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Replacing near twins in a set system

Lax235315.NearTwinReplacement · concepts/Lax235315/NearTwinReplacement.lean · lax-235315

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    Natural Language Statement

    Lemma

    Suppose every set in a finite set system differs from some representative set on at most k vertices. Every order with crossing number at most m for the representatives has crossing number at most m+2k for the original family. This is Lemma 2.2 of Dreier–Kuske.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax195003.WelzlOrders
    2import Mathlib.Data.Set.SymmDiff
    3
    4/-!
    5---
    6title: Replacing near twins in a set system
    7type: lemma
    8---
    9Suppose every set in a finite set system differs from some representative
    10set on at most k vertices. Every order with crossing number at most m for
    11the representatives has crossing number at most m+2k for the original
    12family. This is Lemma 2.2 of Dreier--Kuske.
    13
    14# Formalization notes
    15
    16The claim uses the exact crossing number of Lax195003, including its
    17permutation convention and natural supremum. No sampling or graph hypothesis
    18is needed. The representative relation permits different original sets to
    19share the same representative.
    20-/
    21
    22namespace Lax235315.NearTwinReplacement
    23open scoped symmDiff
    24open Lax195003.WelzlOrders
    25
    26/-- Near-twin representatives incur at most two crossings per changed vertex. -/
    27axiom crossingNumber_le_add_two_mul {n k m : ℕ}
    28 {F R : SetSystem (Fin n)} {π : Equiv.Perm (Fin n)}
    29 (hrep : ∀ X ∈ F, ∃ Y ∈ R, (X ∆ Y).ncard ≤ k)
    30 (hπ : crossingNumber R π ≤ m) :
    31 crossingNumber F π ≤ m + 2 * k
    32
    33end Lax235315.NearTwinReplacement
    34
    Show Proof
    Formalization notes

    The claim uses the exact crossing number of Lax195003, including its permutation convention and natural supremum. No sampling or graph hypothesis is needed. The representative relation permits different original sets to share the same representative.

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