A finite predicate has a maximal element
Provedfinite_predicate_has_maximal__p2m_5e5b150a_02d6_4c7e_b48f_ee76aa630368_a3approximation-algorithmsplanar-graphstreewidth
Every nonempty predicate on a finite partially ordered type contains an element that has no strictly larger element within the predicate.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem finite_predicate_has_maximal__p2m_5e5b150a_02d6_4c7e_b48f_ee76aa630368_a3
{α : Type} [Fintype α] [PartialOrder α] (p : α → Prop)
(hp : ∃ x, p x) :
∃ m, p m ∧ ∀ x, p x → m ≤ x → x ≤ m := by
sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).