`BookProof.ChapterAttentionMasking.maskedDenom_pos` (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} (hS : S.Nonempty) : 0 < ∑ l ∈ S, Real.exp (beta * s l)

Proved

BookProof.ChapterAttentionMasking.maskedDenom_pos

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMasking.

BookProof.ChapterAttentionMasking.maskedDenom_pos (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} (hS : S.Nonempty) : 0 < ∑ l ∈ S, Real.exp (beta * s l)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionMasking.maskedDenom_pos.

Preamble

-- Generated from ChapterAttentionMasking.lean — theorem BookProof.ChapterAttentionMasking.maskedDenom_pos
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionMasking
open BookProof.ChapterAttentionMasking


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionMasking.maskedDenom_pos (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} (hS : S.Nonempty) :
    0 < ∑ l ∈ S, Real.exp (beta * s l) := by sorry

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