Constructive topology and combinatorics

Abstract

We present a method to extract constructive proofs from classical arguments proved by topogical means. Typically, this method will apply to the nonconstructive use of compactness in combinatorics, often in the form of the use of König's lemma (which says that a finitely branching tree that is infinite has an infinite branch.) The method consists roughly of working with the corresponding point-free version of the topological argument, which can be proven constructively using only as primitive the notion of inductive definition. We illustrate here this method on the classical “minimal bad sequence” argument used by Nash-Williams in his proof of Kruskal's theorem. The proofs we get by this method are well-suited for mechanisation in interactive proof systems that allow the user to introduce inductively defined notions, such as NuPrl, or Martin-Löf set theory.