Prove2Me

Telescoping Dyadic Truncation Bound

Proved
LevyStochCalc.Brownian.Continuity.dyadicTrunc_telescope_readable

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

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

Let f : ℝ → ℝ and b : ℕ → ℝ satisfy b(n) ≥ 0 and the consecutive dyadic increment bound at every level n ≥ N. For 0 ≤ x ≤ 1, m with N ≤ m+1, and L ≥ m,

∣f(dyadicTrunc⁡(L,x))−f(dyadicTrunc⁡(m,x))∣≤∑n∈{m+1,…,L}b(n).|f(\operatorname{dyadicTrunc}(L,x))-f(\operatorname{dyadicTrunc}(m,x))|\le\sum_{n\in\{m+1,\ldots,L\}}b(n).

In Lean this sum is written over Finset.Ico (m + 1) (L + 1). The lemma dyadicTrunc_telescope_readable is the multiscale chaining estimate obtained by accumulating the successive truncation bounds.

Original Lean declaration header

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

lemma dyadicTrunc_telescope {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)
    {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) {m : ℕ} (hm : N ≤ m + 1) :
    ∀ L, m ≤ L → |f (dyadicTrunc L x) - f (dyadicTrunc m x)|
      ≤ ∑ n ∈ Finset.Ico (m + 1) (L + 1), b n

Module navigation

Module: LevyStochCalc/Brownian/ContinuityDyadicChaining.

Supporting theorems: dyadicTrunc_succ_step.

Previous: dyadicTrunc_succ_step · Next: dyadicTrunc_near_step.

Source excerpt, lines 248–254 · 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_telescope_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)
    {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) {m : ℕ} (hm : N ≤ m + 1) :
    ∀ L, m ≤ L → |f (dyadicTrunc L x) - f (dyadicTrunc m x)|
      ≤ ∑ n ∈ Finset.Ico (m + 1) (L + 1), b n := by sorry
Source
Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ContinuityDyadicChaining.lean#L245-L268 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ContinuityDyadicChaining`, declaration `LevyStochCalc.Brownian.Continuity.dyadicTrunc_telescope`; 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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