-- Module: Mathlib.Analysis.Fourier.FourierTransformDeriv
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst_2 : NormedAddCommGroup V
inst_3 : NormedSpace ℝ V
inst_4 : NormedAddCommGroup W
inst_5 : NormedSpace ℝ W
L : ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) V (W →L[ℝ] ℝ)
f : V → E
inst_6 : MeasurableSpace V
inst_7 : BorelSpace V
inst_8 : FiniteDimensional ℝ V
μ : Measure V
inst_9 : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N : ℕ∞ω) f
h'f :
  ∀ (n : ℕ),
    (↑n : ℕ∞) ≤ N →
      Integrable (ε := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) (iteratedFDeriv ℝ n f) μ
n : ℕ
hn : (↑n : ℕ∞) ≤ N
zero :
  ∀ (hn : (↑0 : ℕ∞) ≤ N),
    Eq (α := (w : W) → ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin 0) ↦ V) E)
      (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ 0 f)) fun (w : W) ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w 0
succ :
  ∀ (n : ℕ)
    (ih :
      ∀ (hn : (↑n : ℕ∞) ≤ N),
        Eq (α := (w : W) → ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
          (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ n f)) fun (w : W) ↦
          fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w n)
    (hn : (↑(n + 1) : ℕ∞) ≤ N),
    Eq (α := (w : W) → ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin (n + 1)) ↦ V) E)
      (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ (n + 1) f)) fun (w : W) ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1)
⊢ Eq (α := (w : W) → ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
    (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ n f)) fun (w : W) ↦
    fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w n

Tactic:
induction n with
  | zero =>
    ext w m
    simp only [iteratedFDeriv_zero_apply, fourierPowSMulRight_apply, pow_zero,
      Finset.univ_eq_empty, ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply,
      Finset.prod_empty, one_smul, fourierIntegral_continuousMultilinearMap_apply' ((h'f 0 bot_le))]
  | succ n ih =>
    ext w m
    have J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ := by
      specialize h'f (n + 1) hn
      rwa [iteratedFDeriv_succ_eq_comp_left, Function.comp_def,
          LinearIsometryEquiv.integrable_comp_iff (𝕜 := ℝ) (φ := fderiv ℝ (iteratedFDeriv ℝ n f))]
        at h'f
    suffices H : (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (fderiv ℝ (iteratedFDeriv ℝ n f)) w)
          (m 0) (Fin.tail m) =
        (-(2 * π * I)) ^ (n + 1) • (∏ x : Fin (n + 1), -L (m x) w) • ∫ v, 𝐞 (-L v w) • f v ∂μ by
      rw [fourierIntegral_continuousMultilinearMap_apply' (h'f _ hn)]
      simp only [iteratedFDeriv_succ_apply_left, fourierPowSMulRight_apply,
        ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply]
      rw [← fourierIntegral_continuousMultilinearMap_apply' ((J.apply_continuousLinearMap _)),
          ← fourierIntegral_continuousLinearMap_apply' J]
      exact H
    have h'n : n < N := (Nat.cast_lt.mpr n.lt_succ_self).trans_le hn
    rw [fourierIntegral_fderiv _ (h'f n h'n.le)
      (hf.differentiable_iteratedFDeriv (mod_cast h'n)) J]
    simp only [ih h'n.le, fourierSMulRight_apply, ContinuousLinearMap.neg_apply,
      ContinuousLinearMap.flip_apply, neg_smul, smul_neg, neg_neg, smul_apply,
      fourierPowSMulRight_apply, ← coe_smul (E := E), smul_smul]
    congr 1
    simp only [ofReal_prod, ofReal_neg, pow_succ, mul_neg, Fin.prod_univ_succ, neg_mul,
      ofReal_mul, neg_neg, Fin.tail_def]
    ring

Premises used: Finset.prod_empty, neg_smul, pow_succ, Complex.ofReal_neg, ContinuousLinearMap.flip_apply, Finset.univ_eq_empty, smul_smul, neg_neg, Complex.ofReal_prod, smul_neg, pow_zero, one_smul, VectorFourier.fourierSMulRight_apply, iteratedFDeriv_zero_apply, ContinuousMultilinearMap.smul_apply, Fin.prod_univ_succ, ContinuousLinearMap.neg_apply, Complex.coe_smul, VectorFourier.fourierPowSMulRight_apply, neg_mul, Fin.tail_def, OfNat.ofNat, bot_le, Real.fourierIntegral_continuousMultilinearMap_apply', Complex.ofReal_mul, mul_neg