Dyadic Truncation Bound at One Scale

Proved

LevyStochCalc.Brownian.Continuity.dyadicTrunc_near_step_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

Let f : ℝ → ℝ, b : ℕ → ℝ with b(n) ≥ 0, and N : ℕ. Suppose that for every n ≥ N and every integer k with 0 ≤ k and k+1 ≤ 2ⁿ,

\left|f\!\left(\frac{k+1}{2^n}\right)-f\!\left(\frac{k}{2^n}\right)\right|\le b(n).

Then for m ≥ N and 0 ≤ s ≤ t ≤ 1 with t−s ≤ (1/2)^m,

|f(\operatorname{dyadicTrunc}(m,s))-f(\operatorname{dyadicTrunc}(m,t))|\le b(m).

This is the cross-point estimate that connects nearby points at a common dyadic scale.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Continuity. Shown in its original module context. Proof omitted.

lemma dyadicTrunc_near_step {f : ℝ → ℝ} {b : ℕ → ℝ} {N : ℕ}
    (hb : ∀ n, 0 ≤ b n)
    (hf : ∀ n, N ≤ n → ∀ k : ℤ, 0 ≤ k → k + 1 ≤ 2 ^ n →
      |f ((k + 1 : ℤ) / 2 ^ n) - f ((k : ℤ) / 2 ^ n)| ≤ b n)
    {m : ℕ} (hm : N ≤ m) {s t : ℝ} (hs0 : 0 ≤ s) (ht1 : t ≤ 1) (hst : s ≤ t)
    (hclose : t - s ≤ (1 / 2) ^ m) :
    |f (dyadicTrunc m s) - f (dyadicTrunc m t)| ≤ b m

Module navigation

Module: LevyStochCalc/Brownian/ContinuityDyadicChaining.

Previous: dyadicTrunc_telescope · Next: dyadicTrunc_eventually_eq.

Source excerpt, lines 273–279 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ContinuityDyadicChaining

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ContinuityDyadicChaining.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)

Formal statement


lemma LevyStochCalc.Brownian.Continuity.dyadicTrunc_near_step_readable {f : ℝ → ℝ} {b : ℕ → ℝ} {N : ℕ}
    (hb : ∀ n, 0 ≤ b n)
    (hf : ∀ n, N ≤ n → ∀ k : ℤ, 0 ≤ k → k + 1 ≤ 2 ^ n →
      |f ((k + 1 : ℤ) / 2 ^ n) - f ((k : ℤ) / 2 ^ n)| ≤ b n)
    {m : ℕ} (hm : N ≤ m) {s t : ℝ} (hs0 : 0 ≤ s) (ht1 : t ≤ 1) (hst : s ≤ t)
    (hclose : t - s ≤ (1 / 2) ^ m) :
    |f (dyadicTrunc m s) - f (dyadicTrunc m t)| ≤ b m := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ContinuityDyadicChaining.lean#L270-L308 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ContinuityDyadicChaining`, declaration `LevyStochCalc.Brownian.Continuity.dyadicTrunc_near_step`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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