Prove2Me

Successive Dyadic Truncation Step

Proved
LevyStochCalc.Brownian.Continuity.dyadicTrunc_succ_step_readable

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

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

Let f and b satisfy the nonnegative level bounds from dyadicTrunc_near_step_readable: for every n ≥ N and every admissible integer k, consecutive level-n dyadics have f-increment at most b(n). If n+1 ≥ N and 0 ≤ x ≤ 1, then

∣f(dyadicTrunc⁡(n+1,x))−f(dyadicTrunc⁡(n,x))∣≤b(n+1).|f(\operatorname{dyadicTrunc}(n+1,x))-f(\operatorname{dyadicTrunc}(n,x))|\le b(n+1).

The lemma dyadicTrunc_succ_step_readable controls one refinement of the dyadic approximation.

Original Lean declaration header

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

lemma dyadicTrunc_succ_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)
    {n : ℕ} (hn : N ≤ n + 1) {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
    |f (dyadicTrunc (n + 1) x) - f (dyadicTrunc n x)| ≤ b (n + 1)

Module navigation

Module: LevyStochCalc/Brownian/ContinuityDyadicChaining.

Previous: dense_dyadicRationals · Next: dyadicTrunc_telescope.

Source excerpt, lines 212–217 · 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_succ_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)
    {n : ℕ} (hn : N ≤ n + 1) {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
    |f (dyadicTrunc (n + 1) x) - f (dyadicTrunc n x)| ≤ b (n + 1) := by sorry
Source
Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ContinuityDyadicChaining.lean#L209-L243 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ContinuityDyadicChaining`, declaration `LevyStochCalc.Brownian.Continuity.dyadicTrunc_succ_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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