Edge finset cardinality is independent of the finite instance

Proved

WilliamsonShmoys.edgeFinset_card_instance_irrel__p2m_160c891e_c18d_4863_b495_0a238ee51219_a3

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite graph, the number of edges in its edge finset is the same whether the edge-set subtype uses the default finite instance or the graph’s canonical finite instance.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.edgeFinset_card_instance_irrel__p2m_160c891e_c18d_4863_b495_0a238ee51219_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) :
    (letI : DecidableRel G.Adj := Classical.decRel G.Adj
     letI : Fintype ↑G.edgeSet := Fintype.ofFinite _
     G.edgeFinset.card) =
    (letI : DecidableRel G.Adj := Classical.decRel G.Adj
     letI : Fintype ↑G.edgeSet := G.fintypeEdgeSet
     G.edgeFinset.card) := by sorry

Source

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).

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