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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞ω
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n < N
w : W
I₁ : Integrable (fun v ↦ fourierPowSMulRight L f v n) μ
I₂ : Integrable (fun v ↦ ‖v‖ * ‖fourierPowSMulRight L f v n‖) μ
I₃ : Integrable (fun v ↦ fourierPowSMulRight L f v (n + 1)) μ
v : V
⊢ 2 * π * ‖L‖ * ‖v‖ * ‖fourierPowSMulRight L f v n‖ = 2 * π * ‖L‖ * (‖v‖ * ‖fourierPowSMulRight L f v n‖)

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup D
inst✝¹ : NormedSpace ℝ D
inst✝ : NormedSpace 𝕜 F
g : D → E
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
hgrowth : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N g x‖ ≤ C * (1 + ‖x‖) ^ l
kg : ℕ
Cg : ℝ
hg_upper' : ∀ (x : D), ‖x‖ ≤ Cg * (1 + ‖g x‖) ^ kg
hCg : 1 ≤ 1 + Cg
k' : ℕ := kg * (k + l * n)
f : 𝓢(E, F)
x : D
seminorm_f : ℝ := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily 𝕜 E F)) f
hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k'
hbound : ∀ i ≤ n, ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
hgrowth' : ∀ (N : ℕ), 1 ≤ N → N ≤ n → ‖iteratedFDeriv ℝ N g x‖ ≤ ((C + 1) * (1 + ‖x‖) ^ l) ^ N
this : ‖iteratedFDeriv ℝ n (⇑f ∘ g) x‖ ≤ ↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n
hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k
⊢ (1 + ‖x‖) ^ k * (↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) ^ n * (1 + ‖x‖) ^ (l * n))) =
    (1 + ‖x‖) ^ k * (1 + ‖x‖) ^ (l * n) / (1 + ‖g x‖) ^ k' * ((C + 1) ^ n * ↑n ! * 2 ^ k' * seminorm_f)

Tactic:
ring

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

Goal state before:
pp: case pos
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : Integrable f μ
⊢ ContDiff ℝ (↑N) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)

pp: case neg
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : ¬Integrable f μ
⊢ ContDiff ℝ (↑N) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)

Tactic:
· exact (hasFTaylorSeriesUpTo_fourierIntegral' L hf h'f.1).contDiff

Goal state after:
case neg
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ
h'f : ¬Integrable f μ
⊢ ContDiff ℝ (↑N) (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace ℝ H
inst✝ : NormedSpace 𝕜 H
L : F →L[𝕜] G
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k * (‖L‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖) = ‖L‖ * (‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖)

Tactic:
ring

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

Goal state before:
pp: 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 * ∏ x, -↑((L (m x.succ)) w))) =
    (-(2 * ↑π * I)) ^ n * (2 * ↑π * I) * (↑((L (m 0)) w) * ∏ x, -↑((L (m x.succ)) w))

Tactic:
ring

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
hg : Integrable g μ
h'g : ContinuousAt g 0
I : Integrable φ μ
u : Set F
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
hu : 0 ∈ u
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
h'u : ball 0 δ ⊆ u
M : ℝ
Mpos : M > 0
hM : ∀ ⦃x : F⦄, x ∈ (closedBall 0 M)ᶜ → ‖x‖ ^ finrank ℝ F * φ x < δ ^ finrank ℝ F * ε
c : ℝ
hc : M / δ < c
x : F
hx : x ∈ uᶜ
cpos : 0 < c
hδx : δ ≤ ‖x‖
⊢ ‖x‖ ^ finrank ℝ F * (c ^ finrank ℝ F * φ (c • x)) = c ^ finrank ℝ F * ‖x‖ ^ finrank ℝ F * φ (c • x)

Tactic:
ring

Goal state after:
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‖
⊢ eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ * eLpNorm (fun x ↦ (1 + ‖x‖) ^ k • f x) ∞ μ ≤
    eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ *
      (2 ^ k * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f))

Tactic:
by
    gcongr
    refine eLpNormEssSup_le_of_ae_nnnorm_bound (ae_of_all μ fun x ↦ ?_)
    rw [← norm_toNNReal, Real.toNNReal_le_iff_le_coe]
    simpa [norm_smul, abs_of_nonneg (h_one_add x).le] using!
      one_add_le_sup_seminorm_apply (m := (k, 0)) (le_refl k) (le_refl 0) f x

Goal state after:
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‖
⊢ eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ *
      (2 ^ k * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)) =
    eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) p μ * 2 ^ k *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Tactic:
ring

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✝³ : NormedAddCommGroup E
g : α → F
l : Filter α
ι : Type u_4
inst✝² : MeasurableSpace ι
f : ι × α → E
s : Set ι
μ : Measure ι
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
hf : f =O[𝓟 s ×ˢ l] (g ∘ Prod.snd)
hμ : μ s < ⊤
C : ℝ
hC : ∀ᶠ (x : ι × α) in 𝓟 s ×ˢ l, ‖f x‖ ≤ C * ‖(g ∘ Prod.snd) x‖
t : Set (ι × α)
htl : t ∈ 𝓟 s ×ˢ l
ht : ∀ y ∈ t, ‖f y‖ ≤ C * ‖(g ∘ Prod.snd) y‖
u : Set ι
hu : u ∈ 𝓟 s
v : Set α
hv : v ∈ l
huv : u ×ˢ v ⊆ t
x : α
hx : x ∈ v
⊢ C * ‖g x‖ * μ.real s = C * μ.real s * ‖g x‖

Tactic:
ring

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

Goal state before:
pp: 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
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
v : V
w : W
⊢ (2 ^ k * π ^ k * (2 * ↑k + 2) ^ n * ‖L‖ ^ k *
        ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ) *
      ‖v‖ ^ n =
    ‖v‖ ^ n * (2 ^ k * π ^ k * ‖L‖ ^ k) * (2 * ↑k + 2) ^ n *
      ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ

Tactic:
ring

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

Goal state before:
pp: ⊢ π = π / 2 - -(π / 2)

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
x✝ : ℝ
⊢ ↑(rexp (-x✝)) * (s * ↑x✝ ^ (s - 1)) = s * (↑(rexp (-x✝)) * ↑x✝ ^ (s - 1))

Tactic:
ring

Goal state after:
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:
ring

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

Goal state before:
pp: r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π
⊢ ↑r * Complex.cos ↑θ + ↑r * Complex.sin ↑θ * Complex.I = ↑r * (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I)

Tactic:
ring

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

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝¹ : NormedCommRing 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] {f : (i : Fin n) → E i → 𝕜},
    (∀ (i : Fin n), Integrable (f i) (μ i)) → Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
hf : ∀ (i : Fin (n + 1)), Integrable (f i) (μ i)
⊢ Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)

Tactic:
have := ((measurePreserving_piFinSuccAbove μ 0).symm)

Goal state after:
case succ
𝕜 : Type u_1
inst✝¹ : NormedCommRing 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] {f : (i : Fin n) → E i → 𝕜},
    (∀ (i : Fin n), Integrable (f i) (μ i)) → Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
hf : ∀ (i : Fin (n + 1)), Integrable (f i) (μ i)
this :
  MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove E 0).symm) ((μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)))
    (Measure.pi μ)
⊢ Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: p : ℝ × ℝ
⊢ cos p.2 * (p.1 * cos p.2) + p.1 * sin p.2 * sin p.2 = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case e_a.e_f
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
F_der_I :
  ∀ x ∈ Ioo 0 X,
    HasDerivAt (fun x ↦ ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x
cont : Continuous ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s)
der_ible : IntervalIntegrable (fun x ↦ -(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) volume 0 X
int_eval :
  (∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s) + -∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) =
    -(((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) X - ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) 0)
hn : s ≠ 0
x : ℝ
⊢ ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) = s * (↑(rexp (-x)) * ↑x ^ (s - 1))

Tactic:
ring

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

Goal state before:
pp: case e_f.e_a.e_a
b : ℂ
hb : 0 < b.re
p : ℝ × ℝ
⊢ (↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2 =
    (Complex.sin ↑p.2 ^ 2 + Complex.cos ↑p.2 ^ 2) * ↑p.1 ^ 2

Tactic:
ring

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ b⁻¹ * 2⁻¹ * (↑π + ↑π) = ↑π / b

Tactic:
ring

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

Goal state before:
pp: case e'_2
ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ × ℝ) → ℝ≥0∞
⊢ (∫⁻ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target,
      (∏ i, ENNReal.ofReal (p i).1) * f fun i ↦ ↑polarCoord.symm (p i)) =
    ∫⁻ (x : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target,
      ENNReal.ofReal |(fderivPiPolarCoordSymm x).det| * f fun i ↦ ↑polarCoord.symm (x i)

Tactic:
refine setLIntegral_congr_fun measurableSet_pi_polarCoord_target (fun x hx ↦ ?_)

Goal state after:
case e'_2
ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ × ℝ) → ℝ≥0∞
x : ι → ℝ × ℝ
hx : x ∈ Set.univ.pi fun x ↦ polarCoord.target
⊢ ((∏ i, ENNReal.ofReal (x i).1) * f fun i ↦ ↑polarCoord.symm (x i)) =
    ENNReal.ofReal |(fderivPiPolarCoordSymm x).det| * f fun i ↦ ↑polarCoord.symm (x i)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: ⊢ Gamma 0 = 0

Tactic:
Gamma,

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

Goal state before:
pp: case e'_3.e_a.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
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ 2 * ↑π * I * ↑⟪w, v⟫ = 2 * ↑π * ↑⟪w, v⟫ * I

Tactic:
ring

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
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ∫ (x : ℝ), rexp (-1 * x ^ 2) = (π / 1) ^ (1 / 2)

Tactic:
convert! integral_gaussian (1 : ℝ) using 1

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ (π / 1) ^ (1 / 2) = √(π / 1)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ 0 = 2 * |c| * 0

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ Gamma s = ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)

Tactic:
Gamma,

Goal state after:
s : ℝ
hs : 0 < s
⊢ (Complex.Gamma ↑s).re = ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: ⊢ Gamma 1 = 1

Tactic:
Gamma,

Goal state after:
⊢ (Complex.Gamma ↑1).re = 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
⊢ Complex.Gamma ↑s = ↑(Gamma s)

Tactic:
Gamma,

Goal state after:
s : ℝ
⊢ Complex.Gamma ↑s = ↑(Complex.Gamma ↑s).re
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: n : ℕ
⊢ Gamma (↑n + 1) = ↑n !

Tactic:
Gamma,

Goal state after:
n : ℕ
⊢ (Complex.Gamma ↑(↑n + 1)).re = ↑n !
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case e_f.e_a.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
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
B :
  Tendsto (fun c ↦ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
hc : 0 < c
w : V
⊢ ↑π ^ 2 * ↑‖v - w‖ ^ 2 * ↑c = ↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2

Tactic:
ring

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

Goal state before:
pp: case e'_2
x : ℝ
hx : x ∈ Ioi 0
⊢ (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)

Tactic:
have : (x ^ (2 : ℝ)) ^ (1 / (2 : ℝ) - 1) = x⁻¹ := by
      rw [← rpow_mul (le_of_lt hx)]
      norm_num
      rw [rpow_neg (le_of_lt hx), rpow_one]

Goal state after:
case e'_2
x : ℝ
hx : x ∈ Ioi 0
this : (x ^ 2) ^ (1 / 2 - 1) = x⁻¹
⊢ (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case zero
⊢ 1 / 2 * √π = √π / 2

Tactic:
ring

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

Goal state before:
pp: case succ
k : ℕ
ih : Gamma (↑k + 1 + 1 / 2) = ↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)
⊢ (↑k + 1 + 2⁻¹) * (↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)) = (2 * ↑k + 1 + 2) * ↑(2 * k + 1)‼ * √π / 2 ^ (k + 1 + 1)

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ -↑m - 1 = -(1 + ↑m)

Tactic:
ring

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

Goal state before:
pp: case e_a.e_a.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
⊢ ↑(finrank ℝ V) / 2 = -↑(finrank ℝ V) / 2 + ↑2 * (↑(finrank ℝ V) / 2)

Tactic:
ring

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

Goal state before:
pp: case e_a.e_a.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
⊢ ↑(finrank ℝ V) = ↑2 * (↑(finrank ℝ V) / 2)

Tactic:
ring

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℂ
this : b ≠ 0
h : (-↑π * b).re < 0
t x : ℝ
⊢ -2 * ↑π * ↑x * ↑t * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =
    -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0

Tactic:
ring

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

Goal state before:
pp: case e_a
b : ℂ
hb : 0 < b.re
c : ℂ
this✝ : b ≠ 0
h : (-↑π * b).re < 0
t : ℝ
this :
  ∀ (x : ℝ),
    ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =
      -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0
⊢ -(↑π * I * ↑t * c * 8) + ↑π * -1 * ↑t ^ 2 * 4 + ↑π * c ^ 2 * 4 =
    -(↑π * I * ↑t * c * 8) - ↑π * -1 * c ^ 2 * 4 - ↑π * ↑t ^ 2 * 4

Tactic:
ring

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

Goal state before:
pp: ι : Type u_2
inst✝ : Fintype ι
b : ι → ℂ
hb : ∀ (i : ι), 0 < (b i).re
c : ι → ℂ
⊢ Integrable (fun v ↦ cexp (∑ x, (-(b x * ↑(v x) ^ 2) + c x * ↑(v x)))) volume

Tactic:
Complex.exp_sum,

Goal state after:
ι : Type u_2
inst✝ : Fintype ι
b : ι → ℂ
hb : ∀ (i : ι), 0 < (b i).re
c : ι → ℂ
⊢ Integrable (fun v ↦ ∏ x, cexp (-(b x * ↑(v x) ^ 2) + c x * ↑(v x))) volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: ι : Type u_2
inst✝ : Fintype ι
b : ι → ℂ
hb : ∀ (i : ι), 0 < (b i).re
c : ι → ℂ
⊢ ∫ (v : ι → ℝ), cexp (∑ x, (-(b x * ↑(v x) ^ 2) + c x * ↑(v x))) =
    ∏ i, (↑π / b i) ^ (1 / 2) * cexp (c i ^ 2 / (4 * b i))

Tactic:
Complex.exp_sum,

Goal state after:
ι : Type u_2
inst✝ : Fintype ι
b : ι → ℂ
hb : ∀ (i : ι), 0 < (b i).re
c : ι → ℂ
⊢ ∫ (v : ι → ℝ), ∏ x, cexp (-(b x * ↑(v x) ^ 2) + c x * ↑(v x)) = ∏ i, (↑π / b i) ^ (1 / 2) * cexp (c i ^ 2 / (4 * b i))
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) •
        (fourierMultiplierCLM F fun x ↦ inner ℝ x (b i))
          ((2 * ↑π * Complex.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:
← fourierMultiplierCLM_ofReal ℂ (by fun_prop)

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
i : Fin (Module.finrank ℝ E)
x : E
⊢ ((2 * ↑π * Complex.I) •
        (fourierMultiplierCLM F fun x ↦ ↑(inner ℝ x (b i)))
          ((2 * ↑π * Complex.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