`BookProof.HalfLineLimitCircle.integral_deriv2_mul` (f : ℝ → ℂ) (hf : f ∈ testSpace) (w : ℝ → ℂ) (hw : ContDiff ℝ smoothTop w) : ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (deriv...

Proved

BookProof.HalfLineLimitCircle.integral_deriv2_mul

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterHalfLineLimitCircle.

BookProof.HalfLineLimitCircle.integral_deriv2_mul (f : ℝ → ℂ) (hf : f ∈ testSpace) (w : ℝ → ℂ) (hw : ContDiff ℝ smoothTop w) : ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (deriv (deriv f) x) * w x = ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (f x) * deriv (deriv w) x

Formalization note: Lean 4 identifier BookProof.HalfLineLimitCircle.integral_deriv2_mul.

Preamble

-- Generated from ChapterHalfLineLimitCircle.lean — theorem BookProof.HalfLineLimitCircle.integral_deriv2_mul
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterHalfLineLimitCircle
open BookProof.HalfLineLimitCircle



open MeasureTheory Set BookProof.FarisLavine

noncomputable section

local notation "smoothTop" => ((⊤ : ℕ∞) : WithTop ℕ∞)

Formal statement

theorem BookProof.HalfLineLimitCircle.integral_deriv2_mul (f : ℝ → ℂ) (hf : f ∈ testSpace) (w : ℝ → ℂ)
    (hw : ContDiff ℝ smoothTop w) :
    ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (deriv (deriv f) x) * w x
      = ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (f x) * deriv (deriv w) x := by sorry

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