awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (hypothesis)
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
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
w : W
m : (i : Fin (n + 1)) → V
⊢ sorry
have h'f₁ : Integrable (ε := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin (n + 1)) ↦ V) E) (iteratedFDeriv ℝ (n + 1) f)
  μ
Tactic: specialize h'f (n + 1) hn
Cleared hypotheses: h'f
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (hypothesis)
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
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
w : W
m : (i : Fin (n + 1)) → V
h'f :
  Integrable (ε := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin (n + 1)) ↦ V) E)
    (iteratedFDeriv ℝ (n + 1) f) μ
⊢ sorry
have h'f₁ : Integrable (ε := V →L[ℝ] ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
  (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
Tactic: 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
Cleared hypotheses: h'f
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (hypothesis)
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
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
w : W
m : (i : Fin (n + 1)) → V
⊢ Eq.{u_1 + 1} (α := E)
    ((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ (n + 1) f) w : (a : (i : Fin (n + 1)) → V) → E) m : E)
    ((fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1) : (a : (i : Fin (n + 1)) → V) → E)
        m :
      E)
have J : sorry
Tactic: 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
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
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
w : W
m : (i : Fin (n + 1)) → V
J : sorry
H :
  (((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (fderiv ℝ (iteratedFDeriv ℝ n f)) w :
              (a : V) → ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V) E)
            (m 0) :
          ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) :
        (a : (i : Fin n) → V) → E)
      (Fin.tail (α := fun (i : Fin (n + 1)) ↦ V) m) =
    (-(2 * (↑π : ℂ) * I)) ^ (n + 1) •
      HSMul.hSMul (α := ℝ) (β := E)
        (∏ x : Fin (n + 1), -(((L : (a : V) → W →L[ℝ] ℝ) (m x) : W →L[ℝ] ℝ) : (a : W) → ℝ) w)
        (∫ (v : V),
          HSMul.hSMul (α := Circle)
            ((𝐞 : (a : ℝ) → Circle) (-(((L : (a : V) → W →L[ℝ] ℝ) v : W →L[ℝ] ℝ) : (a : W) → ℝ) w) : Circle) (f v) ∂μ)
⊢ Eq.{u_1 + 1} (α := E)
    ((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ (n + 1) f) w : (a : (i : Fin (n + 1)) → V) → E) m : E)
    ((fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1) : (a : (i : Fin (n + 1)) → V) → E)
        m :
      E)
have rw : Eq.{u_1 + 1} (α := E)
  (fourierIntegral 𝐞 μ L.toLinearMap₁₂
      (fun (x : V) ↦ (iteratedFDeriv ℝ (n + 1) f x : (a : (i : Fin (n + 1)) → V) → E) m) w :
    E)
  ((fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1) : (a : (i : Fin (n + 1)) → V) → E)
      m :
    E)
Tactic: rw [fourierIntegral_continuousMultilinearMap_apply' (h'f _ hn)]
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
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
w : W
m : (i : Fin (n + 1)) → V
J : sorry
H :
  (((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (fderiv ℝ (iteratedFDeriv ℝ n f)) w :
              (a : V) → ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V) E)
            (m 0) :
          ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) :
        (a : (i : Fin n) → V) → E)
      (Fin.tail (α := fun (i : Fin (n + 1)) ↦ V) m) =
    (-(2 * (↑π : ℂ) * I)) ^ (n + 1) •
      HSMul.hSMul (α := ℝ) (β := E)
        (∏ x : Fin (n + 1), -(((L : (a : V) → W →L[ℝ] ℝ) (m x) : W →L[ℝ] ℝ) : (a : W) → ℝ) w)
        (∫ (v : V),
          HSMul.hSMul (α := Circle)
            ((𝐞 : (a : ℝ) → Circle) (-(((L : (a : V) → W →L[ℝ] ℝ) v : W →L[ℝ] ℝ) : (a : W) → ℝ) w) : Circle) (f v) ∂μ)
⊢ Eq.{u_1 + 1} (α := E)
    (fourierIntegral 𝐞 μ L.toLinearMap₁₂
        (fun (x : V) ↦ (iteratedFDeriv ℝ (n + 1) f x : (a : (i : Fin (n + 1)) → V) → E) m) w :
      E)
    ((fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1) : (a : (i : Fin (n + 1)) → V) → E)
        m :
      E)
have simp : fourierIntegral 𝐞 μ L.toLinearMap₁₂
    (fun (x : V) ↦
      (((fderiv ℝ (iteratedFDeriv ℝ n f) x : (a : V) → ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V) E) (m 0) :
            ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) :
          (a : (i : Fin n) → V) → E)
        (Fin.tail (α := fun (i : Fin (n + 1)) ↦ V) m))
    w =
  (-(2 * (↑π : ℂ) * I)) ^ (n + 1) •
    HSMul.hSMul (α := ℝ) (∏ x : Fin (n + 1), -(((L : (a : V) → W →L[ℝ] ℝ) (m x) : W →L[ℝ] ℝ) : (a : W) → ℝ) w)
      (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f w)
Tactic: simp only [iteratedFDeriv_succ_apply_left, fourierPowSMulRight_apply,
        ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply]
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
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
w : W
m : (i : Fin (n + 1)) → V
J : sorry
⊢ Eq.{u_1 + 1} (α := E)
    ((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ (n + 1) f) w : (a : (i : Fin (n + 1)) → V) → E) m : E)
    ((fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f) w (n + 1) : (a : (i : Fin (n + 1)) → V) → E)
        m :
      E)
have succ : (((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (fderiv ℝ (iteratedFDeriv ℝ n f)) w :
            (a : V) → ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V) E)
          (m 0) :
        ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) :
      (a : (i : Fin n) → V) → E)
    (Fin.tail (α := fun (i : Fin (n + 1)) ↦ V) m) =
  (-(2 * (↑π : ℂ) * I)) ^ (n + 1) •
    HSMul.hSMul (α := ℝ) (β := E) (∏ x : Fin (n + 1), -(((L : (a : V) → W →L[ℝ] ℝ) (m x) : W →L[ℝ] ℝ) : (a : W) → ℝ) w)
      (∫ (v : V),
        HSMul.hSMul (α := Circle)
          ((𝐞 : (a : ℝ) → Circle) (-(((L : (a : V) → W →L[ℝ] ℝ) v : W →L[ℝ] ℝ) : (a : W) → ℝ) w) : Circle) (f v) ∂μ)
Tactic: 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
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (hypothesis)
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
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
w : W
m : (i : Fin (n + 1)) → V
J : sorry
⊢ (((fourierIntegral 𝐞 μ L.toLinearMap₁₂ (fderiv ℝ (iteratedFDeriv ℝ n f)) w :
              (a : V) → ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V) E)
            (m 0) :
          ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E) :
        (a : (i : Fin n) → V) → E)
      (Fin.tail (α := fun (i : Fin (n + 1)) ↦ V) m) =
    (-(2 * (↑π : ℂ) * I)) ^ (n + 1) •
      HSMul.hSMul (α := ℝ) (β := E)
        (∏ x : Fin (n + 1), -(((L : (a : V) → W →L[ℝ] ℝ) (m x) : W →L[ℝ] ℝ) : (a : W) → ℝ) w)
        (∫ (v : V),
          HSMul.hSMul (α := Circle)
            ((𝐞 : (a : ℝ) → Circle) (-(((L : (a : V) → W →L[ℝ] ℝ) v : W →L[ℝ] ℝ) : (a : W) → ℝ) w) : Circle) (f v) ∂μ)
have h'n : (↑n : ℕ∞) < N
Tactic: have h'n : n < N := (Nat.cast_lt.mpr n.lt_succ_self).trans_le hn
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
⊢ 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
have 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
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
Cleared hypotheses: n, hn
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
K N : ℕ∞
hf : ContDiff ℝ (↑N : ℕ∞ω) f
h'f :
  ∀ (k n : ℕ),
    (↑k : ℕ∞) ≤ K →
      (↑n : ℕ∞) ≤ N →
        Integrable (ε := ℝ)
          (fun (v : V) ↦
            ‖v‖ ^ k *
              Norm.norm (E := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
                (iteratedFDeriv ℝ n f v))
          μ
k n : ℕ
hk : (↑k : ℕ∞) ≤ K
hn : (↑n : ℕ∞) ≤ N
w : W
⊢ Eq (α :=
    ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
      (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
    (fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f)) w n)
    (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ n fun (v : V) ↦ fourierPowSMulRight L f v k) w)
have rw : Eq (α :=
  ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
    (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
  (fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f)) w n)
  ((fun (w : W) ↦
      (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ fun (v : V) ↦ fourierPowSMulRight L f v k) w
          n :
        ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
          (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E)))
    w)
Tactic: rw [fourierIntegral_iteratedFDeriv (N := N) _ (hf.fourierPowSMulRight _ _) _ hn]
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
K N : ℕ∞
hf : ContDiff ℝ (↑N : ℕ∞ω) f
h'f :
  ∀ (k n : ℕ),
    (↑k : ℕ∞) ≤ K →
      (↑n : ℕ∞) ≤ N →
        Integrable (ε := ℝ)
          (fun (v : V) ↦
            ‖v‖ ^ k *
              Norm.norm (E := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
                (iteratedFDeriv ℝ n f v))
          μ
k n : ℕ
hk : (↑k : ℕ∞) ≤ K
hn : (↑n : ℕ∞) ≤ N
w : W
rw :
  Eq (α :=
    ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
      (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
    (fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f)) w n)
    ((fun (w : W) ↦
        (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ fun (v : V) ↦ fourierPowSMulRight L f v k) w
            n :
          ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
            (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E)))
      w)
⊢ Eq (α :=
    ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
      (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
    (fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f)) w n)
    (fourierIntegral 𝐞 μ L.toLinearMap₁₂ (iteratedFDeriv ℝ n fun (v : V) ↦ fourierPowSMulRight L f v k) w)
have rw₁ : ∀ (n : ℕ),
  (↑n : ℕ∞) ≤ N →
    Integrable (ε :=
      ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
        (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
      (iteratedFDeriv ℝ n fun (v : V) ↦ fourierPowSMulRight L f v k) μ
Tactic: rw [fourierIntegral_iteratedFDeriv (N := N) _ (hf.fourierPowSMulRight _ _) _ hn]
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
K N : ℕ∞
hf : ContDiff ℝ (↑N : ℕ∞ω) f
h'f :
  ∀ (k n : ℕ),
    (↑k : ℕ∞) ≤ K →
      (↑n : ℕ∞) ≤ N →
        Integrable (ε := ℝ)
          (fun (v : V) ↦
            ‖v‖ ^ k *
              Norm.norm (E := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
                (iteratedFDeriv ℝ n f v))
          μ
k n : ℕ
hk : (↑k : ℕ∞) ≤ K
hn : (↑n : ℕ∞) ≤ N
w : W
⊢ Eq (α :=
    ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
      (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E))
    (fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f)) w n)
    ((fun (w : W) ↦
        (fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₁₂ fun (v : V) ↦ fourierPowSMulRight L f v k) w
            n :
          ContinuousMultilinearMap ℝ (fun (i : Fin n) ↦ V)
            (ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E)))
      w)
have e_f : Eq (α := W → ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E)
  (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f))
  (fourierIntegral 𝐞 μ L.toLinearMap₁₂ fun (v : V) ↦ fourierPowSMulRight L f v k)
Tactic: congr
-- Lean 4 proof step from Mathlib.Analysis.Fourier.FourierTransformDeriv (goal)
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
K N : ℕ∞
hf : ContDiff ℝ (↑N : ℕ∞ω) f
h'f :
  ∀ (k n : ℕ),
    (↑k : ℕ∞) ≤ K →
      (↑n : ℕ∞) ≤ N →
        Integrable (ε := ℝ)
          (fun (v : V) ↦
            ‖v‖ ^ k *
              Norm.norm (E := ContinuousMultilinearMap.{0, 0, u_2, u_1} ℝ (fun (i : Fin n) ↦ V) E)
                (iteratedFDeriv ℝ n f v))
          μ
k n : ℕ
hk : (↑k : ℕ∞) ≤ K
hn : (↑n : ℕ∞) ≤ N
w : W
⊢ Eq (α := W → ContinuousMultilinearMap.{0, 0, u_3, u_1} ℝ (fun (i : Fin k) ↦ W) E)
    (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₁₂ f))
    (fourierIntegral 𝐞 μ L.toLinearMap₁₂ fun (v : V) ↦ fourierPowSMulRight L f v k)
have rw : ∀ (n : ℕ), (↑n : ℕ∞) ≤ K → Integrable (ε := ℝ) (fun (v : V) ↦ ‖v‖ ^ n * ‖f v‖) μ
Tactic: rw [iteratedFDeriv_fourierIntegral (N := K) _ _ hf.continuous.aestronglyMeasurable hk]