`BookProof.ChapterSinusoidalPosition.peInner_eq_sum_cos` (w : Fin n → ℝ) (p q : ℝ) : peInner w p q = ∑ a, Real.cos (w a * (p - q))

Proved

BookProof.ChapterSinusoidalPosition.peInner_eq_sum_cos

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSinusoidalPosition.

BookProof.ChapterSinusoidalPosition.peInner_eq_sum_cos (w : Fin n → ℝ) (p q : ℝ) : peInner w p q = ∑ a, Real.cos (w a * (p - q))

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {n : ℕ}

Formal statement

theorem BookProof.ChapterSinusoidalPosition.peInner_eq_sum_cos (w : Fin n → ℝ) (p q : ℝ) :
    peInner w p q = ∑ a, Real.cos (w a * (p - q)) := by sorry

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