Maximal element in a nonempty subset of a finite partial order
Provedexists_maximal_mem_of_finite__p2m_d662b61a_1996_4a06_ab90_0abefab2bcb6_a1approximation-algorithmsplanar-graphstreewidth
Every nonempty subset of a finite partially ordered type contains an element with no strictly larger member in that subset.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem exists_maximal_mem_of_finite__p2m_d662b61a_1996_4a06_ab90_0abefab2bcb6_a1
{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).