Prove2Me

`BookProof.ChapterSoftmaxJacobian.hasDerivAt_partition_score` (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : HasDerivAt (fun t : ℝ => partition beta (scorePerturb s i t)) (beta * Real.ex

Proved
BookProof.ChapterSoftmaxJacobian.hasDerivAt_partition_score

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxJacobian.

BookProof.ChapterSoftmaxJacobian.hasDerivAt_partition_score (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : HasDerivAt (fun t : ℝ => partition beta (scorePerturb s i t)) (beta * Real.exp (beta * s i)) 0

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

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ}
Formal statement
theorem BookProof.ChapterSoftmaxJacobian.hasDerivAt_partition_score (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    HasDerivAt (fun t : ℝ => partition beta (scorePerturb s i t))
      (beta * Real.exp (beta * s i)) 0 := by sorry

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