Dialectica interpretation of well-founded induction

From a classical proof that the gcd of natural numbers a1 and a2 is a linear combination of the two, we extract by G¨ odel’s Dialectica interpretation an algorithm computing the coefficients. The proof uses the minimum principle. We show generally how well-founded recursion can be used to Dialectica interpret well-founded induction, which is needed in the proof of the minimum principle. In the special case of the example above it turns out that we obtain a reasonable extracted term, representing an algorithm close to Euclid’s. Copyright line will be provided by the publisher Finding and extracting computational content in existence proofs is a challenging subject, particularly so when the proofs do not seem to have such content. This is regularly the case when what is proved is only a ( ! ") := ", (" ! ) := , (" ! ") := ", ( ◊ ") := , (" ◊ ) := , (" ◊ ") := ". ¯