A nonempty subset of a finite partial order has a maximal element
ProvedWilliamsonShmoys.finite_nonempty_subset_has_maximal__p2m_350700c7_dd24_47f5_9bef_0db864652b8b_a3approximation-algorithmsplanar-graphstreewidth
Every nonempty subset of a finite partially ordered type has an element with no strictly larger element in that subset.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_nonempty_subset_has_maximal__p2m_350700c7_dd24_47f5_9bef_0db864652b8b_a3 {α : Type} [Finite α] [PartialOrder α] (S : Set α) (hS : S.Nonempty) : ∃ a ∈ S, ∀ b ∈ S, a ≤ b → b = a := 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).