`BookProof.ChapterSoftmaxJacobian.softmaxJacobian_posSemidef` {beta : ℝ} (hb : 0 ≤ beta) (s x : Fin m → ℝ) (i : Fin m) : 0 ≤ ∑ i, ∑ j, x i * softmaxJacobian beta s i j * x j

Proved

BookProof.ChapterSoftmaxJacobian.softmaxJacobian_posSemidef

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxJacobian.

BookProof.ChapterSoftmaxJacobian.softmaxJacobian_posSemidef {beta : ℝ} (hb : 0 ≤ beta) (s x : Fin m → ℝ) (i : Fin m) : 0 ≤ ∑ i, ∑ j, x i * softmaxJacobian beta s i j * x j

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxJacobian.softmaxJacobian_posSemidef {beta : ℝ} (hb : 0 ≤ beta) (s x : Fin m → ℝ) (i : Fin m) :
    0 ≤ ∑ i, ∑ j, x i * softmaxJacobian beta s i j * x j := by sorry

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