Formalising Ordinal Partition Relations Using Isabelle/HOL

Abstract

This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

Towards Formalising Schutz' Axioms for Minkowski Spacetime in Isabelle/HOL

“…Estimating thirty lines on each page, this leaves us with a de Bruijn factor [60,7] of roughly 14. This is not exceptional: while many formalisations only report de Bruijn factors as low as 3 to 6, values above 20 can be found [9]. We also note that a recent batch of simplifications and rewritten proofs has cut our formalisation by about eight hundred lines, so this estimated factor may be further reducible.…”

“…The proof of Theorem 9 hinges on three lemmas that are, to any practical purpose, as important as any result of this chapter, and allow us to work with orderings of overlapping sets of events. Theorem 8 presupposes the easy result (not explicitly mentioned by Schutz) that △abc implies no betweenness ordering of a, b, c exists, and extends it to events on the paths defining the triangle (rather than its vertices) 9 . Using some geometric intuition, Theorem 8 might be likened to the statement that no path can cross all three sides of a kinematic triangle internally.…”

Formalising Geometric Axioms for Minkowski Spacetime and Without-Loss-of-Generality Theorems

“…Estimating thirty lines on each page, this leaves us with a de Bruijn factor [29,4] of roughly 14. This is not exceptional: while many formalisations only report de Bruijn factors as low as 3 to 6, values above 20 can be found [5]. One should note that the axiomatisation by itself would have a factor of only around 4.…”

A Formalised Theorem in the Partition Calculus

“…The task for formalising Erdős and Milner's paper arose in the context of a larger project, with Džamonja and Koutsoukou-Argyraki [35], to formalise Larson's proof [34] that ω ω −→ (ω ω , m) for all m < ω. Her proof relies on…”