`BookProof.DampedOscillatorEnergy.hasDerivAt_dampedEnergy` {lam omega : ℝ} {x v a : ℝ → ℝ} (hx : ∀ t, HasDerivAt x (v t) t) (hv : ∀ t, HasDerivAt v (a t) t) (heq : ∀ t, a t + lam *

Proved

BookProof.DampedOscillatorEnergy.hasDerivAt_dampedEnergy

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDampedOscillatorEnergy.

BookProof.DampedOscillatorEnergy.hasDerivAt_dampedEnergy {lam omega : ℝ} {x v a : ℝ → ℝ} (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) (t : ℝ) : HasDerivAt (dampedEnergy omega x v) (-(lam * v t ^ 2)) t

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

Preamble

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



open MeasureTheory

Formal statement

theorem BookProof.DampedOscillatorEnergy.hasDerivAt_dampedEnergy {lam omega : ℝ} {x v a : ℝ → ℝ}
    (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) (t : ℝ) :
    HasDerivAt (dampedEnergy omega x v) (-(lam * v t ^ 2)) t := by sorry

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