`BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom` (s : Fin m → ℝ) (j : Fin m) (hmax : ∀ l, l ≠ j → s l < s j) : Tendsto (fun b : ℝ => ∑ l, Real.exp (b * (s l - s j)))

Proved

BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxSharpness.

BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom (s : Fin m → ℝ) (j : Fin m) (hmax : ∀ l, l ≠ j → s l < s j) : Tendsto (fun b : ℝ => ∑ l, Real.exp (b * (s l - s j))) atTop (𝓝 1)

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom.

Preamble

-- Generated from ChapterSoftmaxSharpness.lean — theorem BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom
import Definitions.Def_ChapterSoftmaxBorn
import Mathlib
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness


open scoped BigOperators

noncomputable section


open Filter Topology BookProof.ChapterSoftmaxBorn

variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxSharpness.tendsto_scoreSoftmax_denom (s : Fin m → ℝ) (j : Fin m)
    (hmax : ∀ l, l ≠ j → s l < s j) :
    Tendsto (fun b : ℝ => ∑ l, Real.exp (b * (s l - s j))) atTop (𝓝 1) := by sorry

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