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Inserting a twin preserves crossings

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

proven

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    Natural Language Statement

    Lemma

    Duplicating a membership entry immediately next to itself leaves the crossing count unchanged. Applying this to every set proves Lemma 2.1 of Dreier–Kuske: an adjacent duplicate of a twin does not change any crossing count.

    Concept map
    2 concepts
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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 Lax235315.SequenceCrossings
    2
    3/-!
    4---
    5title: Inserting a twin preserves crossings
    6type: lemma
    7---
    8Duplicating a membership entry immediately next to itself leaves the crossing
    9count unchanged. Applying this to every set proves Lemma 2.1 of
    10Dreier--Kuske: an adjacent duplicate of a twin does not change any crossing count.
    11
    12# Formalization notes
    13
    14The arbitrary prefix and suffix allow insertion at every position, including
    15the ends. This is a claim about membership sequences, independent of whether
    16the duplicated entries represent distinct vertices.
    17-/
    18
    19namespace Lax235315.TwinInsertion
    20open Lax235315.SequenceCrossings
    21
    22/-- Adjacent repetition of a bit preserves the number of changes. -/
    23axiom crossings_duplicate (pre post : List Bool) (b : Bool) :
    24 crossings (pre ++ b :: b :: post) =
    25 crossings (pre ++ b :: post)
    26
    27end Lax235315.TwinInsertion
    28
    Show Proof
    Formalization notes

    The arbitrary prefix and suffix allow insertion at every position, including the ends. This is a claim about membership sequences, independent of whether the duplicated entries represent distinct vertices.

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