Well Quasi-orders and the Functional Interpretation

A universal algorithm for Krull's theorem

“…We too seek to bring ideas such as the Dialectica interpretation to bear on proofs in abstract algebra, but while [54] focuses on achieving new effective bounds from proofs, we emphasise on the other hand general algorithmic patterns which correspond to the use of Zorn's lemma in a countable setting. Similar work in this direction but in the context of infinitary combinatorics can be found in [30,35].…”

On the computational content of Zorn's lemma

“…Let Ω e denote a fixed point of the primitive recursive defining equation ( 7) -where the closed primitive recursive term e is interpreted as some total object in P ω -and suppose that there exist < and L such that < is compatible with (⊕, ≺) and chain complete w.r. We conclude the paper by showing how our parametrised results can now be implemented in the special case of induction over the lexicographic ordering on sequences. This constitutes a direct counterpart to open induction as presented in [4], and is closely related to the recursive scheme introduced in [20] for extracting a witness from the proof of Higman's lemma. Definition 6.1 (HA ω ).…”

“…Nevertheless, as we will see in Section 6, it in fact generalises open induction, and so in particular can be used to formalize highly non-trivial proofs such as Nash-Williams' minimal bad-sequence construction (cf. [4,20]). To be more specific, our axiom schema will take the shape of a maximum principle of the form ∃xP (x) → ∃y(P (y) ∧ ∀z > y¬P (z))…”

“…In contrast to the aforementioned works, which all focus on variants of countable choice, we give a direct computational interpretation to an axiomatic formulation of Zorn's lemma. Our work is closest in spirit to Berger's realizability interpretation of open induction on the lexicographic ordering via open recursion [4] -an idea which was later transferred to the setting of Gödel's functional interpretation in [20]. However, a crucial difference here is that we do not work with a concrete order, but a general parametrised variant of Zorn's lemma, from which induction on the lexicographic ordering can be considered a special case.…”