Evaluation of the Initial Step Integrand

Proved

LevyStochCalc.Brownian.Ito.stepIoc_zero_eval_readable

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

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

Let $b>0$ , let $s∈ℝ$ , and let $ω∈Ω$ . The elementary step integrand stepIoc₀ Ω hb, associated with the interval $(0,b]$ , evaluates as the indicator of that interval:

\texttt{stepIoc₀}(s,ω)=\mathbf{1}_{(0,b]}(s).

Equivalently, the value is $1$ when $0<s≤b$ and $0$ otherwise. The sample point $ω$ is arbitrary because the step coefficient is constant in $ω$ . This is an algebraic evaluation lemma for simple predictable integrands.

Original Lean declaration header

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

theorem stepIoc₀_eval {b : ℝ} (hb : 0 < b) (s : ℝ) (ω : Ω) :
    (stepIoc₀ Ω hb).eval s ω = (Set.Ioc 0 b).indicator (fun _ => (1 : ℝ)) s

Module navigation

Module: LevyStochCalc/Brownian/ItoAlgebraAssociativity.

Definition modules: LevyStochCalc/Brownian/ItoSimpleIntegrand.

Previous: stepIoc_eval · Next: stochasticIntegralBrownian_indicator_Ioc.

Source excerpt, lines 320–321 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoAlgebraAssociativity

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Modified 2026-10-12 by the KLS formalization project: ported to Lean v4.33.0 and
-- Mathlib db584cd. The changes are listed in vendor/README.md and vendor/port.patch.
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ItoAlgebraAssociativity.lean; Apache-2.0; locally patched, see formalization/vendor/port.patch; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement

theorem LevyStochCalc.Brownian.Ito.stepIoc_zero_eval_readable {b : ℝ} (hb : 0 < b) (s : ℝ) (ω : Ω) :
    (stepIoc₀ Ω hb).eval s ω = (Set.Ioc 0 b).indicator (fun _ => (1 : ℝ)) s := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoAlgebraAssociativity.lean (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoAlgebraAssociativity`, declaration `LevyStochCalc.Brownian.Ito.stepIoc₀_eval`; the repository is private at upload time); vendored copy patched by formalization/vendor/port.patch (project file formalization/vendor/LevyStochCalc/LevyStochCalc/Brownian/ItoAlgebraAssociativity.lean, lines 320-333). 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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