`BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity` {beta : ℝ} (hbeta : beta ≠ 0) (s : Fin m → ℝ) (i : Fin m) : HasDerivAt (fun b : ℝ => attentionEnt

Proved

BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterEntropyTemperature.

BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity {beta : ℝ} (hbeta : beta ≠ 0) (s : Fin m → ℝ) (i : Fin m) : HasDerivAt (fun b : ℝ => attentionEntropy b s) (-(heatCapacity beta s / beta)) beta

Formalization note: Lean 4 identifier BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity.

Preamble

-- Generated from ChapterEntropyTemperature.lean — theorem BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterEntropyTemperature
open BookProof.ChapterEntropyTemperature


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterEntropyTemperature.hasDerivAt_attentionEntropy_neg_heatCapacity {beta : ℝ} (hbeta : beta ≠ 0)
    (s : Fin m → ℝ) (i : Fin m) :
    HasDerivAt (fun b : ℝ => attentionEntropy b s) (-(heatCapacity beta s / beta)) beta := by sorry

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