`BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge` {m : ℕ} {beta : ℝ} (hb : 0 ≤ beta) (w : Fin n → ℝ) (p : ℝ) (k : Fin m → ℝ) (j : Fin m) : Real.exp (-(beta * (2 * n)

Proved

BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSinusoidalPosition.

BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge {m : ℕ} {beta : ℝ} (hb : 0 ≤ beta) (w : Fin n → ℝ) (p : ℝ) (k : Fin m → ℝ) (j : Fin m) : Real.exp (-(beta * (2 * n))) / (m : ℝ) ≤ scoreSoftmax beta (fun l => peInner w p (k l)) j

Formalization note: Lean 4 identifier BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge.

Preamble

-- Generated from ChapterSinusoidalPosition.lean — theorem BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSinusoidalPosition
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSinusoidalPosition


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {n : ℕ}

Formal statement

theorem BookProof.ChapterSinusoidalPosition.scoreSoftmax_sinusoidal_ge {m : ℕ} {beta : ℝ} (hb : 0 ≤ beta) (w : Fin n → ℝ)
    (p : ℝ) (k : Fin m → ℝ) (j : Fin m) :
    Real.exp (-(beta * (2 * n))) / (m : ℝ)
      ≤ scoreSoftmax beta (fun l => peInner w p (k l)) j := by sorry

View graph