On the Constructive Content of Proofs

This thesis aims at exploring the scopes and limits of techniques for extracting programs from proofs. We focus on constructive theories of inductive definitions and classical systems allowing choice principles. Special emphasis is put on optimizations that allow for the extraction of realistic programs. Our main field of application is infinitary combinatorics. Higman’s Lemma, having an elegant non-constructive proof due to Nash-Williams, constitutes an interesting case for the problem of discovering the constructive content behind a classical proof. We give two distinct solutions to this problem. First, we present a proof of Higman’s Lemma for an arbitrary alphabet in a theory of inductive definitions. This proof may be considered as a constructive counterpart to Nash-Williams’ minimal-bad-sequence proof. Secondly, using a refined A-translation method, we directly transform the classical proof into a constructive one and extract a program. The crucial point in the latter is that we do not need to avoid the axiom of classical dependent choice but directly assign a realizer to its translation. A generalization of Higman’s Lemma is Kruskal’s Theorem. We present a constructive proof of Kruskal’s Theorem that is completely formalized in a theory of inductive definitions. As a practical part, we show that these methods can be carried out in an interactive theorem prover. Both approaches to Higman’s Lemma have been implemented in Minlog.