Prove2Me

Evaluation of a Product on a Common Refinement

Proved
LevyStochCalc.Brownian.Ito.SimplePredictable.eval_mul_on_common_readable

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

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

Let H1,H2H₁,H₂ be simple predictable integrands with equal final partition points. Their common-refinement product H₁.mul_on_common H₂ h_eq evaluates pointwise as the product of their evaluations:

eval(H1.mul_on_common H2,s,ω)=eval(H1,s,ω) eval(H2,s,ω).\texttt{eval}(H₁\texttt{.mul\_on\_common}\ H₂,s,ω)=\texttt{eval}(H₁,s,ω)\,\texttt{eval}(H₂,s,ω).

The conclusion holds for every s∈Rs∈ℝ and ω∈Ωω∈Ω. The result is purely finite-dimensional partition algebra and is used to represent products as simple integrands.

Original Lean declaration header

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

theorem SimplePredictable.eval_mul_on_common
    {T : ℝ} (H₁ H₂ : SimplePredictable Ω T)
    (h_eq : H₁.partition (Fin.last H₁.N) = H₂.partition (Fin.last H₂.N))
    (s : ℝ) (ω : Ω) :
    (H₁.mul_on_common H₂ h_eq).eval s ω = H₁.eval s ω * H₂.eval s ω

Module navigation

Module: LevyStochCalc/Brownian/ItoAlgebraProduct.

Definition modules: LevyStochCalc/Brownian/ItoSimpleIntegrand.

Supporting theorems: sum_ite_mul_sum_ite.

Previous: sum_ite_mul_sum_ite · Next: SimplePredictable.sum_xi_mul_simpleIntegral_sub.

Source excerpt, lines 83–87 · Frozen source 408915f.

Preamble
import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoAlgebraProduct

/-
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/ItoAlgebraProduct.lean; Apache-2.0; 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.SimplePredictable.eval_mul_on_common_readable
    {T : ℝ} (H₁ H₂ : SimplePredictable Ω T)
    (h_eq : H₁.partition (Fin.last H₁.N) = H₂.partition (Fin.last H₂.N))
    (s : ℝ) (ω : Ω) :
    (H₁.mul_on_common H₂ h_eq).eval s ω = H₁.eval s ω * H₂.eval s ω := by sorry
Source
Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoAlgebraProduct.lean#L82-L103 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoAlgebraProduct`, declaration `LevyStochCalc.Brownian.Ito.SimplePredictable.eval_mul_on_common`; 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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