Clamping Does Not Increase Distance to a Bounded Target

Proved

LevyStochCalc.Brownian.Ito.abs_clamp_sub_le_readable

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

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

This elementary real-analysis lemma states that clamping a value to $[-C,C]$ cannot increase its distance from a target already in that interval.

For real numbers $C,x,y$ , assume $C\ge0$ and $|y|\le C$ . Then

\left|\max(-C,\min(C,x))-y\right|\le |x-y|.

The inequality is used to control approximation errors after applying SimplePredictable.truncate to coefficients.

Original Lean declaration header

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

theorem abs_clamp_sub_le {C x y : ℝ} (hC : 0 ≤ C) (hy : |y| ≤ C) :
    |max (-C) (min C x) - y| ≤ |x - y|

Module navigation

Module: LevyStochCalc/Brownian/ItoFourthMomentLintegral.

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Source excerpt, lines 95–96 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoFourthMomentLintegral

/-
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/ItoFourthMomentLintegral.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u

Formal statement


theorem LevyStochCalc.Brownian.Ito.abs_clamp_sub_le_readable {C x y : ℝ} (hC : 0 ≤ C) (hy : |y| ≤ C) :
    |max (-C) (min C x) - y| ≤ |x - y| := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoFourthMomentLintegral.lean#L94-L107 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoFourthMomentLintegral`, declaration `LevyStochCalc.Brownian.Ito.abs_clamp_sub_le`; 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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