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Lax56.ValtrFourLayer

Valtr's four-layer lemma

concepts/Lax56/ValtrFourLayer.lean · lax-56

proven

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    Theorem

    This theorem specification is proved in Lax56Proofs.ValtrFourLayerLax56Proofs.ValtrFourLayer. The outer-layer threshold sixteen allows an unoptimized endgame; it is sufficient for the 216-point convex-position application. Chain replacement, all endpoint cases, and the four-layer reduction are proved in Lean. The empty-hexagon proof uses this theorem interface so Lax records the dependency on the proof supplied in this package.

    Reference: Pavel Valtr, "On Empty Hexagons", Section 3 (Section 2 in the author's preprint at https://kam.mff.cuni.cz/~valtr/h.ps).

    Lean source view on GitHub

    1import Lax56.ConvexLayers
    2
    3/-!
    4---
    5title: Valtr's four-layer lemma
    6type: theorem
    7---
    8This theorem specification is proved in `Lax56Proofs.ValtrFourLayer`.
    9The outer-layer threshold sixteen allows an unoptimized endgame; it is
    10sufficient for the 216-point convex-position application. Chain replacement,
    11all endpoint cases, and the four-layer reduction are proved in Lean. The
    12empty-hexagon proof uses this theorem interface so Lax records the dependency
    13on the proof supplied in this package.
    14
    15Reference: Pavel Valtr, "On Empty Hexagons", Section 3 (Section 2 in the
    16author's preprint at https://kam.mff.cuni.cz/~valtr/h.ps).
    17-/
    18
    19namespace Lax56.ValtrFourLayer
    20
    21open Lax56.Geometry Lax56.ConvexLayers
    22
    23/-- The weakened four-layer lemma requested for the unoptimized proof.
    24`layer S 3` is the fourth layer because layer indices start at zero. -/
    25axiom exists_emptyHexagon_of_four_layers
    26 (S : Finset Point) (hgeneral : ¬HasThreeCollinear S)
    27 (hminimal : MinimalOuter S) (hlarge : 16 ≤ (extremeLayer S).card)
    28 (hfourth : (layer S 3).Nonempty) : HasEmptyHexagon S
    29
    30end Lax56.ValtrFourLayer
    31
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