mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
hp : p ≠ ∞
hp₂ : 1 ≤ p
f : E → F
hf : MemLp f p μ
ε : ℝ
hε : 0 < ε
hε₂ : 0 < ε / 2
hε₂' : 0 < ENNReal.ofReal (ε / 2)
g : E → F
hg₁ : HasCompactSupport g
hg₂ : eLpNorm (f - g) p μ ≤ ENNReal.ofReal (ε / 2)
hg₃ : Continuous g
hg₄ : MemLp g p μ
g' : E → F
hg'₁ : HasCompactSupport g'
hg'₂ : ContDiff ℝ ∞ g'
hg'₃ : eLpNorm (g - g') ?m.157 μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε

Tactic:
refine ⟨g', hg'₁, hg'₂, ?_⟩

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
hp : p ≠ ∞
hp₂ : 1 ≤ p
f : E → F
hf : MemLp f p μ
ε : ℝ
hε : 0 < ε
hε₂ : 0 < ε / 2
hε₂' : 0 < ENNReal.ofReal (ε / 2)
g : E → F
hg₁ : HasCompactSupport g
hg₂ : eLpNorm (f - g) p μ ≤ ENNReal.ofReal (ε / 2)
hg₃ : Continuous g
hg₄ : MemLp g p μ
g' : E → F
hg'₁ : HasCompactSupport g'
hg'₂ : ContDiff ℝ ∞ g'
hg'₃ : eLpNorm (g - g') ?m.157 μ ≤ ENNReal.ofReal (ε / 2)
⊢ eLpNorm (f - g') p μ ≤ ENNReal.ofReal ε
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: case a
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : GeneralSchauderBasis β 𝕜 X L
A : Finset β
⊢ Submodule.span 𝕜 (↑b '' ↑A) ≤ (↑(b.proj A)).range

Tactic:
Submodule.span_le

Goal state after:
case a
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : GeneralSchauderBasis β 𝕜 X L
A : Finset β
⊢ ↑b '' ↑A ⊆ ↑(↑(b.proj A)).range
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case e'_5
a b : ℝ
f : ℝ → ℂ
hab : a < b
hL2 : MemLp f 2 (volume.restrict (Ioc a (a + (b - a))))
this : Fact (0 < b - a)
h : MemLp (liftIoc (b - a) a f) 2 haarAddCircle
x✝ : ℤ
⊢ fourierCoeffOn hab f x✝ = fourierCoeff (↑↑(MemLp.toLp (liftIoc (b - a) a f) h)) x✝

Tactic:
simp [fourierCoeff_congr_ae h.coeFn_toLp, fourierCoeff_liftIoc_eq]

Goal state after:
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
inst✝² : SecondCountableTopologyEither V (W →L[ℝ] ℝ)
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
μ : Measure V
hf : AEStronglyMeasurable f μ
aux0 : Continuous[instTopologicalSpaceProd, _] fun p ↦ p.1.smulRight p.2
⊢ AEStronglyMeasurable (fun i ↦ (L i).smulRight (f i)) μ

Tactic:
have aux1 : AEStronglyMeasurable (fun v ↦ (L v, f v)) μ :=
    L.continuous.aestronglyMeasurable.prodMk hf

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
inst✝² : SecondCountableTopologyEither V (W →L[ℝ] ℝ)
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
μ : Measure V
hf : AEStronglyMeasurable f μ
aux0 : Continuous[instTopologicalSpaceProd, _] fun p ↦ p.1.smulRight p.2
aux1 : AEStronglyMeasurable (fun v ↦ (L v, f v)) μ
⊢ AEStronglyMeasurable (fun i ↦ (L i).smulRight (f i)) μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: case h₁
𝕜 : Type u_2
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 : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
m : ℕ × ℕ
k n : ℕ
hk : k ≤ m.1
hn : n ≤ m.2
f : 𝓢(E, F)
x : E
hk' : Finset.range (k + 1) ⊆ Finset.range (m.1 + 1)
i : ℕ
hi : i ∈ Finset.range (m.1 + 1)
⊢ SchwartzMap.seminorm 𝕜 i n ≤ (Finset.Iic m).sup fun m ↦ SchwartzMap.seminorm 𝕜 m.1 m.2

Tactic:
exact Finset.le_sup_of_le (Finset.mem_Iic.2 <|
    Prod.mk_le_mk.2 ⟨Finset.mem_range_succ_iff.mp hi, hn⟩) le_rfl

Goal state after:
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
I₄ : Integrable (fun v ↦ fourierSMulRight L (fun v ↦ fourierPowSMulRight L f v n) v) μ
w' : W
m : Fin n → W
⊢ fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L)
      (fun x ↦ (fourierPowSMulRight L f x (n + 1)) (Fin.cons w' m)) w =
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L)
      (fun x ↦ ((fourierSMulRight L (fun v ↦ fourierPowSMulRight L f v n) x) w') m) w

Tactic:
congr with v

Goal state after:
case e_f
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
I₄ : Integrable (fun v ↦ fourierSMulRight L (fun v ↦ fourierPowSMulRight L f v n) v) μ
w' : W
m : Fin n → W
v : V
⊢ (fourierPowSMulRight L f v (n + 1)) (Fin.cons w' m) =
    ((fourierSMulRight L (fun v ↦ fourierPowSMulRight L f v n) v) w') m
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: case hf.hf
𝕜 : Type u_1
inst✝²² : CommRing 𝕜
V : Type u_2
inst✝²¹ : AddCommGroup V
inst✝²⁰ : Module 𝕜 V
inst✝¹⁹ : MeasurableSpace V
W : Type u_3
inst✝¹⁸ : AddCommGroup W
inst✝¹⁷ : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℂ E
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℂ F
inst✝¹² : NormedAddCommGroup G
inst✝¹¹ : NormedSpace ℂ G
inst✝¹⁰ : TopologicalSpace 𝕜
inst✝⁹ : IsTopologicalRing 𝕜
inst✝⁸ : TopologicalSpace V
inst✝⁷ : BorelSpace V
inst✝⁶ : TopologicalSpace W
inst✝⁵ : MeasurableSpace W
inst✝⁴ : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst✝³ : SigmaFinite μ
inst✝² : SigmaFinite ν
inst✝¹ : SecondCountableTopologyEither W V
σ : ℂ →+* ℂ
inst✝ : RingHomIsometric σ
f : V → E
g : W → F
M : F →L[ℂ] E →SL[σ] G
he : Continuous[inst✝¹⁰, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝¹⁰] fun p ↦ (L p.1) p.2
hf : Integrable f μ
hg : Integrable g ν
this : Integrable (fun p ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ)
⊢ AEStronglyMeasurable (fun p ↦ (M (g p.1)) (e (-(L p.2) p.1) • f p.2)) (ν.prod μ)

Tactic:
have A : AEStronglyMeasurable (fun (p : W × V) ↦ e (-L p.2 p.1) • f p.2) (ν.prod μ) := by
      refine (Continuous.aestronglyMeasurable ?_).fun_smul hf.1.comp_snd
      exact he.comp (hL.comp continuous_swap).neg

Goal state after:
case hf.hf
𝕜 : Type u_1
inst✝²² : CommRing 𝕜
V : Type u_2
inst✝²¹ : AddCommGroup V
inst✝²⁰ : Module 𝕜 V
inst✝¹⁹ : MeasurableSpace V
W : Type u_3
inst✝¹⁸ : AddCommGroup W
inst✝¹⁷ : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℂ E
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℂ F
inst✝¹² : NormedAddCommGroup G
inst✝¹¹ : NormedSpace ℂ G
inst✝¹⁰ : TopologicalSpace 𝕜
inst✝⁹ : IsTopologicalRing 𝕜
inst✝⁸ : TopologicalSpace V
inst✝⁷ : BorelSpace V
inst✝⁶ : TopologicalSpace W
inst✝⁵ : MeasurableSpace W
inst✝⁴ : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst✝³ : SigmaFinite μ
inst✝² : SigmaFinite ν
inst✝¹ : SecondCountableTopologyEither W V
σ : ℂ →+* ℂ
inst✝ : RingHomIsometric σ
f : V → E
g : W → F
M : F →L[ℂ] E →SL[σ] G
he : Continuous[inst✝¹⁰, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝¹⁰] fun p ↦ (L p.1) p.2
hf : Integrable f μ
hg : Integrable g ν
this : Integrable (fun p ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ)
A : AEStronglyMeasurable (fun p ↦ e (-(L p.2) p.1) • f p.2) (ν.prod μ)
⊢ AEStronglyMeasurable (fun p ↦ (M (g p.1)) (e (-(L p.2) p.1) • f p.2)) (ν.prod μ)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
x : ℝ
hx : x ∈ Ioi 0
⊢ rexp (-x) * x ^ (s.re - 1) = rexp (-x) * ‖↑x ^ (s - 1)‖

Tactic:
norm_cpow_eq_rpow_re_of_pos hx _

Goal state after:
case right
s : ℂ
hs : 0 < s.re
x : ℝ
hx : x ∈ Ioi 0
⊢ rexp (-x) * x ^ (s.re - 1) = rexp (-x) * x ^ (s - 1).re
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
rw [isLittleO_exp_comp_exp_comp]
  suffices Tendsto (fun x => x * (b * x ^ (p - 1) + -1)) atTop atTop by
    refine Tendsto.congr' ?_ this
    refine eventuallyEq_of_mem (Ioi_mem_atTop (0 : ℝ)) (fun x hx => ?_)
    rw [mem_Ioi] at hx
    rw [rpow_sub_one hx.ne']
    field
  apply tendsto_id.atTop_mul_atTop₀
  refine tendsto_atTop_add_const_right atTop (-1 : ℝ) ?_
  exact Tendsto.const_mul_atTop hb (tendsto_rpow_atTop (by linarith))

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))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ ContinuousAt (fun x ↦ ↑x ^ (s - 1)) x

Tactic:
refine (?_ : ContinuousAt (fun x : ℂ => x ^ (s - 1)) _).comp continuous_ofReal.continuousAt

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))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ ContinuousAt (fun x ↦ x ^ (s - 1)) ↑x
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ⊢ polarCoord.source =ᵐ[volume] univ

Tactic:
have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_ofPred_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2

Goal state after:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
⊢ polarCoord.source =ᵐ[volume] univ
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ), f p = ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑polarCoord.symm p)

Tactic:
calc
    _ = ∫⁻ p in polarCoord.symm '' polarCoord.target, f p := by
      rw [← setLIntegral_univ, setLIntegral_congr polarCoord_source_ae_eq_univ.symm,
        polarCoord.symm_image_target_eq_source]
    _ = ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal |p.1| • f (polarCoord.symm p) := by
      rw [lintegral_image_eq_lintegral_abs_det_fderiv_mul volume _
        (fun p _ ↦ (hasFDerivAt_polarCoord_symm p).hasFDerivWithinAt)]
      · simp_rw [det_fderivPolarCoordSymm]; rfl
      exacts [polarCoord.symm.injOn, measurableSet_Ioi.prod measurableSet_Ioo]
    _ = ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (polarCoord.symm p) := by
      refine setLIntegral_congr_fun polarCoord.open_target.measurableSet (fun x hx ↦ ?_)
      rw [abs_of_pos hx.1]

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

Goal state before:
pp: case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 0 ≤ n

pp: case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 1 - s.re ≤ 0

Tactic:
· lia

Goal state after:
case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 1 - s.re ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ι → ℂ
⊢ (↑π / b) ^ (↑Finset.univ.card * 2⁻¹) * cexp (∑ x, c x ^ 2 / (4 * b)) =
    (↑π / b) ^ (↑Finset.univ.card / 2) * cexp (∑ i, c i ^ 2 / (4 * b))

Tactic:
div_eq_mul_inv

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

Goal state before:
pp: d✝ : Type u_1
inst✝¹ : Fintype d✝
d : Type u_2
inst✝ : Fintype d
n : d → ℤ
⊢ mFourier n ∈ ↑(star (adjoin ℂ (range mFourier)))

Tactic:
refine subset_adjoin ⟨-n, ?_⟩

Goal state after:
d✝ : Type u_1
inst✝¹ : Fintype d✝
d : Type u_2
inst✝ : Fintype d
n : d → ℤ
⊢ mFourier (-n) = star (mFourier n)
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
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
v v' : V
⊢ v ≠ v' →
    ∃ f ∈ (fun f ↦ ⇑f) '' ↑(StarSubalgebra.map (toContinuousMapStarₐ ℂ) (charPoly he hL)).toSubalgebra, f v ≠ f v'

Tactic:
hvv'

Goal state after:
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
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
v v' : V
hvv' : v ≠ v'
⊢ ∃ f ∈ (fun f ↦ ⇑f) '' ↑(StarSubalgebra.map (toContinuousMapStarₐ ℂ) (charPoly he hL)).toSubalgebra, f v ≠ f v'
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
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
⊢ (StarSubalgebra.map (toContinuousMapStarₐ ℂ) (charPoly he hL)).SeparatesPoints

Tactic:
by
  intro v v' hvv'
  obtain ⟨w, hw⟩ : ∃ w, char he hL w v ≠ char he hL w v' := by
    contrapose! hvv'
    exact ext_of_char_eq he he' hL hL' hvv'
  use char he hL w
  simp only [StarSubalgebra.coe_toSubalgebra, StarSubalgebra.coe_map, Set.mem_image,
    SetLike.mem_coe, exists_exists_and_eq_and, ne_eq]
  exact ⟨⟨char he hL w, char_mem_charPoly w, rfl⟩, hw⟩

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) = 𝓕 (Lp.toTemperedDistribution f')
⊢ 𝓕 (Lp.toTemperedDistribution f') = Lp.toTemperedDistribution (𝓕 f')

Tactic:
Lp.fourier_toTemperedDistribution_eq f'

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' : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = 𝓕 (Lp.toTemperedDistribution f')
⊢ Lp.toTemperedDistribution (𝓕 f') = Lp.toTemperedDistribution (𝓕 f')
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: case left
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
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
⊢ AEStronglyMeasurable (fun x ↦ (1 + ‖x‖ ^ 2) ^ (-s / 2)) volume

Tactic:
have : (fun x : E ↦ (1 + ‖x‖ ^ 2) ^ (-s / 2)).HasTemperateGrowth := by
        fun_prop
      exact this.1.continuous.aestronglyMeasurable

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

Goal state before:
pp: E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : InnerProductSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace ℂ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace ℂ F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace ℂ F₃
inst✝² : CompleteSpace F₃
inst✝¹ : CompleteSpace F₁
inst✝ : CompleteSpace F₂
B : F₁ →L[ℂ] F₂ →L[ℂ] F₃
f : 𝓢(E, F₁)
g : 𝓢(E, F₂)
x : E
⊢ 𝓕⁻ (fun y ↦ 𝓕 (⇑f ⋆[B, volume] ⇑g) y) x = (⇑f ⋆[B, volume] ⇑g) x

Tactic:
by
    rw [Continuous.fourierInv_fourier_eq]
    · refine BddAbove.continuous_convolution_right_of_integrable B ?_ f.integrable g.continuous
      exact ⟨SchwartzMap.seminorm ℝ 0 0 g, fun x ⟨y, hy⟩ ↦ hy ▸ norm_le_seminorm ℝ g y⟩
    · exact f.integrable.integrable_convolution B g.integrable
    · have : Integrable (fun ξ ↦ B (𝓕 f ξ) (𝓕 g ξ)) volume := (pairing B (𝓕 f) (𝓕 g)).integrable
      convert! this
      rw [← fourier_convolution_apply B f g, fourier_convolution, pairing_apply_apply]

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 = (b * a).gcd m

Tactic:
apply Coprime.factor_eq_gcd_right cop am bn

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

Goal state before:
pp: l : ℕ
hl : Squarefree l
⊢ μ l ^ 2 = 1

Tactic:
by
  rw [moebius_apply_of_squarefree hl, ← pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : ArithmeticFunction R
hf : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∏ p ∈ n.primeFactors, (1 + ((↑μ).pmul f) p) = ∑ d ∈ n.divisors, ↑(μ d) * f d

Tactic:
(isMultiplicative_moebius.intCast.pmul hf).prodPrimeFactors_one_add_of_squarefree hn

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : ArithmeticFunction R
hf : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∑ d ∈ n.divisors, ((↑μ).pmul f) d = ∑ d ∈ n.divisors, ↑(μ d) * f d
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
k : ℕ
ζ : M
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
⊢ IsPrimitiveRoot (ζ ^ i) k

Tactic:
by_cases h0 : k = 0
  · subst k; simp_all only [pow_one, Nat.coprime_zero_right]
  rcases h.isUnit h0 with ⟨ζ, rfl⟩
  rw [← Units.val_pow_eq_pow_val]
  rw [coe_units_iff] at h ⊢
  refine
    { pow_eq_one := by rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]
      dvd_of_pow_eq_one := fun l hl ↦ h.dvd_of_pow_eq_one l ?_ }
  rw [← pow_one ζ, ← zpow_natCast ζ, ← hi.gcd_eq_one, Nat.gcd_eq_gcd_ab, zpow_add, mul_pow,
    ← zpow_natCast, ← zpow_mul, mul_right_comm]
  simp only [zpow_mul, hl, h.pow_eq_one, one_zpow, one_pow, one_mul, zpow_natCast]

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
⊢ IsCoprime (if i * 2 ≤ n then ↑i else ↑i - ↑n) ↑n

Tactic:
replace hin := Nat.isCoprime_iff_coprime.mpr hin
    split_ifs
    · exact hin
    · convert! hin.add_mul_left_left (-1) using 1
      rw [mul_neg_one, sub_eq_add_neg]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case h.inr
K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
k : ℕ
ihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)
ζ : K
h : IsPrimitiveRoot ζ k
hpos : k > 0
B : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K
⊢ cyclotomic' k K ∈ lifts (Int.castRingHom K)

Tactic:
have Bmo : B.Monic := by
    apply monic_prod_of_monic
    intro i _
    exact cyclotomic'.monic i K

Goal state after:
case h.inr
K : Type u_2
inst✝¹ : CommRing K
inst✝ : IsDomain K
k : ℕ
ihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)
ζ : K
h : IsPrimitiveRoot ζ k
hpos : k > 0
B : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K
Bmo : B.Monic
⊢ cyclotomic' k K ∈ lifts (Int.castRingHom K)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
q : R
n : ℕ
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
⊢ eval₂ f (f q) (cyclotomic n R) = f (eval q (cyclotomic n R))

Tactic:
eval₂_at_apply

Goal state after:
R : Type u_1
S : Type u_2
q : R
n : ℕ
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
⊢ f (eval q (cyclotomic n R)) = f (eval q (cyclotomic n R))
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
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B : ℝ≥0
hB_pos : 0 < B
hB_vol : volume A ≤ ↑B
δ : ℝ
hδ1 : δ > 0
w : V
hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖
hw_ne : w ≠ 0
hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)
this : ‖1 / 2‖ = 2⁻¹
hδ2 : ∀ ⦃a b : V⦄, ‖a - b‖ < δ → ‖f a - f b‖ < ε / ↑B
x : V
x✝ : x ∈ A
⊢ ‖x - (x + (1 / (2 * ‖w‖ ^ 2)) • w)‖ < δ

Tactic:
rw [sub_add_cancel_left, norm_neg, hw'_nm, ← div_div, div_lt_iff₀ (norm_pos_iff.mpr hw_ne), ←
      div_lt_iff₀' hδ1, div_div]

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
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B : ℝ≥0
hB_pos : 0 < B
hB_vol : volume A ≤ ↑B
δ : ℝ
hδ1 : δ > 0
w : V
hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖
hw_ne : w ≠ 0
hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)
this : ‖1 / 2‖ = 2⁻¹
hδ2 : ∀ ⦃a b : V⦄, ‖a - b‖ < δ → ‖f a - f b‖ < ε / ↑B
x : V
x✝ : x ∈ A
⊢ 1 / (2 * δ) < ‖w‖