-- Mathlib/Analysis/Convolution.lean
theorem convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z)) {x₀ : G}
    (hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) (hk : AEStronglyMeasurable k μ)
    (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
    (hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ)
    (hfgk :
      ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀
        (mul ℝ ℝ) ν) :
    ((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] g ⋆[L₄, μ] k) x₀

Proof state:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀ (mul ℝ ℝ) ν
h_meas : AEStronglyMeasurable (uncurry fun x y => (L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y => ∫ (x : G), ‖(L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))‖ ∂μ) ν
h3 : Measure.map (fun z => (z.1 - z.2, z.2)) (μ.prod ν) = μ.prod ν
⊢ Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)

Tactic:
suffices Integrable (uncurry fun x y => L₃ (f y) (L₄ (g x) (k (x₀ - y - x)))) (μ.prod ν) by
  rw [← h3] at this
  convert this.comp_measurable (measurable_sub.prod_mk measurable_snd)
  ext ⟨x, y⟩
  simp (config := { unfoldPartialApp := true }) only [uncurry, Function.comp_apply,
    sub_sub_sub_cancel_right]

Result:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀ (mul ℝ ℝ) ν
h_meas : AEStronglyMeasurable (uncurry fun x y => (L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y => ∫ (x : G), ‖(L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))‖ ∂μ) ν
h3 : Measure.map (fun z => (z.1 - z.2, z.2)) (μ.prod ν) = μ.prod ν
⊢ Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g x)) (k (x₀ - y - x)))) (μ.prod ν)