Lax Lean Archive

Crossbar-or-pseudo-grid dichotomy

Lax17.CrossbarOrPseudoGrid · concepts/Lax17/CrossbarOrPseudoGrid.lean · lax-17

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    Natural Language Statement

    Theorem

    Theorem 4.1: two full disjoint linkage families yield either a crossbar or a pseudo-grid.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax17.Crossbar
    2import Lax17.Degree
    3
    4/-!
    5---
    6title: Crossbar-or-pseudo-grid dichotomy
    7type: theorem
    8---
    9Theorem 4.1: two full disjoint linkage families yield either a crossbar or a
    10pseudo-grid.
    11-/
    12
    13namespace Lax17.CrossbarOrPseudoGrid
    14
    15universe u
    16
    17/-- The self-contained crossbar-or-pseudo-grid form of Theorem 4.1. -/
    18axiom crossbarOrPseudoGrid :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) {A B X : Finset V} {g κ D : ℕ},
    21 2 ≤ g →
    22 Lax17.Crossbar.IsPowerOfTwo g →
    23 A.card = κ → B.card = κ → X.card = κ →
    24 Disjoint A B → Disjoint A X → Disjoint B X →
    25 (∀ x ∈ X, Lax17.Degree.Exactly G x 1) →
    26 Lax17.Paths.VertexLinkage G A B κ →
    27 Lax17.Paths.VertexLinkage G A X κ →
    28 1 ≤ D → D ≤ κ / (2 * g ^ 2) →
    29 Nonempty
    30 (Lax17.Crossbar.System G A B X (g ^ 2)) ∨
    31 Nonempty
    32 (Lax17.Crossbar.PseudoGrid
    33 G A B X g D κ)
    34
    35end Lax17.CrossbarOrPseudoGrid
    36
    Show Proof
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