`BookProof.DampedOscillatorEnergy.dampedEnergy_antitone` {lam omega : ℝ} {x v a : ℝ → ℝ} (hlam : 0 ≤ lam) (hx : ∀ t, HasDerivAt x (v t) t) (hv : ∀ t, HasDerivAt v (a t) t) (heq : ∀

Proved

BookProof.DampedOscillatorEnergy.dampedEnergy_antitone

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDampedOscillatorEnergy.

BookProof.DampedOscillatorEnergy.dampedEnergy_antitone {lam omega : ℝ} {x v a : ℝ → ℝ} (hlam : 0 ≤ lam) (hx : ∀ t, HasDerivAt x (v t) t) (hv : ∀ t, HasDerivAt v (a t) t) (heq : ∀ t, a t + lam * v t + omega ^ 2 * x t = 0) : Antitone (dampedEnergy omega x v)

Formalization note: Lean 4 identifier BookProof.DampedOscillatorEnergy.dampedEnergy_antitone.

Preamble

-- Generated from ChapterDampedOscillatorEnergy.lean — theorem BookProof.DampedOscillatorEnergy.dampedEnergy_antitone
import Mathlib
import Definitions.Def_ChapterDampedOscillatorEnergy
open BookProof.DampedOscillatorEnergy



open MeasureTheory

Formal statement

theorem BookProof.DampedOscillatorEnergy.dampedEnergy_antitone {lam omega : ℝ} {x v a : ℝ → ℝ} (hlam : 0 ≤ lam)
    (hx : ∀ t, HasDerivAt x (v t) t) (hv : ∀ t, HasDerivAt v (a t) t)
    (heq : ∀ t, a t + lam * v t + omega ^ 2 * x t = 0) :
    Antitone (dampedEnergy omega x v) := by sorry

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