We illustrate the effectiveness of proof transformations which expose the computational content of classical proofs even in cases where it is not apparent. We state without proof a theorem that these transformations apply to proofs in a fragment of type theory and discuss their implementation in Nuprl. We end with a discussion of the applications to Higman's lemma by the second author using the implemented system.

For every natural number n there is a prime p greater than n. To prove this, notice first that every number m has a least prime factor; to find it, just try dividing it by 2, 3, …, m and take the first divisor. In particular n! + 1 has a least prime factor. Call it p. Clearly p cannot be any number between 2 and n since none of those divide n! + 1 evenly. Therefore p > n. QED