`BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition` (beta : ℝ) (s : Fin m → ℝ) : HasDerivAt (fun b : ℝ => partition b s) (∑ l, s l * Real.exp (beta * s l)) beta

Proved

BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxFluctuation.

BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition (beta : ℝ) (s : Fin m → ℝ) : HasDerivAt (fun b : ℝ => partition b s) (∑ l, s l * Real.exp (beta * s l)) beta

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition.

Preamble

-- Generated from ChapterSoftmaxFluctuation.lean — theorem BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSoftmaxFluctuation
open BookProof.ChapterSoftmaxFluctuation


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxFluctuation.hasDerivAt_partition (beta : ℝ) (s : Fin m → ℝ) :
    HasDerivAt (fun b : ℝ => partition b s)
      (∑ l, s l * Real.exp (beta * s l)) beta := by sorry

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