mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
hiφ : ∀ (n : ℕ), ∫ (x : α) in s, φ n x ∂μ = 1
u : Set α
u_open : IsOpen[inst✝⁶] u
x₀u : x₀ ∈ u
t : ℝ
t_pos : 0 ≤ t
tx₀ : t < c x₀
ht : ∀ x ∈ s \ u, c x ≤ t
t' : ℝ
tt' : t < t'
t'x₀ : t' < c x₀
t'_pos : 0 < t'
v : Set α
v_open : IsOpen[inst✝⁶] v
x₀_v : x₀ ∈ v
hv : v ∩ s ⊆ c ⁻¹' Ioi t'
M : ∀ (n : ℕ), ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n
N : Tendsto (fun n ↦ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n) atTop (𝓝 0)
ε : ℝ
εpos : ε > 0
n : ℕ
⊢ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n < ε → ∀ x ∈ s \ u, dist (0 x) (φ n x) < ε

Tactic:
hn

Goal state after:
α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
hiφ : ∀ (n : ℕ), ∫ (x : α) in s, φ n x ∂μ = 1
u : Set α
u_open : IsOpen[inst✝⁶] u
x₀u : x₀ ∈ u
t : ℝ
t_pos : 0 ≤ t
tx₀ : t < c x₀
ht : ∀ x ∈ s \ u, c x ≤ t
t' : ℝ
tt' : t < t'
t'x₀ : t' < c x₀
t'_pos : 0 < t'
v : Set α
v_open : IsOpen[inst✝⁶] v
x₀_v : x₀ ∈ v
hv : v ∩ s ⊆ c ⁻¹' Ioi t'
M : ∀ (n : ℕ), ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n
N : Tendsto (fun n ↦ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n) atTop (𝓝 0)
ε : ℝ
εpos : ε > 0
n : ℕ
hn : (μ.real (v ∩ s))⁻¹ * (t / t') ^ n < ε
⊢ ∀ x ∈ s \ u, dist (0 x) (φ n x) < ε
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
⊢ 0 ≤ Cg

Tactic:
specialize hg_upper' 0

Goal state after:
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' : ‖0‖ ≤ Cg * (1 + ‖g 0‖) ^ kg
⊢ 0 ≤ Cg
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case fderiv
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
⊢ ↑n < N →
    ∀ (x : W),
      HasFDerivAt
        (fun y ↦ fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n) y)
        (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n.succ)
            x).curryLeft
        x

Tactic:
hn

Goal state after:
case fderiv
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
⊢ ∀ (x : W),
    HasFDerivAt
      (fun y ↦ fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n) y)
      (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n.succ) x).curryLeft
      x
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case cont
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
⊢ ↑n ≤ N →
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
      fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n) x

Tactic:
hn

Goal state after:
case cont
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
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun v ↦ fourierPowSMulRight L f v n) x
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: H : Type u_8
V : Type u_9
inst✝⁷ : NormedAddCommGroup H
inst✝⁶ : NormedSpace ℝ H
inst✝⁵ : FiniteDimensional ℝ H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℂ V
f g : 𝓢(H, V)
μ : Measure H
inst✝ : μ.HasTemperateGrowth
⊢ inner ℂ (f.toLp 2 μ) (g.toLp 2 μ) = ∫ (x : H), inner ℂ (f x) (g x) ∂μ

Tactic:
by
  apply integral_congr_ae
  have hf_ae := f.coeFn_toLp 2 μ
  have hg_ae := g.coeFn_toLp 2 μ
  filter_upwards [hf_ae, hg_ae] with _ hf hg
  rw [hf, hg]

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

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
x : ℝ
hx : x ∈ Ioi c
⊢ (fun t ↦ t ^ a.re) x = ‖↑x ^ a‖

Tactic:
Complex.norm_cpow_eq_rpow_re_of_pos (hc.trans hx)

Goal state after:
a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
x : ℝ
hx : x ∈ Ioi c
⊢ (fun t ↦ t ^ a.re) x = x ^ a.re
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
⊢ IntegrableOn f (Iic ?m.106 ∪ Ici ?m.106) μ

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
⊢ α

Tactic:
integrableOn_union

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
⊢ IntegrableOn f (Iic ?m.106) μ ∧ IntegrableOn f (Ici ?m.106) μ

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
⊢ α
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
h : 0 < s
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioc 0 1 ∪ Ioi 1) volume

Tactic:
integrableOn_union

Goal state after:
s : ℝ
h : 0 < s
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume ∧
    IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioi 1) volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
⊢ (fun x ↦ x ^ s * rexp (-b * x ^ 2)) =o[atTop] fun x ↦ rexp (-(1 / 2) * x)

Tactic:
exact rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg s one_lt_two hb

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

Goal state before:
pp: case left
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))
⊢ AEStronglyMeasurable (fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))) (volume.restrict (Ioc 0 Y))

Tactic:
refine (continuousOn_const.mul ?_).aestronglyMeasurable measurableSet_Ioc

Goal state after:
case left
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))
⊢ ContinuousOn (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) (Ioc 0 Y)
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n✝ : ℕ
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume
hn : ↑n✝ ≤ N
x : ℝ
n : ℕ
⊢ ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume

Tactic:
hn

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n✝ : ℕ
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume
hn✝ : ↑n✝ ≤ N
x : ℝ
n : ℕ
hn : ↑n ≤ N
⊢ Integrable (iteratedFDeriv ℝ n f) volume
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 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),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
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
⊢ ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ) =
    (∫ (x : E 0), f 0 x ∂μ 0) * ∫ (x : (i : Fin n) → E i.succ), ∏ i, f i.succ (x i) ∂Measure.pi fun i ↦ μ i.succ

Tactic:
← integral_prod_mul

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 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),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
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
⊢ ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ) =
    ∫ (z : E 0 × ((i : Fin n) → E i.succ)), f 0 z.1 * ∏ i, f i.succ (z.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) * ∫ (x : ℝ), cexp (-b * ↑x ^ 2) =
    ∫ (p : ℝ × ℝ), cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2)

Tactic:
← integral_prod_mul

Goal state after:
b : ℂ
hb : 0 < b.re
⊢ ∫ (z : ℝ × ℝ), cexp (-b * ↑z.1 ^ 2) * cexp (-b * ↑z.2 ^ 2) ∂volume.prod volume =
    ∫ (p : ℝ × ℝ), cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ↑(∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * rexp (-(r * t))) = (↑1 / ↑r) ^ ↑a * Complex.Gamma ↑a

Tactic:
convert! integral_cpow_mul_exp_neg_mul_Ioi (by rwa [ofReal_re] : 0 < (a : ℂ).re) hr

Goal state after:
case e'_2
a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ↑(∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * rexp (-(r * t))) = ∫ (t : ℝ) in Ioi 0, ↑t ^ (↑a - 1) * cexp (-(↑r * ↑t))
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
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 c ↦ ∫ (w : V), (c ^ finrank ℝ V * φ (c • (v - w))) • f w) atTop (𝓝 (f v))
B : Tendsto (fun c ↦ ∫ (w : V), ((c ^ (1 / 2)) ^ finrank ℝ V * φ (c ^ (1 / 2) • (v - w))) • f w) atTop (𝓝 (f v))
c : ℝ
hc : 0 < c
w : V
⊢ ↑((c ^ (1 / 2)) ^ finrank ℝ V) * ↑(π ^ (↑(finrank ℝ V) / 2)) = (↑π * ↑c) ^ (↑(finrank ℝ V) / 2)

Tactic:
rw [mul_cpow_ofReal_nonneg pi_nonneg hc.le, ← rpow_natCast, ← rpow_mul hc.le, mul_comm,
      ofReal_cpow pi_nonneg, ofReal_cpow hc.le]
    simp [div_eq_inv_mul]

Goal state after:
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : InnerProductSpace ℝ E
inst✝⁶ : NormedSpace ℂ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℂ 𝕜 F
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : CompleteSpace F
g₁ g₂ : E → 𝕜
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
f : 𝓢(E, F)
⊢ (fourierMultiplierCLM F g₁) ((fourierMultiplierCLM F g₂) f) = (fourierMultiplierCLM F (g₁ * g₂)) f

Tactic:
by
  simp [fourierMultiplierCLM_apply, smulLeftCLM_smulLeftCLM_apply hg₁ hg₂]

Goal state after:
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
m : E
f : 𝓢(E, F)
⊢ ∂_{m} f = (2 * ↑π * Complex.I) • (fourierMultiplierCLM F fun x ↦ inner ℝ x m) f

Tactic:
rw [fourierMultiplierCLM_apply, ← FourierTransform.fourierInv_smul, ← fourier_lineDerivOp_eq,
    FourierTransform.fourierInv_fourier_eq]

Goal state after:
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
g₁ g₂ : E → ℂ
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
f : 𝓢'(E, F)
⊢ (fourierMultiplierCLM F g₂) ((fourierMultiplierCLM F g₁) f) = (fourierMultiplierCLM F (g₁ * g₂)) f

Tactic:
by
  simp [fourierMultiplierCLM_apply, smulLeftCLM_smulLeftCLM_apply hg₁ hg₂]

Goal state after:
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
m : E
f : 𝓢'(E, F)
⊢ ∂_{m} f = (2 * ↑π * Complex.I) • (fourierMultiplierCLM F fun x ↦ ↑(inner ℝ x m)) f

Tactic:
rw [fourierMultiplierCLM_apply, ← FourierTransform.fourierInv_smul, ← fourier_lineDerivOp_eq,
    FourierTransform.fourierInv_fourier_eq]

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : PseudoMetricSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
⊢ (dsupport f)ᶜ =
    ⋃₀ {a | IsVanishingOn f a ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧ Bornology.IsBounded a}

Tactic:
ext x
  grind [(Metric.hasBasis_nhds_isOpen_isBounded x).notMem_dsupport]

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : NormedSpace ℂ F
s s' : ℝ
f : 𝓢'(E, F)
⊢ (besselPotential E F s') ((besselPotential E F s) f) = (besselPotential E F (s + s')) f

Tactic:
simp only [besselPotential]
  rw [fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop) (by fun_prop)]
  congr
  ext x
  simp only [Pi.mul_apply]
  norm_cast
  calc
    _ = (1 + ‖x‖ ^ 2) ^ (s / 2 + s' / 2) := by
      rw [← Real.rpow_add (by positivity)]
    _ = _ := by congr; ring

Goal state after:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
w : ι → ℝ
a : 𝕜
f : ι → 𝕜
⊢ ⟪const ι a, f⟫_[𝕜, w] = (∑ i, w i • f i) * (starRingEnd 𝕜) a

Tactic:
by simp [wInner, const_apply, sum_mul]

Goal state after:
Module: Mathlib.Data.PNat.Prime

Goal state before:
pp: a b m n : ℕ+
cop : m.Coprime n
am : a ∣ m
bn : b ∣ n
⊢ a = (a * b).gcd m

Tactic:
← gcd_eq_left_iff_dvd

Goal state after:
a b m n : ℕ+
cop : m.Coprime n
am : a.gcd m = a
bn : b ∣ n
⊢ a = (a * b).gcd m
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: n : ℕ
⊢ |μ n| ≤ 1

Tactic:
by
  rw [abs_moebius, apply_ite (· ≤ 1)]
  simp

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: case mp
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := fun n ↦ ∑ i ∈ n.divisors, f i
n : ℕ
⊢ n > 0 → n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n

Tactic:
hn

Goal state after:
case mp
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := fun n ↦ ∑ i ∈ n.divisors, f i
n : ℕ
hn : n > 0
⊢ n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := fun n ↦ ∑ i ∈ n.divisors, f i
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, μ (n / d) • G d = f n
⊢ ∑ i ∈ n.divisors, μ (n / i) • g i = ∑ d ∈ n.divisors, μ (n / d) • G d

Tactic:
sum_congr rfl

Goal state after:
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := fun n ↦ ∑ i ∈ n.divisors, f i
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, μ (n / d) • G d = f n
⊢ n.divisors.sum ?m.113 = ∑ d ∈ n.divisors, μ (n / d) • G d

R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := fun n ↦ ∑ i ∈ n.divisors, f i
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, μ (n / d) • G d = f n
⊢ ∀ x ∈ n.divisors, μ (n / x) • g x = ?m.113 x

R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n
G : ℕ → R := ⋯
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, μ (n / d) • G d = f n
⊢ ℕ → R
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: case e'_3
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x ↦ |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ (b_1 : ℝ), d ≤ b_1 → ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + max 0 (-2 * R) ≤ x ∧ d - R ≤ x
hx' : 0 < x ∧ -2 * R < x
y : ↑(Icc (x + R) (x + S))
A : ∀ (x : ℝ), 0 ≤ |x| ^ (-b)
⊢ |↑y| = ↑y

Tactic:
apply abs_of_nonneg; linarith [y.2.1]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: case mpr
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
F : ℕ → R := fun n ↦ ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2
n : ℕ
⊢ n > 0 → n ∈ s → ∑ i ∈ n.divisors, f i = g n

Tactic:
hn

Goal state after:
case mpr
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
F : ℕ → R := fun n ↦ ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2
n : ℕ
hn : n > 0
⊢ n ∈ s → ∑ i ∈ n.divisors, f i = g n