`BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos` {beta : ℝ} (hb : 0 ≤ beta) (s : Fin m → ℝ) {i j : Fin m} (hij : j ≠ i) : softmaxJacobian beta s i j ≤ 0

Proved

BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxJacobian.

BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos {beta : ℝ} (hb : 0 ≤ beta) (s : Fin m → ℝ) {i j : Fin m} (hij : j ≠ i) : softmaxJacobian beta s i j ≤ 0

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos.

Preamble

-- Generated from ChapterSoftmaxJacobian.lean — theorem BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSoftmaxJacobian
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxJacobian


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxJacobian.softmaxJacobian_offDiag_nonpos {beta : ℝ} (hb : 0 ≤ beta) (s : Fin m → ℝ)
    {i j : Fin m} (hij : j ≠ i) : softmaxJacobian beta s i j ≤ 0 := by sorry

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