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

Goal state before:
pp: α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝³] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
u : Set α
u_open : IsOpen[inst✝³] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball 0 δ
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < δ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < δ
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
I : IntegrableOn (φ i) t μ
B : ‖∫ (x : α) in s ∩ u, φ i x • g x ∂μ‖ ≤ 2 * δ
⊢ s \ u ≤ᵐ[μ] s

Tactic:
· filter_upwards using sdiff_subset (s := s) (t := u)

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

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k n : ℕ
⊢ ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C

Tactic:
rcases f.decay' k n with ⟨C, hC⟩
  exact ⟨max C 1, by positivity, fun x => (hC x).trans (le_max_left _ _)⟩

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

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k : ℕ
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ (-↑k)

Tactic:
Asymptotics.IsBigO,

Goal state after:
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k : ℕ
d : ℝ
left✝ : 0 < d
hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d
⊢ ∃ c, IsBigOWith c (cocompact E) ⇑f fun x ↦ ‖x‖ ^ (-↑k)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : 𝓢(E, F)
inst✝ : ProperSpace E
s : ℝ
k : ℕ := ⌈-s⌉₊
hk : -↑k ≤ s
⊢ (fun x ↦ x ^ (-↑k)) =O[atTop] fun x ↦ x ^ s

Tactic:
Asymptotics.IsBigO,

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : 𝓢(E, F)
inst✝ : ProperSpace E
s : ℝ
k : ℕ := ⌈-s⌉₊
hk : -↑k ≤ s
⊢ ∃ c, IsBigOWith c atTop (fun x ↦ x ^ (-↑k)) fun x ↦ x ^ s
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
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
h4 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, ‖F' w' v‖ ≤ B v
h5 : Integrable B μ
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w

Tactic:
have h6 : ∀ᵐ v ∂μ, ∀ w', w' ∈ Metric.ball w 1 → HasFDerivAt (fun x ↦ F x v) (F' w' v) w' :=
    ae_of_all _ (fun v w' _ ↦ hasFDerivAt_fourierChar_smul L f v w')

Goal state after:
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
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
h4 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, ‖F' w' v‖ ≤ B v
h5 : Integrable B μ
h6 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, HasFDerivAt (fun x ↦ F x v) (F' w' v) w'
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ (k, n).1 * ‖iteratedFDeriv ℝ (k, n).2 (fun x ↦ (L x).smulRight (f x)) x‖ ≤
    ‖x‖ ^ k *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖

Tactic:
gcongr 1

Goal state after:
case hbc
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖iteratedFDeriv ℝ (k, n).2 (fun x ↦ (L x).smulRight (f x)) x‖ ≤
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ‖x‖ ^ k *
      ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖ ≤
    ‖x‖ ^ k * (‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖)

Tactic:
gcongr 1

Goal state after:
case hbc
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
inst✝⁴ : NormedAddCommGroup D
inst✝³ : NormedSpace ℝ D
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace 𝕜 F
L : E →L[ℝ] G →L[ℝ] ℝ
k n : ℕ
f : 𝓢(E, F)
x : E
⊢ ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i (⇑L) x‖ * ‖iteratedFDeriv ℝ (n - i) (⇑f) x‖ ≤
    ‖L x‖ * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ↑n * ‖L‖ * ‖iteratedFDeriv ℝ (n - 1) (⇑f) x‖
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: case hab
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
⊢ 1 ≤ 1 * (1 + ‖g x‖) ^ kg

Tactic:
← one_mul (1 : ℝ)

Goal state after:
case hab
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
⊢ 1 * 1 ≤ 1 * (1 + ‖g x‖) ^ kg
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 * ‖iteratedFDeriv ℝ n (⇑L ∘ ⇑f) x‖ =
    ‖x‖ ^ k * ‖(ContinuousLinearMap.restrictScalars ℝ L).compContinuousMultilinearMap (iteratedFDeriv ℝ n (⇑f) x)‖

Tactic:
congr
      exact (L.restrictScalars ℝ).iteratedFDeriv_comp_left f.smooth'.contDiffAt (mod_cast le_top)

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

Goal state before:
pp: case h
E : Type u_5
F : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
inst✝³ : SecondCountableTopologyEither E F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
p : ℝ≥0∞
hp : p ≠ ∞
hp' : Fact (1 ≤ p)
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : IsFiniteMeasureOnCompacts μ
f : ↥(Lp F p μ)
ε : ℝ
hε : 0 < ε
g : E → F
hg₁ : HasCompactSupport g
hg₂ : ContDiff ℝ ∞ g
hg₃ : eLpNorm (↑↑f - g) p μ ≤ ENNReal.ofReal ε
this : ↑↑f - ↑↑((hg₁.toSchwartzMap hg₂).toLp p μ) =ᵐ[μ] ↑↑f - g
⊢ (eLpNorm (↑↑f - g) p μ).toReal ≤ ε

Tactic:
hg₃,

Goal state after:
case h
E : Type u_5
F : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
inst✝³ : SecondCountableTopologyEither E F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
p : ℝ≥0∞
hp : p ≠ ∞
hp' : Fact (1 ≤ p)
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : IsFiniteMeasureOnCompacts μ
f : ↥(Lp F p μ)
ε : ℝ
hε : 0 < ε
g : E → F
hg₁ : HasCompactSupport g
hg₂ : ContDiff ℝ ∞ g
hg₃ : eLpNorm (↑↑f - g) p μ ≤ ENNReal.ofReal ε
this : ↑↑f - ↑↑((hg₁.toSchwartzMap hg₂).toLp p μ) =ᵐ[μ] ↑↑f - g
⊢ (ENNReal.ofReal ε).toReal ≤ ε

case hb
E : Type u_5
F : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
inst✝³ : SecondCountableTopologyEither E F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
p : ℝ≥0∞
hp : p ≠ ∞
hp' : Fact (1 ≤ p)
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : IsFiniteMeasureOnCompacts μ
f : ↥(Lp F p μ)
ε : ℝ
hε : 0 < ε
g : E → F
hg₁ : HasCompactSupport g
hg₂ : ContDiff ℝ ∞ g
hg₃ : eLpNorm (↑↑f - g) p μ ≤ ENNReal.ofReal ε
this : ↑↑f - ↑↑((hg₁.toSchwartzMap hg₂).toLp p μ) =ᵐ[μ] ↑↑f - g
⊢ ENNReal.ofReal ε ≠ ∞
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f : V → E
w : V
⊢ ∫ (v : V), ↑(𝐞 ⟪v, w⟫) • f v = ∫ (v : V), Complex.exp (↑(2 * π * ⟪v, w⟫) * Complex.I) • f v

Tactic:
Real.fourierChar_apply

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
⊢ HasDerivAt (𝓕 f) (𝓕 (fun x ↦ (-2 * ↑π * I * ↑x) • f x) w) w

Tactic:
let L := ContinuousLinearMap.mul ℝ ℝ |>.flip

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
⊢ HasDerivAt (𝓕 f) (𝓕 (fun x ↦ (-2 * ↑π * I * ↑x) • f x) w) w
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

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

Tactic:
simp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.ofReal_cos,
        Complex.ofReal_sin]

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

Goal state before:
pp: case refine_2
b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun x ↦ x * rexp (-b.re * x ^ 2)) volume
⊢ HasFiniteIntegral (fun x ↦ ↑x * cexp (-b * ↑x ^ 2)) volume

Tactic:
rw [← hasFiniteIntegral_norm_iff] at this ⊢

Goal state after:
case refine_2
b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
⊢ HasFiniteIntegral (fun a ↦ ‖↑a * cexp (-b * ↑a ^ 2)‖) volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
this : (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)
⊢ GammaAux n ((starRingEnd ℂ) (s + 1)) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s

Tactic:
IH

Goal state after:
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
this : (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)
⊢ (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ (∫ (x : ℝ), cexp (-c * ↑x ^ 2)) ^ 2 = ((↑π / c) ^ (1 / 2)) ^ 2

Tactic:
rw [integral_gaussian_sq_complex hc, sq]

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ ↑π / c = (↑π / c) ^ (1 / 2) * (↑π / c) ^ (1 / 2)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : s ≠ 0
⊢ Gamma (s + 1) = s * Gamma s

Tactic:
simp_rw [Gamma]
  rw [Complex.ofReal_add, Complex.ofReal_one, Complex.Gamma_add_one, Complex.re_ofReal_mul]
  rwa [Complex.ofReal_ne_zero]

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

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ 0 < Gamma s

Tactic:
rw [Gamma_eq_integral hs]

Goal state after:
s : ℝ
hs : 0 < s
⊢ 0 < ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
convert! integral_gaussian_complex_Ioi (by rwa [ofReal_re] : 0 < (b : ℂ).re)

Goal state after:
case e'_2.e'_7
b : ℝ
hb : 0 < b
x✝ : ℝ
⊢ ↑(rexp (-b * x✝ ^ 2)) = cexp (-↑b * ↑x✝ ^ 2)

case e'_3
b : ℝ
hb : 0 < b
⊢ ↑(√(π / b) / 2) = (↑π / ↑b) ^ (1 / 2) / 2
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
⊢ (∂_{b i} (∂_{b i} f)) x = -(2 * π) ^ 2 • ((fourierMultiplierCLM F fun x ↦ inner ℝ x (b i) ^ 2) f) x

Tactic:
lineDeriv_eq_fourierMultiplierCLM

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
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
hL : Continuous fun p ↦ (L p.1) p.2
w : AddMonoidAlgebra ℂ W
v : V
⊢ ((charAlgHom he hL) w) v = w.coeff.sum fun a z ↦ z • ↑(e ((L v) a))

Tactic:
by
  simp [charAlgHom, charMonoidHom, char, AddMonoidAlgebra.lift_apply]
  simp [Finsupp.sum]

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

Goal state before:
pp: case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
f' : ↥(Lp F 2 volume)
hf' : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution f'
⊢ 𝓕⁻ ((smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f)) = Lp.toTemperedDistribution (𝓕⁻ f')

Tactic:
apply_fun 𝓕⁻ at hf'

Goal state after:
case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
f' : ↥(Lp F 2 volume)
hf' : 𝓕⁻ ((smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f)) = 𝓕⁻ (Lp.toTemperedDistribution f')
⊢ 𝓕⁻ ((smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f)) = Lp.toTemperedDistribution (𝓕⁻ f')
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
E : ι → Type u_4
inst✝¹⁰ : Fintype ι
inst✝⁹ : RCLike 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → InnerProductSpace 𝕜 (E i)
𝕝 : Type u_5
inst✝⁶ : CommSemiring 𝕝
inst✝⁵ : StarRing 𝕝
inst✝⁴ : Algebra 𝕝 𝕜
inst✝³ : StarModule 𝕝 𝕜
inst✝² : SMulCommClass ℝ 𝕝 𝕜
inst✝¹ : (i : ι) → Module 𝕝 (E i)
inst✝ : ∀ (i : ι), IsScalarTower 𝕝 𝕜 (E i)
c : 𝕝
w : ι → ℝ
f g : (i : ι) → E i
⊢ ⟪c • f, g⟫_[𝕜, w] = star c • ⟪f, g⟫_[𝕜, w]

Tactic:
simp_rw [wInner, Pi.smul_apply, inner_smul_left_eq_star_smul, starRingEnd_apply, smul_sum,
    smul_comm (w _)]

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
f : ι → 𝕜
a : 𝕜
⊢ ⟪f, const ι a⟫_[𝕜, w] = a * ∑ i, w i • (starRingEnd 𝕜) (f i)

Tactic:
simp [wInner, const_apply, mul_sum]

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s s' : ℝ
h : s' ≤ s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
⊢ (s' - s) / 2 ≤ 0

Tactic:
rw [div_le_iff₀ (by norm_num)]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s s' : ℝ
h : s' ≤ s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
⊢ s' - s ≤ 0 * 2
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
⊢ (starRingEnd ℝ) (1 / (2 * ‖w‖ ^ 2)) * ⟪w, w⟫ = 1 / 2

Tactic:
inner_self_eq_norm_sq_to_K,

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
⊢ (starRingEnd ℝ) (1 / (2 * ‖w‖ ^ 2)) * ↑‖w‖ ^ 2 = 1 / 2
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
h_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
h_sum : Summable fun n ↦ 𝓕 ⇑f ↑n
x : ℝ
⊢ ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x

Tactic:
let F : C(UnitAddCircle, ℂ) :=
    ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩

Goal state after:
f : C(ℝ, ℂ)
h_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
h_sum : Summable fun n ↦ 𝓕 ⇑f ↑n
x : ℝ
F : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }
⊢ ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
⊢ (-1) ^ 1 = -1

Tactic:
pow_one

Goal state after:
p : ℕ
hp : Nat.Prime p
⊢ -1 = -1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: R : Type u_4
k : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
h0 : 0 < k
⊢ ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k

Tactic:
rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]

Goal state after:
R : Type u_4
k : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
h0 : 0 < k
⊢ IsPrimitiveRoot ζ k → ζ ^ k = 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
k : ℕ
ζ ζ' : M
k' : ℕ
hζ : IsPrimitiveRoot ζ k
hζ' : IsPrimitiveRoot ζ' k'
hk : k ≠ 0
hk' : k' ≠ 0
⊢ IsPrimitiveRoot (ζ ^ (k / k.factorizationLCMLeft k') * ζ' ^ (k' / k.factorizationLCMRight k')) (k.lcm k')

Tactic:
convert! IsPrimitiveRoot.orderOf _
  convert!
    ((Commute.all ζ ζ').orderOf_mul_pow_eq_lcm (by simpa [← hζ.eq_orderOf])
        (by simpa [← hζ'.eq_orderOf])).symm using 2
  all_goals simp [hζ.eq_orderOf, hζ'.eq_orderOf]

Goal state after: