Norm Bound Before a Continuous-Path Exit Time

Proved

LevyStochCalc.Probability.norm_le_of_le_exitTime_readable

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

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

This bounds a continuous path before its exit time.

Let $E$ be a normed additive commutative group, let $X:\Omega\to\mathbb R\to E$ have a continuous path at $\omega$ , and let $s>0$ . If

s\le\operatorname{exitTime}(X,k,\omega),

then

\|X(\omega,s)\|\le k.

The restriction $s>0$ is part of the statement. The lemma is used to obtain uniform bounds on paths before localization.

Original Lean declaration header

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

theorem norm_le_of_le_exitTime {X : Ω → ℝ → E} {ω : Ω} (hX : Continuous (X ω)) {k s : ℝ}
    (hs : 0 < s) (hle : (s : WithTop ℝ) ≤ exitTime X k ω) : ‖X ω s‖ ≤ k

Module navigation

Module: LevyStochCalc/Probability/ExitTime.

Previous: exitTime_le_iff_forall · Next: exists_lt_exitTime.

Source excerpt, lines 194–195 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Probability_ExitTime

/-
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/Probability/ExitTime.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u v
variable {Ω : Type u} {mΩ : MeasurableSpace Ω} {E : Type v} [NormedAddCommGroup E]

Formal statement


theorem LevyStochCalc.Probability.norm_le_of_le_exitTime_readable {X : Ω → ℝ → E} {ω : Ω} (hX : Continuous (X ω)) {k s : ℝ}
    (hs : 0 < s) (hle : (s : WithTop ℝ) ≤ exitTime X k ω) : ‖X ω s‖ ≤ k := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Probability/ExitTime.lean#L191-L213 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Probability.ExitTime`, declaration `LevyStochCalc.Probability.norm_le_of_le_exitTime`; 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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