A bounded strict-progress process has a terminal state
Provedbounded_strict_progress_has_terminal__p2m_a9a5aac4_cf42_46e5_abca_ec8765edbc1f_a1approximation-algorithmsplanar-graphstreewidth
If there is an initial state satisfying P, all P-states have a natural-number measure bounded by N, and every P-state failing Q has a P-state of strictly larger measure, then some P-state satisfies Q.
Preamble
import Mathlib
Formal statement
theorem bounded_strict_progress_has_terminal__p2m_a9a5aac4_cf42_46e5_abca_ec8765edbc1f_a1
{α : Type} (P Q : α → Prop) (μ : α → Nat) (N : Nat)
(hbase : ∃ a : α, P a)
(hbound : ∀ a : α, P a → μ a ≤ N)
(hprogress : ∀ a : α, P a → ¬ Q a → ∃ b : α, P b ∧ μ a < μ b) :
∃ a : α, P a ∧ Q 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).