Roth's Theorem on Arithmetic Progressions

We formalise a proof of Roth's Theorem on Arithmetic Progressions, a major result in additive combinatorics on the existence of 3-term arithmetic progressions in subsets of natural numbers. To this end, we follow a proof using graph regularity. We employ our recent formalisation of Szemerédi's Regularity Lemma, a major result in extremal graph theory, which we use here to prove the Triangle Counting Lemma and the Triangle Removal Lemma. Our sources are Yufei Zhao's MIT lecture notes "Graph Theory and Additive Combinatorics" (revised version here) and W.T. Gowers's Cambridge lecture notes "Topics in Combinatorics". We also refer to the University of Georgia notes by Stephanie Bell and Will Grodzicki, "Using Szemerédi's Regularity Lemma to Prove Roth's Theorem".