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Polynomial order bound from the sampled width inequality

Lax263307.WidthGrowth · concepts/Lax263307/WidthGrowth.lean · lax-263307

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    Natural Language Statement

    Theorem

    Suppose W(m)a≤A(m+1)a−1W(m)^a\le A(m+1)^{a-1} and W(m)≥1W(m)\ge1. For a source graph of order tt and at most 2t2t edges, take route cutoff 24cℓ24c\ell. If its linear width lower bound and discarded-route estimate are both compatible with the sampled width upper bound, then t≤(12c)aA(6(24c)2+1)a−1ℓ2(a−1)t\le (12c)^a A(6(24c)^2+1)^{a-1}\ell^{2(a-1)}. This integer inequality makes the sampling contradiction polynomial in ell.

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    1import Mathlib.Data.Nat.Basic
    2
    3/-!
    4---
    5title: Polynomial order bound from the sampled width inequality
    6type: theorem
    7---
    8Suppose W(m)a≤A(m+1)a−1W(m)^a\le A(m+1)^{a-1} and W(m)≥1W(m)\ge1. For a source graph of
    9order tt and at most 2t2t edges, take route cutoff 24cℓ24c\ell. If its linear
    10width lower bound and discarded-route estimate are both compatible with
    11the sampled width upper bound, then
    12t≤(12c)aA(6(24c)2+1)a−1ℓ2(a−1)t\le (12c)^a A(6(24c)^2+1)^{a-1}\ell^{2(a-1)}.
    13This integer inequality makes the sampling contradiction polynomial in ell.
    14
    15# Attribution
    16
    17The growth calculation in Bonnet–Chang, proof of Lemma 11.
    18-/
    19
    20namespace Lax263307.WidthGrowth
    21
    22def threshold (a A c : ℕ) : ℕ := (12 * c) ^ a * A * (6 * (24 * c) ^ 2 + 1) ^ (a - 1)
    23
    24axiom order_bound (a A c ℓ t m q : ℕ) (a_positive : 0 < a) (ℓ_positive : 0 < ℓ)
    25 (t_positive : 0 < t) (W : ℕ → ℕ)
    26 (W_positive : ∀ M, 1 ≤ W M) (W_power : ∀ M, (W M) ^ a ≤ A * (M + 1) ^ (a - 1))
    27 (edges : m ≤ 2 * t) (discarded : (24 * c * ℓ) * q ≤ 4 * m * ℓ)
    28 (width : t ≤ c * (max 1 (6 * W (3 * m * (24 * c * ℓ) ^ 2)) + q)) :
    29 t ≤ threshold a A c * ℓ ^ (2 * (a - 1))
    30
    31end Lax263307.WidthGrowth
    32
    Show Proof

    Attribution

    The growth calculation in Bonnet–Chang, proof of Lemma 11.

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