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Subexponential container counts at the paper sample scale

Lax342547.ContainerAsymptotics · concepts/Lax342547/ContainerAsymptotics.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The logarithmic fingerprint count is bounded by a degree-three polynomial times 2^(100gN). Its ratio to 2^(1000gN) tends to zero, with an actual eventual strict exponent margin.

    Concept map
    1 concept; 5 descendants hidden
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    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 container_exponent_little_o proven

    2 eventually_container_exponent proven

    Lean source view on GitHub

    1import Mathlib.Analysis.SpecificLimits.Normed
    2import Mathlib.Tactic
    3
    4/-!
    5---
    6title: Subexponential container counts at the paper sample scale
    7type: lemma
    8---
    9The logarithmic fingerprint count is bounded by a degree-three polynomial times 2^(100gN). Its ratio to 2^(1000gN) tends to zero, with an actual eventual strict exponent margin.
    10-/
    11
    12namespace Lax342547.ContainerAsymptotics
    13
    14open Filter Asymptotics
    15
    16def exponentBudget (A g N : ℕ) : ℕ := (2*A*N^2+1)*(4000*g*N*2^(100*g*N)+2)
    17
    18axiom container_exponent_little_o (A g : ℕ) (hg : 0 < g) :
    19 (fun N : ℕ => (exponentBudget A g N : ℝ)) =o[atTop] fun N => (2 : ℝ)^(1000*g*N)
    20
    21axiom eventually_container_exponent (A g : ℕ) (hg : 0 < g) :
    22 ∀ᶠ N : ℕ in atTop, exponentBudget A g N+2 < 2^(1000*g*N)
    23
    24end Lax342547.ContainerAsymptotics
    25
    Show ProofShow Proof
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