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Local routing-or-grid alternative

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

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    Natural Language Statement

    Theorem

    In one strong cluster, large equal terminal sets either admit the desired local routing or force the target grid minor.

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

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    1import Lax17.GridMinor
    2import Lax17.PathOfSets
    3
    4/-!
    5---
    6title: Local routing-or-grid alternative
    7type: theorem
    8---
    9In one strong cluster, large equal terminal sets either admit the desired
    10local routing or force the target grid minor.
    11-/
    12
    13namespace Lax17.LocalRoutingOrGrid
    14
    15universe u
    16
    17/-- In one connected cluster, large linked terminal sets either force an
    18`h × h` grid minor or admit `q` disjoint routes with pairwise bridges. -/
    19axiom localRoutingOrGrid :
    20 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    21 (G : SimpleGraph V) {C A B : Finset V} {h q w : ℕ},
    22 1 < h →
    23 1 < q →
    24 Lax17.PathOfSets.IsCluster G C →
    25 Lax17.Linkedness.NodeLinkedIn G C A B →
    26 A.card = w →
    27 B.card = w →
    28 (16 * h + 10) * q ≤ w →
    29 Lax17.GridMinor.ContainsGridMinor G h ∨
    30 ∃ Q : Lax17.Paths.VertexLinkage G A B q,
    31 (∀ i : Fin q, (Q.path i).StaysIn C) ∧
    32 Q.HasPairwiseBridgesIn C
    33
    34end Lax17.LocalRoutingOrGrid
    35
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