l3lab/ntp-mathlib huggingface dataset:
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S2 : ∫ (x : E), (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), f x * (g (x + t • -v) - g x) ∂μ
⊢ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ

Next tactic:
simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
⊢ ∫ (x : E), (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), f x * (g (x + t • -v) - g x) ∂μ

Next tactic:
have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
⊢ ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ

Next tactic:
rw [← integral_add_right_eq_self _ (t • (-v))]
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; 

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
⊢ ∫ (x : E), f (x + t • -v + t • v) * g (x + t • -v) ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ

Next tactic:
simp
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
  

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (x : E), (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), f x * (g (x + t • -v) - g x) ∂μ

Next tactic:
simp_rw [_root_.sub_mul, _root_.mul_sub]
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
  simp_rw [_root_.sub_mul, _root_.mul_sub]
  

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (x : E), f (x + t • v) * g x - f x * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) - f x * g x ∂μ

Next tactic:
rw [integral_sub, integral_sub, S3]
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
  simp_rw [_root_.sub_mul, _root_.mul_sub]
  rw [integral_sub, integral_sub, S3]
  · 

Proof state:
case hf
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Integrable fun x => f x * g (x + t • -v)

Next tactic:
apply Continuous.integrable_of_hasCompactSupport
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
  simp_rw [_root_.sub_mul, _root_.mul_sub]
  rw [integral_sub, integral_sub, S3]
  · apply Continuous.integrable_of_hasCompactSupport
    · 

Proof state:
case hf.hf
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Continuous fun x => f x * g (x + t • -v)

Next tactic:
exact hf.continuous.mul (hg.continuous.comp (continuous_add_right _))
/-- Integration by parts formula for the line derivative of Lipschitz functions, assuming one of
them is compactly supported. -/
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ := by
  /- Write down the line derivative as the limit of `(f (x + t v) - f x) / t` and
  `(g (x - t v) - g x) / t`, and therefore the integrals as limits of the corresponding integrals
  thanks to the dominated convergence theorem. At fixed positive `t`, the integrals coincide
  (with the change of variables `y = x + t v`), so the limits also coincide. -/
  have A : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
      hf (hg.continuous.integrable_of_hasCompactSupport h'g) v
  have B : Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ) (𝓝[>] 0)
              (𝓝 (∫ x, lineDeriv ℝ g x (-v) * f x ∂μ)) :=
    integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' hg h'g hf.continuous (-v)
  suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B
  intro t
  suffices S2 : ∫ x, (f (x + t • v) - f x) * g x ∂μ = ∫ x, f x * (g (x + t • (-v)) - g x) ∂μ by
    simp only [smul_eq_mul, mul_assoc, integral_mul_left, S2, mul_neg, mul_comm (f _)]
  have S3 : ∫ x, f (x + t • v) * g x ∂μ = ∫ x, f x * g (x + t • (-v)) ∂μ := by
    rw [← integral_add_right_eq_self _ (t • (-v))]; simp
  simp_rw [_root_.sub_mul, _root_.mul_sub]
  rw [integral_sub, integral_sub, S3]
  · apply Continuous.integrable_of_hasCompactSupport
    · exact hf.continuous.mul (hg.continuous.comp (continuous_add_right _))
    · 

Proof state:
case hf.hcf
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ HasCompactSupport fun x => f x * g (x + t • -v)

Next tactic:
exact (h'g.comp_homeomorph (Homeomorph.addRight (t • (-v)))).mul_left