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Crossbar stitching

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

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    Natural Language Statement

    Theorem

    Compatible local routed row families in the designated clusters of a strong path-of-sets system can be stitched into global rows. The pairwise local bridges survive in every designated even cluster.

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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 Lax17.PathOfSets
    2
    3/-!
    4---
    5title: Crossbar stitching
    6type: theorem
    7---
    8Compatible local routed row families in the designated clusters of a strong
    9path-of-sets system can be stitched into global rows. The pairwise local
    10bridges survive in every designated even cluster.
    11-/
    12
    13namespace Lax17.CrossbarStitching
    14
    15universe u
    16
    17/-- The row-stitching conclusion used before sparse-grid assembly. -/
    18axiom crossbarStitching :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) {g : ℕ}
    21 (P : Lax17.PathOfSets.StrongSystem G
    22 (2 * g * (g - 1)) (16 * g ^ 2 + 10 * g)),
    23 2 ≤ g →
    24 (∀ i : Fin (2 * g * (g - 1)),
    25 ∃ localRows :
    26 Lax17.Paths.VertexLinkage G
    27 (P.left i) (P.right i) g,
    28 (∀ row : Fin g,
    29 (localRows.path row).StaysIn (P.cluster i)) ∧
    30 localRows.HasPairwiseBridgesIn (P.cluster i)) →
    31 ∃ rows :
    32 Lax17.Paths.VertexLinkage G
    33 (P.left P.toSystem.firstIndex)
    34 (P.right P.toSystem.lastIndex) g,
    35 ∀ i : Fin (g * (g - 1)),
    36 ∃ clusterIndex : Fin (2 * g * (g - 1)),
    37 clusterIndex.1 = 2 * i.1 + 1 ∧
    38 rows.HasPairwiseBridgesIn
    39 (P.cluster clusterIndex)
    40
    41end Lax17.CrossbarStitching
    42
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