Maximum rank in a finite nonempty feasible family
Provedfinite_exists_max_rank__p2m_9e5fb9d2_aa0e_41c4_9a80_cb078400c4e8_a1approximation-algorithmsplanar-graphstreewidth
On a finite type, any nonempty feasible family has an element whose natural-number rank is at least the rank of every feasible element. This is a generic finite maximization lemma, provable by enumerating the type and taking a maximum over the feasible elements.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem finite_exists_max_rank__p2m_9e5fb9d2_aa0e_41c4_9a80_cb078400c4e8_a1 {α : Type} [Finite α] (P : α → Prop) (rank : α → Nat) (hex : ∃ a, P a) : ∃ a, P a ∧ ∀ b, P b → rank b ≤ rank 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).