mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
x : M₁
hx : x ∈ closedBall 0 1
t : M₂
ht : t ∈ T
htx : x ∈ ⇑v ⁻¹' (t +ᵥ V)
⊢ x ∈ ⇑u ⁻¹' (t +ᵥ U)

Tactic:
mem_preimage,

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
x : M₁
hx : x ∈ closedBall 0 1
t : M₂
ht : t ∈ T
htx : v x ∈ t +ᵥ V
⊢ u x ∈ t +ᵥ U
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike 𝕜
A M : Matrix n n 𝕜
hM : M.PosSemidef
x y : n → 𝕜
⊢ (starRingEnd 𝕜) (M *ᵥ x ⬝ᵥ star y) = M *ᵥ y ⬝ᵥ star x

Tactic:
by
    rw [dotProduct_comm, star_dotProduct, starRingEnd_apply, star_star,
      star_mulVec, dotProduct_comm (M *ᵥ y), dotProduct_mulVec, hM.isHermitian.eq]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ v ∈ eigenspace T μ

Tactic:
simp_rw [v, unsortedEigenvectorBasis]

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ (DirectSum.IsInternal.subordinateOrthonormalBasis hn ⋯ ⋯) i ∈ eigenspace T μ
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝³ : FiniteDimensional 𝕜 E
n : ℕ
E' : Type u_3
inst✝² : NormedAddCommGroup E'
inst✝¹ : InnerProductSpace 𝕜 E'
inst✝ : FiniteDimensional 𝕜 E'
T' : E' →ₗ[𝕜] E'
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
hT' : T'.IsSymmetric
hn' : finrank 𝕜 E' = n
h : hT.eigenvalues hn = hT'.eigenvalues hn'
⊢ T.charpoly = T'.charpoly

Tactic:
by rw [hT.charpoly_eq hn, hT'.charpoly_eq hn', h]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : SecondCountableTopologyEither α E
inst✝ : IsFiniteMeasure μ
f : α →ᵇ E
⊢ ‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖₊

Tactic:
by
  apply Lp.nnnorm_le_of_ae_bound
  refine (f.toContinuousMap.coeFn_toAEEqFun μ).mono ?_
  intro x hx
  rw [← NNReal.coe_le_coe, coe_nnnorm, coe_nnnorm]
  convert! f.norm_coe_le_norm x using 2

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
⊢ abs a ^ x = (star a * a) ^ (x / 2)

Tactic:
by
  simp only [← abs_nnrpow_two_mul, mul_div_left_comm, ne_eq, OfNat.ofNat_ne_zero,
    not_false_eq_true, div_self, mul_one]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
⊢ MemLp f 2 μ ↔ Integrable (fun x ↦ f x ^ 2) μ

Tactic:
convert! memLp_two_iff_integrable_sq_norm hf using 3
  simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Algebra ℝ A
inst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
U : ↥(unitary A)
⊢ abs ↑U = 1

Tactic:
by simp [abs]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
⊢ ∃ g, ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ MemLp (⇑g) (ENNReal.ofReal p) μ

Tactic:
rcases hf.exists_boundedContinuous_eLpNorm_sub_le ENNReal.coe_ne_top A with ⟨g, hg, g_mem⟩

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
g : α →ᵇ E
hg : eLpNorm (f - ⇑g) (↑p.toNNReal) μ ≤ ENNReal.ofReal (ε ^ (1 / p))
g_mem : MemLp (⇑g) (↑p.toNNReal) μ
⊢ ∃ g, ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ MemLp (⇑g) (ENNReal.ofReal p) μ
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
inst✝ : Fact (0 < T)
n : ℤ
⊢ ‖fourier n‖ = 1

Tactic:
rw [ContinuousMap.norm_eq_iSup_norm]

Goal state after:
T : ℝ
inst✝ : Fact (0 < T)
n : ℤ
⊢ ⨆ x, ‖(fourier n) x‖ = 1
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
x : E
s : Finset ι
i : ι
⊢ ↑(b.repr x) i • b i ∈ (span 𝕜 (Set.range ⇑b)).toAddSubmonoid

Tactic:
refine smul_mem _ _ ?_

Goal state after:
ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
x : E
s : Finset ι
i : ι
⊢ b i ∈ span 𝕜 (Set.range ⇑b)
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case e_a
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
n : ℤ
hab : a < b
this : Fact (0 < b - a)
⊢ ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x • liftIoc (b - a) a f ↑x = ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x • f x

Tactic:
intervalIntegral.integral_of_le hab.le

Goal state after:
case e_a
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
n : ℤ
hab : a < b
this : Fact (0 < b - a)
⊢ ∫ (x : ℝ) in Ioc a b, (fourier (-n)) ↑x • liftIoc (b - a) a f ↑x ∂volume =
    ∫ (x : ℝ) in Ioc a b, (fourier (-n)) ↑x • f x ∂volume
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
A : Tendsto (fun δ ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0) (𝓝 0)
this : Ioo 0 1 ∈ 𝓝[>] 0
⊢ ∃ δ, δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1

Tactic:
rcases (((tendsto_order.1 A).2 ε εpos).and this).exists with ⟨δ, hδ, h'δ⟩

Goal state after:
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
A : Tendsto (fun δ ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0) (𝓝 0)
this : Ioo 0 1 ∈ 𝓝[>] 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
h'δ : 0 < δ ∧ δ < 1
⊢ ∃ δ, δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
⊢ ContDiff ℝ ∞ (-⇑f)

Tactic:
exact (f.smooth _).neg

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
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
⊢ fderiv ℝ (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f) =
    fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f)

Tactic:
ext w : 1
  exact (hasFDerivAt_fourierIntegral L hf hf' w).fderiv

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

Goal state before:
pp: case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
m : ℕ
hm : m ≤ n
⊢ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m = ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)

Tactic:
show ‖L‖ ^ n = ‖L‖ ^ (m + (n - m)) by rw [Nat.add_sub_cancel' hm],

Goal state after:
case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
K : ℕ∞ω
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : (W →L[ℝ] ℝ) [×n]→L[ℝ] W [×n]→L[ℝ] ℝ :=
  (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x ↦ L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ :
  ∀ (m : ℕ),
    ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x ↦ L)) v‖ ≤
      ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
m : ℕ
hm : m ≤ n
⊢ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m =
    ↑(n.descFactorial m) * ‖L‖ ^ (m + (n - m)) * ‖v‖ ^ (n - m)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 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 : NontriviallyNormedField 𝕜
inst✝¹ : NormedAlgebra ℝ 𝕜
inst✝ : NormedSpace 𝕜 F
g₁ g₂ : E → 𝕜
hg₁ : Function.HasTemperateGrowth g₁
hg₂ : Function.HasTemperateGrowth g₂
⊢ smulLeftCLM F (g₁ + g₂) = smulLeftCLM F g₁ + smulLeftCLM F g₂

Tactic:
by
  ext f x
  simp [hg₁, hg₂, hg₁.add hg₂, add_smul]

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

Goal state before:
pp: case hab
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 := ⋯
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'
n : ℕ
x : α
hx : x ∈ s \ u
B : t' ^ n * μ.real (v ∩ s) ≤ ∫ (y : α) in s, c y ^ n ∂μ
this : 0 < (μ (v ∩ s)).toReal
⊢ c x ≤ t

Tactic:
· exact ht x hx

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

Goal state before:
pp: E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
hg : Integrable g μ
h'g : ContinuousAt g 0
I : Integrable φ μ
u : Set F
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
hu : 0 ∈ u
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
h'u : ball 0 δ ⊆ u
M : ℝ
Mpos : M > 0
hM : ∀ ⦃x : F⦄, x ∈ (closedBall 0 M)ᶜ → ‖x‖ ^ finrank ℝ F * φ x < δ ^ finrank ℝ F * ε
c : ℝ
hc : M / δ < c
x : F
hx : x ∈ uᶜ
cpos : 0 < c
hδx : δ ≤ ‖x‖
⊢ ‖x‖ ^ finrank ℝ F * (c ^ finrank ℝ F * φ (c • x)) = ‖c • x‖ ^ finrank ℝ F * φ (c • x)

Tactic:
simp [norm_smul, abs_of_pos cpos, mul_pow]

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

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

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ IntegrableOn (fun x ↦ s * ↑x ^ (s - 1)) (Ioi t) volume

Tactic:
refine (integrableOn_Ioi_cpow_of_lt ?_ ht).const_mul _

Goal state after:
s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ (s - 1).re < -1
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1) = ∫ (x : ℝ) in c..id x, ↑x ^ a

Tactic:
rw [integral_cpow, id]

Goal state after:
a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ -1 < a.re ∨ a ≠ -1 ∧ 0 ∉ uIcc c (id x)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
⊢ (↑s).GammaIntegral = ↑(∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1))

Tactic:
GammaIntegral

Goal state after:
s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1))
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: case left
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ 0 < x ^ 2 + y ^ 2

pp: case right
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0

Tactic:
· rcases hxy with hxy | hxy
      · dsimp at hxy; linarith [sq_pos_of_ne_zero hxy.ne', sq_nonneg y]
      · linarith [sq_nonneg x, sq_pos_of_ne_zero hxy]

Goal state after:
case right
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
E : Fin n → Type u_3
mE : (i : Fin n) → MeasurableSpace (E i)
μ : (i : Fin n) → Measure (E i)
inst✝ : ∀ (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

Tactic:
by
  induction n with
  | zero => simp [measureReal_def]
  | succ n n_ih =>
      calc
        _ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
            f 0 x.1 * ∏ i : Fin n, f (Fin.succ i) (x.2 i)
            ∂((μ 0).prod (Measure.pi (fun i ↦ μ i.succ))) := by
          rw [← ((measurePreserving_piFinSuccAbove μ 0).symm).integral_comp']
          simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
            Fin.prod_univ_succ, Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.cons_succ,
            Fin.zero_succAbove, cast_eq, Fin.cons_zero]
        _ = (∫ x, f 0 x ∂μ 0)
            * ∏ i : Fin n, ∫ (x : E (Fin.succ i)), f (Fin.succ i) x ∂(μ i.succ) := by
          rw [← n_ih, ← integral_prod_mul]
        _ = ∏ i, ∫ x, f i x ∂(μ i) := by rw [Fin.prod_univ_succ]

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

Goal state before:
pp: b : ℂ
c T : ℝ
⊢ ‖cexp (-b * (↑T + ↑c * I) ^ 2)‖ = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))

Tactic:
rw [Complex.norm_exp, neg_mul, neg_re, ← re_add_im b]
  simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im]
  ring_nf

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

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : (ι → ℂ) → E
⊢ (∫ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ Complex.polarCoord.target,
      (∏ i, (p i).1) • f fun i ↦ ↑Complex.polarCoord.symm (p i)) =
    ∫ (p : ι → ℂ), f p

Tactic:
by
  let e := MeasurableEquiv.piCongrRight (fun _ : ι ↦ measurableEquivRealProd.symm)
  have := volume_preserving_pi (fun _ : ι ↦ Complex.volume_preserving_equiv_real_prod.symm)
  rw [← MeasurePreserving.integral_comp this e.measurableEmbedding f]
  exact integral_comp_pi_polarCoord_symm (f ∘ e)

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

Goal state before:
pp: case zero.hb
s : ℂ
h1 : s = 0
⊢ s.re ≤ 0

Tactic:
rw [h1]

Goal state after:
case zero.hb
s : ℂ
h1 : s = 0
⊢ re 0 ≤ 0
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → E
hf : Integrable f volume
⊢ Tendsto (fun c ↦ ∫ (v : V), cexp (-↑c⁻¹ * ↑‖v‖ ^ 2) • f v) atTop (𝓝 (∫ (v : V), f v))

Tactic:
by
  apply tendsto_integral_filter_of_dominated_convergence _ _ _ hf.norm
  · filter_upwards with v
    nth_rewrite 2 [show f v = cexp (- (0 : ℝ) * ‖v‖ ^ 2) • f v by simp]
    apply (Tendsto.cexp _).smul_const
    exact tendsto_inv_atTop_zero.ofReal.neg.mul_const _
  · filter_upwards with c using
      AEStronglyMeasurable.smul (Continuous.aestronglyMeasurable (by fun_prop)) hf.1
  · filter_upwards [Ici_mem_atTop (0 : ℝ)] with c (hc : 0 ≤ c)
    filter_upwards with v
    simp only [ofReal_inv, neg_mul, norm_smul]
    norm_cast
    conv_rhs => rw [← one_mul (‖f v‖)]
    gcongr
    simp only [norm_eq_abs, abs_exp, exp_le_one_iff, Left.neg_nonpos_iff]
    positivity

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

Goal state before:
pp: b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
⊢ Integrable (fun v ↦ cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫)) volume

Tactic:
rw [← (PiLp.volume_preserving_toLp ι).integrable_comp_emb
    (MeasurableEquiv.toLp 2 _).measurableEmbedding]

Goal state after:
b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
⊢ Integrable ((fun v ↦ cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫)) ∘ WithLp.toLp 2) volume