mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case succ
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
h'n : ↑n < N
⊢ (2 * ↑π * I * (↑((L (m 0)) w) * ((-(2 * ↑π * I)) ^ n * ↑(∏ i, -(L (Fin.tail m i)) w)))) •
      fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f w =
    ((-(2 * ↑π * I)) ^ (n + 1) * ↑(∏ x, -(L (m x)) w)) • ∫ (v : V), 𝐞 (-(L v) w) • f v ∂μ

Tactic:
congr 1

Goal state after:
case succ.e_a
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ
n : ℕ
ih :
  ↑n ≤ N →
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (iteratedFDeriv ℝ n f) = fun w ↦
      fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n
hn : ↑(n + 1) ≤ N
w : W
m : Fin (n + 1) → V
J : Integrable (fderiv ℝ (iteratedFDeriv ℝ n f)) μ
h'n : ↑n < N
⊢ 2 * ↑π * I * (↑((L (m 0)) w) * ((-(2 * ↑π * I)) ^ n * ↑(∏ i, -(L (Fin.tail m i)) w))) =
    (-(2 * ↑π * I)) ^ (n + 1) * ↑(∏ x, -(L (m x)) w)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
k : ℕ
hk : eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ < ∞
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ Continuous fun x ↦ 1 + ‖x‖

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
x₀ : F
hg : Integrable g μ
h'g : ContinuousAt g x₀
f : F → E := fun x ↦ g (x₀ - x)
If : Integrable f μ
A : ContinuousAt g (x₀ - 0)
⊢ ContinuousAt (HSub.hSub x₀) 0

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
h_map_neg : Measure.map Neg.neg (μ.restrict (Ici 0)) = μ.restrict (Iic 0)
⊢ Integrable f (Measure.map Neg.neg (μ.restrict (Ici 0)))

Tactic:
measurableEmbedding_neg.integrable_map_iff

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
h_map_neg : Measure.map Neg.neg (μ.restrict (Ici 0)) = μ.restrict (Iic 0)
⊢ Integrable (f ∘ Neg.neg) (μ.restrict (Ici 0))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ HasFiniteIntegral (fun a ↦ ‖s * (↑(rexp (-a)) * ↑a ^ (s - 1))‖) (volume.restrict (Ioc 0 Y))

Tactic:
norm_mul

Goal state after:
case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ HasFiniteIntegral (fun a ↦ ‖s‖ * (‖↑(rexp (-a))‖ * ‖↑a ^ (s - 1)‖)) (volume.restrict (Ioc 0 Y))
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: case fst
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π
⊢ √((r * cos θ) ^ 2 + (r * sin θ) ^ 2) = √((sin θ ^ 2 + cos θ ^ 2) * r ^ 2)

Tactic:
congr 1

Goal state after:
case fst
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π
⊢ (r * cos θ) ^ 2 + (r * sin θ) ^ 2 = (sin θ ^ 2 + cos θ ^ 2) * r ^ 2
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ ContinuousOn (fun q ↦ √(q.1 ^ 2 + q.2 ^ 2)) ({q | 0 < q.1} ∪ {q | q.2 ≠ 0})

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ ContinuousOn (fun p ↦ (p.1 * cos p.2, p.1 * sin p.2)) (Ioi 0 ×ˢ Ioo (-π) π)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ Continuous fun x ↦ x.1 * cos x.2

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ Continuous fun x ↦ x.1 * sin x.2

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume

Tactic:
by
  simp_rw [← rpow_two]
  exact integrableOn_rpow_mul_exp_neg_mul_rpow hs two_pos hb

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ AEStronglyMeasurable (fun x ↦ cexp (-b * ↑x ^ 2)) volume

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : ∀ (i : ι), IsFiniteMeasure (μ i)
i : ι
f : X i → E
hf : Integrable f (μ i)
⊢ Measurable fun x ↦ x i

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ Measurable (Function.eval i)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), (algebraMap ℝ ℂ) (rexp (-b * x ^ 2))) ^ 2 = (algebraMap ℝ ℂ) (π / b)

Tactic:
coe_algebraMap,

Goal state after:
case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), ↑(rexp (-b * x ^ 2))) ^ 2 = ↑(π / b)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x ↦ cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_le_bd : ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)
⊢ ∀ (x : ℂ), AEStronglyMeasurable (fun x_1 ↦ cexp (-x * ↑x_1 ^ 2)) volume

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x ↦ cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_le_bd : ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)
⊢ ∀ (a : ℝ), ContinuousAt (fun x ↦ cexp (-x * ↑a ^ 2)) b

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ 2 * (b.im * y) ≤ 2 * |b.im| * |c|

Tactic:
(conv_rhs => rw [mul_assoc])

Goal state after:
case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ 2 * (b.im * y) ≤ 2 * (|b.im| * |c|)
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → E
hf : Integrable f volume
c : ℝ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ cexp (-↑c⁻¹ * ↑‖a‖ ^ 2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b✝ : ℂ
hb✝ : 0 < b✝.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
b : ℂ
hb : b ∈ {c | 0 < c.re}
⊢ 0 < π * b.re / normSq b + 0

Tactic:
add_zero

Goal state after:
b✝ : ℂ
hb✝ : 0 < b✝.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
b : ℂ
hb : b ∈ {c | 0 < c.re}
⊢ 0 < π * b.re / normSq b
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: ⊢ GammaAux ⌊1 - 0⌋₊ 0 = 0

Tactic:
sub_zero,

Goal state after:
⊢ GammaAux ⌊1⌋₊ 0 = 0
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case e'_3
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w = cexp (-↑c⁻¹ * ↑‖w‖ ^ 2) • 𝐞 ⟪w, v⟫ • 𝓕 f w

pp: case e'_5
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
⊢ 𝓕⁻ (𝓕 f) v = ∫ (v_1 : V), 𝐞 ⟪v_1, v⟫ • 𝓕 f v_1

Tactic:
· rw [Submonoid.smul_def, Real.fourierChar_apply, smul_smul, ← Complex.exp_add, real_inner_comm]
      congr 3
      simp only [ofReal_mul, ofReal_ofNat]
      ring

Goal state after:
case e'_5
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
⊢ 𝓕⁻ (𝓕 f) v = ∫ (v_1 : V), 𝐞 ⟪v_1, v⟫ • 𝓕 f v_1
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ AEStronglyMeasurable (fun x ↦ cexp (-b * (↑x + ↑c * I) ^ 2)) volume

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)
full_integral : (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)
h_eq : ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f x
⊢ (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)

Tactic:
simpa using full_integral

Goal state after:
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun w ↦ π ^ 2 * ‖w‖ ^ 2) (cobounded V) atTop
B :
  Tendsto
    (fun x ↦
      π ^ (-↑(finrank ℝ V) / 2) * ((fun x ↦ x ^ (↑(finrank ℝ V) / 2) * rexp (-1 * x)) ∘ fun w ↦ π ^ 2 * ‖w‖ ^ 2) x)
    (cobounded V) (𝓝 0)
x : V
⊢ ‖x‖ ^ ↑(finrank ℝ V) * π ^ (↑(finrank ℝ V) / 2) * rexp (-(π ^ ↑2 * ‖x‖ ^ ↑2)) =
    π ^ (-↑(finrank ℝ V) / 2) * (π ^ ↑2 * ‖x‖ ^ ↑2) ^ (↑(finrank ℝ V) / 2) * rexp (-(π ^ ↑2 * ‖x‖ ^ ↑2))

Tactic:
congr 1

Goal state after:
case e_a
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
A : Tendsto (fun w ↦ π ^ 2 * ‖w‖ ^ 2) (cobounded V) atTop
B :
  Tendsto
    (fun x ↦
      π ^ (-↑(finrank ℝ V) / 2) * ((fun x ↦ x ^ (↑(finrank ℝ V) / 2) * rexp (-1 * x)) ∘ fun w ↦ π ^ 2 * ‖w‖ ^ 2) x)
    (cobounded V) (𝓝 0)
x : V
⊢ ‖x‖ ^ ↑(finrank ℝ V) * π ^ (↑(finrank ℝ V) / 2) =
    π ^ (-↑(finrank ℝ V) / 2) * (π ^ ↑2 * ‖x‖ ^ ↑2) ^ (↑(finrank ℝ V) / 2)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
x : ℝ
⊢ cexp (b * ↑x ^ 2 + c * ↑x + d) = cexp (- -b * (↑x + c / (2 * b)) ^ 2 + (d - c ^ 2 / (4 * b)))

Tactic:
congr 1

Goal state after:
b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
x : ℝ
⊢ b * ↑x ^ 2 + c * ↑x + d = - -b * (↑x + c / (2 * b)) ^ 2 + (d - c ^ 2 / (4 * b))
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SMulCommClass ℂ 𝕜 F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
⊢ 𝓕⁻ ((smulLeftCLM F fun x ↦ ↑(g x)) (𝓕 f)) = 𝓕⁻ ((smulLeftCLM F g) (𝓕 f))

Tactic:
congr 1

Goal state after:
𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : NormedSpace ℂ F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SMulCommClass ℂ 𝕜 F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
⊢ (smulLeftCLM F fun x ↦ ↑(g x)) (𝓕 f) = (smulLeftCLM F g) (𝓕 f)
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
⊢ ∑ i, ∂_{b i} (∂_{b i} f) = ∑ x, -(2 * π) ^ 2 • (fourierMultiplierCLM F fun x_1 ↦ inner ℝ x_1 (b x) ^ 2) f

Tactic:
congr 1

Goal state after:
case e_f
E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
⊢ (fun i ↦ ∂_{b i} (∂_{b i} f)) = fun x ↦ -(2 * π) ^ 2 • (fourierMultiplierCLM F fun x_1 ↦ inner ℝ x_1 (b x) ^ 2) f
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
i : Fin (Module.finrank ℝ E)
x : E
⊢ Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
x w : V
⊢ 𝓕 (fun v ↦ cexp (-b * ↑‖v‖ ^ 2 + 2 * ↑π * I * ↑⟪x, v⟫)) w =
    (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖x - w‖ ^ 2 / b)

Tactic:
simp only [neg_mul, fourier_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,
    smul_eq_mul, ← Complex.exp_add, real_inner_comm w]

Goal state after:
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
x w : V
⊢ ∫ (v : V), cexp (-(2 * ↑π * ↑⟪w, v⟫ * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑⟪x, v⟫)) =
    (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-(↑π ^ 2 * ↑‖x - w‖ ^ 2) / b)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case e'_2.e'_7
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
x w v : V
⊢ cexp (-(2 * ↑π * ↑⟪w, v⟫ * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑⟪x, v⟫)) =
    cexp (-b * ↑‖v‖ ^ 2 + 2 * ↑π * I * ↑⟪x - w, v⟫)

Tactic:
congr 1

Goal state after:
case e'_2.e'_7
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
x w v : V
⊢ -(2 * ↑π * ↑⟪w, v⟫ * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑⟪x, v⟫) = -b * ↑‖v‖ ^ 2 + 2 * ↑π * I * ↑⟪x - w, v⟫
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: case e_f
E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
i : Fin (Module.finrank ℝ E)
x : E
⊢ (2 * ↑π * Complex.I * (2 * ↑π * Complex.I)) •
      ((fourierMultiplierCLM F fun x ↦ ↑(inner ℝ x (b i))) ((fourierMultiplierCLM F fun x ↦ inner ℝ x (b i)) f)) x =
    -(2 * π) ^ 2 • ((fourierMultiplierCLM F fun x ↦ inner ℝ x (b i) ^ 2) f) x

Tactic:
congr 1

Goal state after:
case e_f.e_a
E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
i : Fin (Module.finrank ℝ E)
x : E
⊢ 2 * ↑π * Complex.I * (2 * ↑π * Complex.I) = ↑(algebraMap ℝ ℂ).toMonoidWithZeroHom (-(2 * π) ^ 2)

case e_f.e_a
E : Type u_3
F : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : NormedSpace ℂ F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
f : 𝓢(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ inner ℝ x (b i) ^ 2
i : Fin (Module.finrank ℝ E)
x : E
⊢ ((fourierMultiplierCLM F fun x ↦ ↑(inner ℝ x (b i))) ((fourierMultiplierCLM F fun x ↦ inner ℝ x (b i)) f)) x =
    ((fourierMultiplierCLM F fun x ↦ inner ℝ x (b i) ^ 2) f) x
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
k n : ℕ
f : 𝓢(V, E)
x : V
p : ℕ × ℕ
a✝ : p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1)
⊢ (2 ^ volume.integrablePower * ∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower) ∂volume) *
      ((SchwartzMap.seminorm 𝕜 0 p.2) f + (SchwartzMap.seminorm 𝕜 (p.1 + volume.integrablePower) p.2) f) ≤
    (2 ^ volume.integrablePower * ∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower)) * 2 *
      ((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E)) f

Tactic:
simp only [mul_assoc, two_mul]

Goal state after:
case h
𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
k n : ℕ
f : 𝓢(V, E)
x : V
p : ℕ × ℕ
a✝ : p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1)
⊢ 2 ^ volume.integrablePower *
      ((∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower) ∂volume) *
        ((SchwartzMap.seminorm 𝕜 0 p.2) f + (SchwartzMap.seminorm 𝕜 (p.1 + volume.integrablePower) p.2) f)) ≤
    2 ^ volume.integrablePower *
      ((∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower)) *
        (((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E))
            f +
          ((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E))
            f))