/-- Compute the total derivative of `f ⋆ g` if `g` is `C^1` with compact support and `f` is locally
integrable. To write down the total derivative as a convolution, we use
`ContinuousLinearMap.precompR`. -/
theorem HasCompactSupport.hasFDerivAt_convolution_right (hcg : HasCompactSupport g)
    (hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G) :
    HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ := by
  rcases hcg.eq_zero_or_finiteDimensional 𝕜 hg.continuous with (rfl | fin_dim)
  · have : fderiv 𝕜 (0 : G → E') = 0 := fderiv_const (0 : E')
    simp only [this, convolution_zero, Pi.zero_apply]
    exact hasFDerivAt_const (0 : F) x₀
  have : ProperSpace G := FiniteDimensional.proper_isROrC 𝕜 G
  set L' := L.precompR G
  have h1 : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (fun t => L (f t) (g (x - t))) μ :=
    eventually_of_forall
      (hf.aestronglyMeasurable.convolution_integrand_snd L hg.continuous.aestronglyMeasurable)
  have h2 : ∀ x, AEStronglyMeasurable (fun t => L' (f t) (fderiv 𝕜 g (x - t))) μ :=
    hf.aestronglyMeasurable.convolution_integrand_snd L'
      (hg.continuous_fderiv le_rfl).aestronglyMeasurable
  have h3 : ∀ x t, HasFDerivAt (fun x => g (x - t)) (fderiv 𝕜 g (x - t)) x := fun x t ↦ by
    simpa using
      (hg.differentiable le_rfl).differentiableAt.hasFDerivAt.comp x
        ((hasFDerivAt_id x).sub (hasFDerivAt_const t x))
  let K' := -tsupport (fderiv 𝕜 g) + closedBall x₀ 1
  have hK' : IsCompact K' := (hcg.fderiv 𝕜).neg.add (isCompact_closedBall x₀ 1)
  -- porting note: was
  -- `refine' hasFDerivAt_integral_of_dominated_of_fderiv_le zero_lt_one h1 _ (h2 x₀) _ _ _`
  -- but it failed; surprisingly, `apply` works
  apply hasFDerivAt_integral_of_dominated_of_fderiv_le zero_lt_one h1 _ (h2 x₀)
  · refine' eventually_of_forall fun t x hx =>
      (hcg.fderiv 𝕜).convolution_integrand_bound_right L' (hg.continuous_fderiv le_rfl)
        (ball_subset_closedBall hx)
  · rw [integrable_indicator_iff hK'.measurableSet]
    

Proof state:
case inr.bound_integrable
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedAddCommGroup E''
inst✝¹³ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : NormedSpace 𝕜 E'
inst✝⁹ : NormedSpace 𝕜 E''
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedSpace 𝕜 F
n : ℕ∞
inst✝⁶ : CompleteSpace F
inst✝⁵ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SigmaFinite μ
inst✝ : IsAddLeftInvariant μ
hcg : HasCompactSupport g
hf : LocallyIntegrable f
hg : ContDiff 𝕜 1 g
x₀ : G
fin_dim : FiniteDimensional 𝕜 G
this : ProperSpace G
L' : E →L[𝕜] (G →L[𝕜] E') →L[𝕜] G →L[𝕜] F := precompR G L
h1 : ∀ᶠ (x : G) in 𝓝 x₀, AEStronglyMeasurable (fun t => (L (f t)) (g (x - t))) μ
h2 : ∀ (x : G), AEStronglyMeasurable (fun t => (L' (f t)) (fderiv 𝕜 g (x - t))) μ
h3 : ∀ (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv 𝕜 g (x - t)) x
K' : Set G := -tsupport (fderiv 𝕜 g) + closedBall x₀ 1
hK' : IsCompact K'
⊢ IntegrableOn (fun t => ‖L'‖ * ‖f t‖ * ⨆ i, ‖fderiv 𝕜 g i‖) K'

Next tactic:
exact ((hf.integrableOn_isCompact hK').norm.const_mul _).mul_const _