Kruskal's tree theorem in Type theory
We present a Coq mechanisation of a purely inductive proof of Kruskal's
theorem.
Contrary to classical proofs, there are few instances of intuitionistic proofs for Kruskal's theorem. Some require the assumption that the ground relation is decidable (e.g. [1,2]). Veldman's [3] is the only published proof that does not require that decidability, but it requires "Brouwer's thesis". Moreover, no intuitionistic proof had been mechanized before.
We implement a typed variant of Veldman's intuitionistic proof where the axiom called "Brouwer's thesis" is not necessary: our proof is "axiom free".
We use Coquand's [4] inductive definition of Almost Full relations as an alternative to Veldman's. We present the architecture of the proof:
the Ramsey and FAN theorems, combinatorial principles and evaluation maps. We replace Veldman's "stump" based induction by lexicographic products of relations well-founded upto a projection.
[1] J. Goubault-Larrecq. A Constructive Proof of the Topological Kruskal Theorem.
[2] M. Seisenberger. On the Constructive Content of Proofs.
[3] W. Veldman. An intuitionistic proof of Kruskal's theorem.
[4] D. Vytiniotis et al. Stop When You Are Almost-Full.