`BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score` (beta : ℝ) (s : Fin m → ℝ) (y i : Fin m) : HasDerivAt (fun t : ℝ => crossEntropyLoss beta (scorePerturb s

Proved

BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCrossEntropyGradient.

BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score (beta : ℝ) (s : Fin m → ℝ) (y i : Fin m) : HasDerivAt (fun t : ℝ => crossEntropyLoss beta (scorePerturb s i t) y) (crossEntropyGradient beta s y i) 0

Formalization note: Lean 4 identifier BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score.

Preamble

-- Generated from ChapterCrossEntropyGradient.lean — theorem BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterCrossEntropyGradient
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxJacobian
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterCrossEntropyGradient


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterSoftmaxJacobian

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterCrossEntropyGradient.hasDerivAt_crossEntropyLoss_score (beta : ℝ) (s : Fin m → ℝ) (y i : Fin m) :
    HasDerivAt (fun t : ℝ => crossEntropyLoss beta (scorePerturb s i t) y)
      (crossEntropyGradient beta s y i) 0 := by sorry

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