss its power and its limitations.