`BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta` (beta : ℝ) (s : Fin m → ℝ) (j : Fin m) : deriv (fun b : ℝ => scoreSoftmax b s j) beta = scoreSoftmax beta s j

Proved

BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxTemperatureMonotone.

BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta (beta : ℝ) (s : Fin m → ℝ) (j : Fin m) : deriv (fun b : ℝ => scoreSoftmax b s j) beta = scoreSoftmax beta s j * (s j - meanScore beta s)

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta.

Preamble

-- Generated from ChapterSoftmaxTemperatureMonotone.lean — theorem BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSoftmaxTemperatureMonotone
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxFluctuation
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxTemperatureMonotone


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxTemperatureMonotone.deriv_scoreSoftmax_beta (beta : ℝ) (s : Fin m → ℝ) (j : Fin m) :
    deriv (fun b : ℝ => scoreSoftmax b s j) beta
      = scoreSoftmax beta s j * (s j - meanScore beta s) := by sorry

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