-- Module: Mathlib.Analysis.Calculus.ContDiff.Convolution
Goal:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst : NormedAddCommGroup E
inst_1 : NormedAddCommGroup E'
inst_2 : NormedAddCommGroup F
f : G → E
inst_3 : RCLike 𝕜
inst_4 : NormedSpace 𝕜 E
inst_5 : NormedSpace 𝕜 E'
inst_6 : NormedSpace ℝ F
inst_7 : NormedSpace 𝕜 F
inst_8 : MeasurableSpace G
inst_9 : NormedAddCommGroup G
inst_10 : BorelSpace G
inst_11 : NormedSpace 𝕜 G
inst_12 : NormedAddCommGroup P
inst_13 : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (SProd.sprod (β := Set G) s univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : (x : P × G) → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
A' : ∀ (q : P × G), q.1 ∈ s → SProd.sprod (β := Set G) s univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
obtain :
  ∀ (ε C : ℝ) (εpos : 0 < ε) (h₀ε : ball q₀.1 ε ⊆ s) (hε : ∀ (p : P) (x : G), ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C),
    HasFDerivAt (𝕜 := 𝕜) (F := F) (fun (q : P × G) ↦ convolution.{u𝕜, uG, uE, uE', uF} (𝕜 := 𝕜) f (g q.1) L μ q.2)
      (convolution (𝕜 := 𝕜) (E' := P × G →L[𝕜] E') f (fun (x : G) ↦ fderiv 𝕜 ↿g (q₀.1, x)) (precompR (P × G) L) μ q₀.2)
      q₀
⊢ HasFDerivAt (𝕜 := 𝕜) (F := F) (fun (q : P × G) ↦ convolution.{u𝕜, uG, uE, uE', uF} (𝕜 := 𝕜) f (g q.1) L μ q.2)
    (convolution (𝕜 := 𝕜) (E' := P × G →L[𝕜] E') f (fun (x : G) ↦ fderiv 𝕜 ↿g (q₀.1, x)) (precompR (P × G) L) μ q₀.2) q₀

Tactic:
obtain ⟨ε, C, εpos, h₀ε, hε⟩ :
      ∃ ε C, 0 < ε ∧ ball q₀.1 ε ⊆ s ∧ ∀ p x, ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C := by
    have A : IsCompact ({q₀.1} ×ˢ k) := isCompact_singleton.prod hk
    obtain ⟨t, kt, t_open, ht⟩ : ∃ t, {q₀.1} ×ˢ k ⊆ t ∧ IsOpen t ∧ IsBounded (g' '' t) := by
      have B : ContinuousOn g' (s ×ˢ univ) :=
        hg.continuousOn_fderiv_of_isOpen (hs.prod isOpen_univ) le_rfl
      apply exists_isOpen_isBounded_image_of_isCompact_of_continuousOn A (hs.prod isOpen_univ) _ B
      simp only [prod_subset_prod_iff, hq₀, singleton_subset_iff, subset_univ, and_self_iff,
        true_or]
    obtain ⟨ε, εpos, hε, h'ε⟩ :
      ∃ ε : ℝ, 0 < ε ∧ thickening ε ({q₀.fst} ×ˢ k) ⊆ t ∧ ball q₀.1 ε ⊆ s := by
      obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ thickening ε (({q₀.fst} : Set P) ×ˢ k) ⊆ t :=
        A.exists_thickening_subset_open t_open kt
      obtain ⟨δ, δpos, hδ⟩ : ∃ δ : ℝ, 0 < δ ∧ ball q₀.1 δ ⊆ s := Metric.isOpen_iff.1 hs _ hq₀
      refine ⟨min ε δ, lt_min εpos δpos, ?_, ?_⟩
      · exact Subset.trans (thickening_mono (min_le_left _ _) _) hε
      · exact Subset.trans (ball_subset_ball (min_le_right _ _)) hδ
    obtain ⟨C, Cpos, hC⟩ : ∃ C, 0 < C ∧ g' '' t ⊆ closedBall 0 C := ht.subset_closedBall_lt 0 0
    refine ⟨ε, C, εpos, h'ε, fun p x hp => ?_⟩
    have hps : p ∈ s := h'ε (mem_ball_iff_norm.2 hp)
    by_cases hx : x ∈ k
    · have H : (p, x) ∈ t := by
        apply hε
        refine mem_thickening_iff.2 ⟨(q₀.1, x), ?_, ?_⟩
        · simp only [hx, singleton_prod, mem_image, Prod.mk_inj, true_and, exists_eq_right]
        · rw [← dist_eq_norm] at hp
          simpa only [Prod.dist_eq, εpos, dist_self, max_lt_iff, and_true] using hp
      have : g' (p, x) ∈ closedBall (0 : P × G →L[𝕜] E') C := hC (mem_image_of_mem _ H)
      rwa [mem_closedBall_zero_iff] at this
    · have : g' (p, x) = 0 := g'_zero _ _ hps hx
      rw [this]
      simpa only [norm_zero] using Cpos.le