mathlib-initiative/mathlib-tactics huggingface dataset:
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
⊢ ∀ (f : 𝓢(E, F)), ‖f.toBoundedContinuousFunction‖ ≤ 1 * ({0}.sup (schwartzSeminormFamily 𝕜 E F)) f

Tactic:
simpa using

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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
⊢ ∀ (f : 𝓢(E, F)) (x : E), ‖f x‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) 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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : ProperSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
⊢ ∀ (f : 𝓢(E, F)), ‖f.toZeroAtInfty‖ ≤ 1 * ({0}.sup (schwartzSeminormFamily 𝕜 E F)) f

Tactic:
simpa using

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 : ProperSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
⊢ ∀ (f : 𝓢(E, F)) (x : E), ‖f x‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case h
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
w : W
m : ℕ
hm : ↑m ≤ N
I : Integrable (fun v ↦ ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (m + 1), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ
v : V
i : ℕ
hi : i ≤ m
j : ℕ
hj : j ≤ k
⊢ (j, i) ∈ Finset.range (k + 1) ×ˢ Finset.range (m + 1)

Tactic:
simpa using

Goal state after:
case h
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
w : W
m : ℕ
hm : ↑m ≤ N
I : Integrable (fun v ↦ ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (m + 1), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ
v : V
i : ℕ
hi : i ≤ m
j : ℕ
hj : j ≤ k
⊢ j ≤ k ∧ i ≤ m
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)
⊢ Integrable f μ

Tactic:
simpa using

Goal state after:
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)
⊢ Integrable (fun x ↦ g (x₀ + -x)) μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
μ : Measure E
C : ℝ
left✝ : 0 < C
hC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑f) x‖ ≤ C
x : E
⊢ ‖f x‖ ≤ C

Tactic:
simpa using

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
μ : Measure E
C : ℝ
left✝ : 0 < C
hC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑f) x‖ ≤ C
x : E
⊢ ‖f x‖ ≤ C
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
⊢ ContinuousAt g (x₀ - 0)

Tactic:
simpa using

Goal state after:
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
⊢ ContinuousAt g x₀
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
p : ℝ≥0∞
μ : Measure E
hp₁ : p ≠ 0
hp₂ : p ≠ ∞
hμ : μ.HasTemperateGrowth
⊢ (eLpNorm (⇑f) p μ).toReal = (∫ (x : E), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹

Tactic:
MeasureTheory.MemLp.eLpNorm_eq_integral_rpow_norm hp₁ hp₂ (f.memLp p μ),

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
p : ℝ≥0∞
μ : Measure E
hp₁ : p ≠ 0
hp₂ : p ≠ ∞
hμ : μ.HasTemperateGrowth
⊢ (ENNReal.ofReal ((∫ (a : E), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)).toReal =
    (∫ (x : E), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
μ : Measure E
hμ : μ.HasTemperateGrowth
⊢ ‖f.toLp 1 μ‖ = ∫ (x : E), ‖f x‖ ∂μ

Tactic:
simpa using

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : SecondCountableTopologyEither E F
f : 𝓢(E, F)
μ : Measure E
hμ : μ.HasTemperateGrowth
⊢ ‖f.toLp 1 μ‖ = ∫ (x : E), ‖f x‖ ∂μ
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
w : W
p : ℕ × ℕ
hp : p.1 ≤ k ∧ p.2 ≤ n
⊢ ↑p.1 ≤ ↑k

Tactic:
simpa using

Goal state after:
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
w : W
p : ℕ × ℕ
hp : p.1 ≤ k ∧ p.2 ≤ n
⊢ p.1 ≤ k
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : MeasurableSpace E
inst✝² : OpensMeasurableSpace E
inst✝¹ : SecondCountableTopologyEither E F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : μ.IsOpenPosMeasure
f g : 𝓢(E, F)
⊢ (fun f ↦ f.toLp p μ) f = (fun f ↦ f.toLp p μ) g → f = g

Tactic:
simpa using

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : MeasurableSpace E
inst✝² : OpensMeasurableSpace E
inst✝¹ : SecondCountableTopologyEither E F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
inst✝ : μ.IsOpenPosMeasure
f g : 𝓢(E, F)
⊢ ⇑f =ᵐ[μ] ⇑g → f = g
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
w : W
p : ℕ × ℕ
hp : p.1 ≤ k ∧ p.2 ≤ n
⊢ ↑p.2 ≤ ↑n

Tactic:
simpa using

Goal state after:
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
w : W
p : ℕ × ℕ
hp : p.1 ≤ k ∧ p.2 ≤ n
⊢ p.2 ≤ n
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
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 : NormedField 𝕜
inst✝² : NormedSpace 𝕜 F
inst✝¹ : SMulCommClass ℝ 𝕜 F
inst✝ : SecondCountableTopologyEither E F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
k : ℕ
C : ℝ≥0
hC :
  ∀ (f : 𝓢(E, F)), eLpNorm (⇑f) p μ ≤ ↑C * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)
f : 𝓢(E, F)
⊢ eLpNorm (⇑f) p μ ≤ ENNReal.ofReal ↑C * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_2
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 : NormedField 𝕜
inst✝² : NormedSpace 𝕜 F
inst✝¹ : SMulCommClass ℝ 𝕜 F
inst✝ : SecondCountableTopologyEither E F
p : ℝ≥0∞
μ : Measure E
hμ : μ.HasTemperateGrowth
k : ℕ
C : ℝ≥0
hC :
  ∀ (f : 𝓢(E, F)), eLpNorm (⇑f) p μ ≤ ↑C * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)
f : 𝓢(E, F)
⊢ eLpNorm (⇑f) p μ ≤ ↑C * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: case refine_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
hf : Summable fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
n : ℤ
x : ↑(Icc (↑n) (↑n + 1))
this :
  ‖(ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))) ⟨↑x - ↑n, ⋯⟩‖ ≤
    ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
⊢ ‖f ↑x‖ ≤ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖

Tactic:
simpa using

Goal state after:
case refine_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
hf : Summable fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
n : ℤ
x : ↑(Icc (↑n) (↑n + 1))
this :
  ‖(ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))) ⟨↑x - ↑n, ⋯⟩‖ ≤
    ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
⊢ ‖f ↑x‖ ≤ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 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)
⊢ f ∈ closure (Set.range ⇑(toLpCLM ℝ F p μ))

Tactic:
refine (mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall).2 fun ε hε ↦ ?_

Goal state after:
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
⊢ ∃ y ∈ Set.range ⇑(toLpCLM ℝ F p μ), y ∈ Metric.closedBall f ε
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
c : ℝ
f : ℝ → E
⊢ ∫ (x : ℝ) in Ioi c, f (-x) = ∫ (x : ℝ) in Iic (-c), f x

Tactic:
by
  rw [← neg_neg c, ← integral_comp_neg_Iic]
  simp only [neg_neg]

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

Goal state before:
pp: c : ℝ
⊢ ∫ (x : ℝ) in Ioi c, rexp (-x) = rexp (-c)

Tactic:
simpa using

Goal state after:
c : ℝ
⊢ ∫ (x : ℝ) in Iic (-c), rexp x = rexp (-c)
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: ⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) = 1

Tactic:
simpa using

Goal state after:
⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) = 1
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : 0 < a.re
c : ℝ
⊢ IntegrableOn (fun x ↦ Complex.exp (a * ↑x)) (Iic c) volume

Tactic:
simpa using

Goal state after:
a : ℂ
ha : 0 < a.re
c : ℝ
⊢ IntegrableOn (fun x ↦ Complex.exp (a * ↑x)) (Iic c) volume
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < 0
c : ℝ
⊢ IntegrableOn (fun x ↦ rexp (a * x)) (Ioi c) volume

Tactic:
have := Integrable.norm <| integrableOn_exp_mul_complex_Ioi (a := a) (by simpa using! ha) c

Goal state after:
a : ℝ
ha : a < 0
c : ℝ
this : Integrable (fun a_1 ↦ ‖Complex.exp (↑a * ↑a_1)‖) (volume.restrict (Ioi c))
⊢ IntegrableOn (fun x ↦ rexp (a * x)) (Ioi c) volume
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : a.re < 0
c : ℝ
this : Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)
⊢ Tendsto (fun i ↦ (Complex.exp (a * ↑i) - Complex.exp (a * ↑c)) / a) atTop (𝓝 (-Complex.exp (a * ↑c) / a))

Tactic:
simpa using

Goal state after:
a : ℂ
ha : a.re < 0
c : ℝ
this : Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)
⊢ Tendsto (fun i ↦ (Complex.exp (a * ↑i) - Complex.exp (a * ↑c)) / a) atTop (𝓝 (-Complex.exp (a * ↑c) / a))
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : a.re < 0
c : ℝ
⊢ Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)

Tactic:
simpa using

Goal state after:
a : ℂ
ha : a.re < 0
c : ℝ
⊢ Tendsto (fun x ↦ a.re * x) atTop atBot
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : 0 < a.re
c : ℝ
⊢ ∫ (x : ℝ) in Iic c, Complex.exp (a * ↑x) = Complex.exp (a * ↑c) / a

Tactic:
simpa using

Goal state after:
a : ℂ
ha : 0 < a.re
c : ℝ
⊢ ∫ (x : ℝ) in Iic c, Complex.exp (a * ↑x) = Complex.exp (a * ↑c) / a
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < 0
c : ℝ
⊢ (↑a).re < 0

Tactic:
simpa using

Goal state after:
a : ℝ
ha : a < 0
c : ℝ
⊢ a < 0
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : 0 < a
c : ℝ
⊢ ∫ (x : ℝ) in Iic c, rexp (a * x) = rexp (a * c) / a

Tactic:
simpa using

Goal state after:
a : ℝ
ha : 0 < a
c : ℝ
⊢ ∫ (x : ℝ) in Iic c, rexp (a * x) = rexp (a * c) / a
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a c : ℝ
ha : a < -1
hc : 0 < c
⊢ IntegrableOn (fun t ↦ t ^ a) (Ioi c) volume

Tactic:
simpa using

Goal state after:
a c : ℝ
ha : a < -1
hc : 0 < c
⊢ IntegrableOn (fun t ↦ t ^ a) (Ioi c) volume
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ Tendsto (fun a_1 ↦ a_1 ^ (a + 1)) atTop (𝓝 0)

Tactic:
simpa using

Goal state after:
a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ Tendsto (fun a_1 ↦ a_1 ^ (a + 1)) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)

Tactic:
have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
    intro x hx
    convert! (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
    simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
  have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1))) := by
    apply Tendsto.div_const
    simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))
  convert! integral_Ioi_of_hasDerivAt_of_tendsto' hd (integrableOn_Ioi_rpow_of_lt ha hc) ht using 1
  simp only [neg_div, zero_div, zero_sub]

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
| μ.restrict (Iic 0)

Tactic:
← Measure.map_neg_eq_self μ,

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
| (Measure.map Neg.neg μ).restrict (Iic 0)
Module: Mathlib.MeasureTheory.Integral.ExpDecay

Goal state before:
pp: a b : ℝ
h : 0 < b
this : Tendsto (fun x ↦ -rexp (-b * x) / b) atTop (𝓝 (-0 / b))
x : ℝ
x✝ : x ∈ Ici a
⊢ HasDerivAt (fun x ↦ -rexp (-b * x) / b) (rexp (-b * x)) x

Tactic:
simpa using

Goal state after:
a b : ℝ
h : 0 < b
this : Tendsto (fun x ↦ -rexp (-b * x) / b) atTop (𝓝 (-0 / b))
x : ℝ
x✝ : x ∈ Ici a
⊢ HasDerivAt (fun x ↦ -rexp (-(b * x)) / b) (rexp (-(b * x))) x
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
hf : MemLp f 1 volume
x : V
⊢ (fourierTransformInv (MemLp.toLp f hf)) x = 𝓕⁻ f x

Tactic:
simpa using

Goal state after:
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
hf : MemLp f 1 volume
x : V
⊢ 𝓕⁻ (↑↑(MemLp.toLp f hf)) x = 𝓕⁻ f x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
⊢ ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral

Tactic:
rw [GammaIntegral, GammaIntegral, ← integral_conj]

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

Goal state before:
pp: ⊢ GammaIntegral 1 = 1

Tactic:
simpa using

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

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

Tactic:
simpa using

Goal state after:
s b p : ℝ
hp : 1 < p
hb : 0 < b
⊢ (fun x ↦ x ^ s * rexp (-x)) =o[atTop] fun x ↦ rexp (x * -(1 / 2))
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 : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume
hn : ↑n ≤ N
x : ℝ
A : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume
⊢ AEStronglyMeasurable f volume

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n : ℕ
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume
hn : ↑n ≤ N
x : ℝ
A : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume
⊢ AEStronglyMeasurable f volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
this : MeasurableSet (Ioi 0)
x : ℝ
hx : x ∈ Ioi 0
h'x : 0 ≤ x
⊢ |(-x) ^ s| ≤ x ^ s

Tactic:
simpa using

Goal state after:
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
this : MeasurableSet (Ioi 0)
x : ℝ
hx : x ∈ Ioi 0
h'x : 0 ≤ x
⊢ |(-x) ^ s| ≤ x ^ s