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Factorial estimates for iterated analytic vector fields

Lax825748.FactorialEstimate · concepts/Lax825748/FactorialEstimate.lean · lax-825748

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    Natural Language Statement

    Theorem

    Suppose the output and all vector-field coefficients admit the same bound αC∣I∣\alpha C^{|I|}, where α,C>0\alpha,C>0. For a word of length pp, its generated coefficient has absolute value at most α(2dαC)pp!\alpha(2d\alpha C)^p p!. Consequently every analytic representation on a finite alphabet has a uniform factorial bound on its word coefficients.

    The central binomial estimate used in the majorant calculation is (2pp)≤4p\binom{2p}{p}\leq 4^p, rather than the printed 2p2^p.

    Concept map
    7 concepts; 4 descendants hidden
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 factorial_bound proven

    2 word_majorant proven

    In the paper

    • page 6 of this submission's paper

    Lean source view on GitHub

    1import Lax825748.AnalyticRepresentations
    2
    3/-!
    4---
    5title: Factorial estimates for iterated analytic vector fields
    6type: theorem
    7---
    8Suppose the output and all vector-field coefficients admit the same bound
    9αC∣I∣\alpha C^{|I|}, where α,C>0\alpha,C>0. For a word of length pp, its generated
    10coefficient has absolute value at most α(2dαC)pp!\alpha(2d\alpha C)^p p!.
    11Consequently every analytic representation on a finite alphabet has a
    12uniform factorial bound on its word coefficients.
    13
    14The central binomial estimate used in the majorant calculation is
    15(2pp)≤4p\binom{2p}{p}\leq 4^p, rather than the printed 2p2^p.
    16-/
    17
    18noncomputable section
    19
    20namespace Lax825748.FactorialEstimate
    21
    22open Lax825748.Series Lax825748.DifferentialRepresentation
    23open ConvergentSeries AnalyticRepresentations
    24
    25def HasFactorialBound {X : Type} (g : NCSeries X) : Prop :=
    26 ∃ α : ℝ, 0 < α ∧ ∃ C : ℝ, 0 < C ∧ ∀ w : Word X,
    27 |g w| ≤ α * C ^ w.toList.length * (w.toList.length.factorial : ℝ)
    28
    29axiom word_majorant {X : Type} {d : ℕ} (R : Representation X d)
    30 (α C : ℝ) (hα : 0 < α) (hC : 0 < C)
    31 (hh : HasGeometricBound R.output α C)
    32 (hA : ∀ x i, HasGeometricBound (R.fields x i) α C) (w : Word X) :
    33 |generatingSeries R w| ≤
    34 α * (2 * (d : ℝ) * α * C) ^ w.toList.length * (w.toList.length.factorial : ℝ)
    35
    36axiom factorial_bound {X : Type} [Finite X] {d : ℕ}
    37 (R : Representation X d) (hR : IsAnalytic R) :
    38 HasFactorialBound (generatingSeries R)
    39
    40end Lax825748.FactorialEstimate
    41
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