`BookProof.DampedOscillatorEnergy.coupledEnergy_antitone` {lam1 lam2 omega1 omega2 c : ℝ} {x1 x2 v1 v2 a1 a2 : ℝ → ℝ} (hlam1 : 0 ≤ lam1) (hlam2 : 0 ≤ lam2) (hx1 : ∀ t, HasDerivAt x

Proved

BookProof.DampedOscillatorEnergy.coupledEnergy_antitone

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDampedOscillatorEnergy.

BookProof.DampedOscillatorEnergy.coupledEnergy_antitone {lam1 lam2 omega1 omega2 c : ℝ} {x1 x2 v1 v2 a1 a2 : ℝ → ℝ} (hlam1 : 0 ≤ lam1) (hlam2 : 0 ≤ lam2) (hx1 : ∀ t, HasDerivAt x1 (v1 t) t) (hx2 : ∀ t, HasDerivAt x2 (v2 t) t) (hv1 : ∀ t, HasDerivAt v1 (a1 t) t) (hv2 : ∀ t, HasDerivAt v2 (a2 t) t) (heq1 : ∀ t, a1 t + lam1 * v1 t + omega1 ^ 2 * x1 t - c * x2 t = 0) (heq2 : ∀ t, a2 t + lam2 * v2 t + omega2 ^ 2 * x2 t - c * x1 t = 0) : Antitone (coupledEnergy omega1 omega2 c x1 x2 v1 v2)

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

Preamble

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



open MeasureTheory

Formal statement

theorem BookProof.DampedOscillatorEnergy.coupledEnergy_antitone {lam1 lam2 omega1 omega2 c : ℝ} {x1 x2 v1 v2 a1 a2 : ℝ → ℝ}
    (hlam1 : 0 ≤ lam1) (hlam2 : 0 ≤ lam2)
    (hx1 : ∀ t, HasDerivAt x1 (v1 t) t) (hx2 : ∀ t, HasDerivAt x2 (v2 t) t)
    (hv1 : ∀ t, HasDerivAt v1 (a1 t) t) (hv2 : ∀ t, HasDerivAt v2 (a2 t) t)
    (heq1 : ∀ t, a1 t + lam1 * v1 t + omega1 ^ 2 * x1 t - c * x2 t = 0)
    (heq2 : ∀ t, a2 t + lam2 * v2 t + omega2 ^ 2 * x2 t - c * x1 t = 0) :
    Antitone (coupledEnergy omega1 omega2 c x1 x2 v1 v2) := by sorry

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