Product of Step Sums on a Strict Partition

Proved

LevyStochCalc.Brownian.Ito.sum_ite_mul_sum_ite_readable

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

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

Let $π:Fin(M+1)→ℝ$ be strictly monotone, let $f,g:Fin\,M→ℝ$ , and let $s∈ℝ$ . At most one partition cell $(π(j),π(j+1)]$ contains $s$ . Consequently,

\left(\sum_j\mathbf{1}_{(π_j,π_{j+1}]}(s)f_j\right)\left(\sum_j\mathbf{1}_{(π_j,π_{j+1}]}(s)g_j\right) =\sum_j\mathbf{1}_{(π_j,π_{j+1}]}(s)f_jg_j.

This is a finite algebraic identity for step functions and is independent of measure-theoretic assumptions. It underlies evaluation of products of simple predictable integrands.

Original Lean declaration header

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

theorem sum_ite_mul_sum_ite {M : ℕ} {π : Fin (M + 1) → ℝ} (hπ : StrictMono π)
    (f g : Fin M → ℝ) (s : ℝ) :
    (∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then f j else 0)
      * (∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then g j else 0)
      = ∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then f j * g j else 0

Module navigation

Module: LevyStochCalc/Brownian/ItoAlgebraProduct.

Next: SimplePredictable.eval_mul_on_common.

Source excerpt, lines 28–32 · 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

Formal statement


theorem LevyStochCalc.Brownian.Ito.sum_ite_mul_sum_ite_readable {M : ℕ} {π : Fin (M + 1) → ℝ} (hπ : StrictMono π)
    (f g : Fin M → ℝ) (s : ℝ) :
    (∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then f j else 0)
      * (∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then g j else 0)
      = ∑ j : Fin M, if π j.castSucc < s ∧ s ≤ π j.succ then f j * g j else 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoAlgebraProduct.lean#L26-L57 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoAlgebraProduct`, declaration `LevyStochCalc.Brownian.Ito.sum_ite_mul_sum_ite`; 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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