Caccioppoli weighted on ball of pos Part Approx

theorem caccioppoli_weighted_on_ball_of_posPartApprox    {u η : E → ℝ} {x₀ : E} {s Cη k : ℝ}    (A : EllipticCoeff d (Metric.ball x₀ s))    (hs : 0 < s)    (hsub : IsSubsolution A u)    (hu : MemW1pWitness 2 u (Metric.ball x₀ s))    (happroxBallShift :      ∃ ψ : ℕ → E → ℝ,        (∀ n, ContDiff ℝ 1 (ψ n)) ∧        (∀ n, HasCompactSupport (ψ n)) ∧        Tendsto          (fun n =>            eLpNorm (fun x => ψ n x - (u x - k)) 2              (volume.restrict (Metric.ball x₀ s)))          atTop (nhds 0) ∧        (∀ i : Fin d,          Tendsto            (fun n =>              eLpNorm                (fun x =>                  (fderiv ℝ (ψ n) x) (EuclideanSpace.single i 1) - hu.weakGrad x i)                2 (volume.restrict (Metric.ball x₀ s)))            atTop (nhds 0)))    (hη : ContDiff ℝ (⊤ : ℕ∞) η)    (hη_nonneg : ∀ x, 0 ≤ η x)    (hη_bound : ∀ x, |η x| ≤ 1)    (hCη : 0 ≤ Cη)    (hη_grad_bound : ∀ x, ‖fderiv ℝ η x‖ ≤ Cη)    (hη_sub_ball : tsupport η ⊆ Metric.ball x₀ s) :    ∫ x in Metric.ball x₀ s, η x ^ 2 * ‖(positivePartSub_memW1pWitness_on_ball        hs hu k happroxBallShift).weakGrad x‖ ^ 2 ∂volume ≤      4 * ellipticityRatio A *        ∫ x in Metric.ball x₀ s, ‖fderiv ℝ η x‖ ^ 2 * |positivePartSub u k x| ^ 2 ∂volume := by  haveI : IsFiniteMeasure (volume.restrict (Metric.ball x₀ s)) := by    rw [isFiniteMeasure_restrict]    exact measure_ball_lt_top.ne  let hw_trunc : MemW1pWitness 2 (positivePartSub u k) (Metric.ball x₀ s) :=    positivePartSub_memW1pWitness_on_ball hs hu k happroxBallShift  have hη_admissible :      MemW01p 2 (deGiorgiCutoffTestGeneral η u k) (Metric.ball x₀ s) :=    deGiorgiCutoffTest_memW01p_on_ball_of_ballPosPartApprox      hs hu hη hη_bound hCη hη_grad_bound hη_sub_ball k happroxBallShift  change    ∫ x in Metric.ball x₀ s, η x ^ 2 * ‖hw_trunc.weakGrad x‖ ^ 2 ∂volume ≤      4 * ellipticityRatio A *        ∫ x in Metric.ball x₀ s, ‖fderiv ℝ η x‖ ^ 2 * |positivePartSub u k x| ^ 2 ∂volume  exact caccioppoli_weighted_of_subsolution isOpen_ball A hsub hu hw_trunc hη hη_nonneg    (by norm_num) hCη hη_bound hη_grad_bound hη_admissible