Suppose a formal proof of ∀x∃y Spec(x, y) is given, where Spec(x, y) is an atomic formula expressing some specification for natural numbers x, y. For any particular number n we then obtain a formal proof of ∃ySpec(n, y). Now the proof–theoretic normalization procedure yields another proof of ∃ySpec(n, y) which is in normal form. In particular, it does not use induction axioms any more, and it also does not contain non–evaluated terms. Hence we can read off, linearly in the size of the normal proof, an instance m for y such that Spec(n, m) holds. In this way a formal proof can be seen as a program, and the central part in implementing this programming language consists in an implementation of the proof–theoretic normalization procedure.

There are many ways to implement normalization. As usual, a crucial point is a good choice of the data structures. One possibility is to represent a term as a function (i.e. a SCHEME–procedure) of its free variables, and similarly to represent a derivation (in a Gentzen–style system of natural deduction) as a function of its free assumption and object variables. Then substitution is realized as application, and normalization is realized as the built–in evaluation process of SCHEME (or any other language of the LISP–family). We presently experiment with an implementation along these lines, and the results up to now are rather promising. Some details are given in an appendix.