mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.Holder

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p q r : ℝ≥0∞
hpqr : p.HolderTriple q r
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(Lp 𝕜 p μ)
g₁ g₂ : ↥(Lp E q μ)
⊢ ↑↑f • ↑↑(g₁ + g₂) =ᵐ[μ] ↑↑f • ↑↑g₁ + ↑↑f • ↑↑g₂

Tactic:
filter_upwards [AEEqFun.coeFn_add g₁.val g₂.val] with x hx

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p q r : ℝ≥0∞
hpqr : p.HolderTriple q r
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(Lp 𝕜 p μ)
g₁ g₂ : ↥(Lp E q μ)
x : α
hx : ↑(↑g₁ + ↑g₂) x = (↑↑g₁ + ↑↑g₂) x
⊢ (↑↑f • ↑↑(g₁ + g₂)) x = (↑↑f • ↑↑g₁ + ↑↑f • ↑↑g₂) x
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case right
𝕜 : Type u_2
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
n k1 : ℕ
h1 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k1
⊢ ∃ k, (iteratedFDeriv ℝ n fun x ↦ (B (f x)) (g x)) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
rcases hg.isBigO_uniform n with ⟨k2, h2⟩

Goal state after:
case right
𝕜 : Type u_2
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
n k1 : ℕ
h1 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k1
k2 : ℕ
h2 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k2
⊢ ∃ k, (iteratedFDeriv ℝ n fun x ↦ (B (f x)) (g x)) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖iteratedDerivWithin n (fun x ↦ x ^ r) t x‖ ≤
    (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k

Tactic:
iteratedDerivWithin_eq_iteratedDeriv hunique this hx,

Goal state after:
case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖iteratedDeriv n (fun x ↦ x ^ r) x‖ ≤
    (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
hx' : 1 / 2 < x
hx'' : 0 < x
h✝ : r - ↑n < 0
h : 0 < ↑n - r
⊢ 2 ^ (↑n - r) = Real.exp (Real.log 2 * (↑n - r))

Tactic:
by
      rw [Real.rpow_def_of_pos]
      norm_num

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
j : n
⊢ A *ᵥ (hA.eigenvectorBasis j).ofLp = hA.eigenvalues j • (hA.eigenvectorBasis j).ofLp

Tactic:
by
  simpa only [eigenvectorBasis, OrthonormalBasis.reindex_apply, toLpLin_apply,
    RCLike.real_smul_eq_coe_smul (K := 𝕜)] using!
      congr(⇑$((isSymmetric_toEuclideanLin_iff.mpr hA).apply_eigenvectorBasis
        finrank_euclideanSpace ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm j)))

Goal state after:
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ IsClosedEmbedding ⇑hA.cfcAux

Tactic:
have h0 : FiniteDimensional ℝ C(spectrum ℝ A, ℝ) :=
    FiniteDimensional.of_injective (ContinuousMap.coeFnLinearMap ℝ (M := ℝ)) DFunLike.coe_injective
  refine LinearMap.isClosedEmbedding_of_injective (𝕜 := ℝ) (E := C(spectrum ℝ A, ℝ))
    (F := Matrix n n 𝕜) (f := hA.cfcAux) <| LinearMap.ker_eq_bot'.mpr fun f hf ↦ ?_
  have h2 :
      diagonal (RCLike.ofReal ∘ f ∘ fun i ↦ ⟨hA.eigenvalues i, hA.eigenvalues_mem_spectrum_real i⟩)
        = (0 : Matrix n n 𝕜) := by
    simp only [LinearMap.coe_coe, cfcAux_apply, conjStarAlgAut_apply] at hf
    replace hf := congr($hf * (hA.eigenvectorUnitary : Matrix n n 𝕜))
    simp only [mul_assoc, SetLike.coe_mem, Unitary.star_mul_self_of_mem, mul_one, zero_mul] at hf
    simpa [← mul_assoc] using congr((star hA.eigenvectorUnitary : Matrix n n 𝕜) * $hf)
  ext x
  simp only [ContinuousMap.zero_apply]
  obtain ⟨x, hx⟩ := x
  obtain ⟨i, rfl⟩ := hA.spectrum_real_eq_range_eigenvalues ▸ hx
  rw [← diagonal_zero] at h2
  have := diagonal_eq_diagonal_iff.mp h2
  exact RCLike.ofReal_eq_zero.mp (this i)

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
μ : 𝕜
⊢ {i | ↑(hT.unsortedEigenvalues hn i) = μ}.card = finrank 𝕜 ↥(eigenspace T μ)

Tactic:
by_cases hμ : HasEigenvalue T μ

Goal state after:
case pos
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
μ : 𝕜
hμ : HasEigenvalue T μ
⊢ {i | ↑(hT.unsortedEigenvalues hn i) = μ}.card = finrank 𝕜 ↥(eigenspace T μ)

case neg
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
μ : 𝕜
hμ : ¬HasEigenvalue T μ
⊢ {i | ↑(hT.unsortedEigenvalues hn i) = μ}.card = finrank 𝕜 ↥(eigenspace T μ)
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
ι : Type u_3
A B : Matrix ι ι 𝕜
hA : A.PosSemidef
hB : B.PosSemidef
⊢ (A ⊙ B).PosSemidef

Tactic:
by
  classical
  refine ⟨hA.isHermitian.hadamard hB.isHermitian, fun x ↦ ?_⟩
  have hAB : ((A ⊙ B).submatrix (↑) (↑) : Matrix x.support _ _).PosSemidef := by
    have hAs := hA.submatrix ((↑) : x.support → ι)
    have hBs := hB.submatrix ((↑) : x.support → ι)
    rw [submatrix_hadamard, posSemidef_iff_dotProduct_mulVec]
    refine ⟨hAs.isHermitian.hadamard hBs.isHermitian, fun y ↦ ?_⟩
    rw [star_dotProduct_hadamard_mulVec_eq_kronecker]
    exact (hAs.kronecker hBs).dotProduct_mulVec_nonneg _
  simpa [Finsupp.sum, ← Finset.sum_attach x.support, ← Finset.subtype_mem_eq_attach,
    ← Finsupp.subtypeDomain_apply, ← Finsupp.support_subtypeDomain] using hAB.2 _

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
x : α
a✝ : ↑(ContinuousMap.toAEEqFun μ f.toContinuousMap) x = f.toContinuousMap x
⊢ ‖↑(ContinuousMap.toAEEqFun μ f.toContinuousMap) x‖ ≤ ‖f‖

Tactic:
convert! f.norm_coe_le_norm x using 2

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
x : α
hx : ↑(ContinuousMap.toAEEqFun μ f.toContinuousMap) x = f.toContinuousMap x
⊢ ‖↑↑⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩ x‖ ≤ ‖f‖

Tactic:
convert! f.norm_coe_le_norm x using 2

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_8
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 F
f : 𝓢(E, F)
⊢ (laplacianCLM 𝕜 E 𝓢(E, F)) f = Δ f

Tactic:
by
  simp [laplacianCLM, laplacian_eq_sum (stdOrthonormalBasis ℝ E)]

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
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ (f g : α → E),
    Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ MemLp f p μ ∧ IsBounded (range f) →
      Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ IsBounded (range g) →
        Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f + g) ∧
          MemLp (f + g) p μ ∧ IsBounded (range (f + g))

Tactic:
rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩

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
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ∞
f✝ : α → E
hf : MemLp f✝ p μ
ε : ℝ≥0∞
hε : ε ≠ 0
f g : α → E
f_cont : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
f_mem : MemLp f p μ
f_bd : IsBounded (range f)
g_cont : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g p μ
g_bd : IsBounded (range g)
⊢ Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f + g) ∧
    MemLp (f + g) p μ ∧ IsBounded (range (f + g))
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
x : AddCircle T
⊢ (fourier (-n)) x = (starRingEnd ℂ) ((fourier n) x)

Tactic:
by
  induction x using QuotientAddGroup.induction_on
  simp_rw [fourier_apply, toCircle]
  rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_zsmul]
  simp_rw [Function.Periodic.lift_coe, ← Circle.coe_inv_eq_conj, ← Circle.exp_neg,
    neg_smul, mul_neg]

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

Goal state before:
pp: case refine_2
T : ℝ
x : C(AddCircle T, ℂ)
hx : x ∈ ↑(Submonoid.closure (range fourier))
m n : ℤ
⊢ fourier m * fourier n ∈ range fourier

Tactic:
refine ⟨m + n, ?_⟩

Goal state after:
case refine_2
T : ℝ
x : C(AddCircle T, ℂ)
hx : x ∈ ↑(Submonoid.closure (range fourier))
m n : ℤ
⊢ fourier (m + n) = fourier m * fourier n
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
inst✝ : DecidableEq ι
hV : IsHilbertSum 𝕜 G V
i : ι
x : G i
⊢ hV.linearIsometryEquiv.symm (lp.single 2 i x) = (V i) x

Tactic:
by
  simp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply_single]

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

Goal state before:
pp: case insert
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ι : Type u_2
f : ι → AddCircle T → E
a : ι
s : Finset ι
ha : a ∉ s
iha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)
hf : ∀ i ∈ insert a s, Integrable (f i) haarAddCircle
hf₁ : Integrable (f a) haarAddCircle
hf₂ : ∀ a ∈ s, Integrable (f a) haarAddCircle
⊢ fourierCoeff (∑ i ∈ insert a s, f i) = ∑ i ∈ insert a s, fourierCoeff (f i)

Tactic:
rw [s.sum_insert ha, s.sum_insert ha,
        fourierCoeff.add hf₁ (integrable_finsetSum' s hf₂), iha hf₂]

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

Goal state before:
pp: a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =
    1 / (-2 * ↑π * I * ↑n) *
      ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f' x))

Tactic:
conv => pattern (occs := 1 2 3) fourier _ _ * _ <;> (rw [mul_comm])

Goal state after:
a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: case refine_1
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
⊢ IntegrableOn (fun x ↦ ‖φ i x • g x‖) (s ∩ u) μ

Tactic:
exact IntegrableOn.mono_set h''i.norm inter_subset_left

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

Goal state before:
pp: α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
⊢ Tendsto (fun n ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) atTop (𝓝 (g x₀))

Tactic:
have J : ∀ n, 0 ≤ᵐ[μ.restrict s] fun x : α => c x ^ n := by
    intro n
    filter_upwards [ae_restrict_mem hs.measurableSet] with x hx
    exact pow_nonneg (hnc x hx) n

Goal state after:
α : Type u_1
E : Type u_2
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
x₀ : α
s : Set α
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen[inst✝⁶] u → x₀ ∈ u → 0 < μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s μ
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x ↦ c x ^ n) s μ
J : ∀ (n : ℕ), 0 ≤ᵐ[μ.restrict s] fun x ↦ c x ^ n
⊢ Tendsto (fun n ↦ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) atTop (𝓝 (g x₀))
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:
by simpa using hp.2

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

Goal state before:
pp: 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

Tactic:
simpa [Complex.norm_exp] using! this

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

Goal state before:
pp: a : ℝ
ha : 0 < a
c : ℝ
this : Integrable (fun a_1 ↦ ‖Complex.exp (↑a * ↑a_1)‖) (volume.restrict (Iic c))
⊢ IntegrableOn (fun x ↦ rexp (a * x)) (Iic c) volume

Tactic:
simpa [Complex.norm_exp] using! this

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

Goal state before:
pp: case e'_9
a c m : ℝ
ha : a < -1
hc : -m < c
x : ℝ
hx : x ∈ Ici c
⊢ (x + m) ^ a = 1 * (a + 1) * (id x + m) ^ (a + 1 - 1) / (a + 1)

Tactic:
simp [show a + 1 ≠ 0 by linarith]

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

Goal state before:
pp: s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
⊢ False

Tactic:
rcases le_or_gt s.re (-1) with hs | hs

Goal state after:
case inl
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : s.re ≤ -1
⊢ False

case inr
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : -1 < s.re
⊢ False
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
⊢ 𝓕 f w = ∫ (v : V), Complex.exp (↑(-2 * π * ⟪v, w⟫) * Complex.I) • f v

Tactic:
simp_rw [fourier_eq, Circle.smul_def, Real.fourierChar_apply, mul_neg, neg_mul]

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

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

Tactic:
exact Or.inr hxy

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this✝ : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
this : Tendsto (fun X ↦ -↑X ^ s * ↑(rexp (-X))) atTop (𝓝 0)
⊢ Tendsto (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) atTop (𝓝 (s * s.GammaIntegral))

Tactic:
by
    simpa using! Tendsto.add (Tendsto.const_mul s (tendsto_partialGamma hs)) this

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ Tendsto (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) atTop (𝓝 (s * s.GammaIntegral))

Tactic:
suffices Tendsto (fun X => -X ^ s * (-X).exp : ℝ → ℂ) atTop (𝓝 0) by
    simpa using! Tendsto.add (Tendsto.const_mul s (tendsto_partialGamma hs)) this

Goal state after:
s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ Tendsto (fun X ↦ -↑X ^ s * ↑(rexp (-X))) atTop (𝓝 0)
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:
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.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
⊢ Tendsto (fun c ↦ ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop
    (𝓝 (f v))

Tactic:
let φ : V → ℝ := fun w ↦ π ^ (finrank ℝ V / 2 : ℝ) * Real.exp (-π ^ 2 * ‖w‖ ^ 2)

Goal state after:
V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
v : V
h'f : ContinuousAt f v
φ : V → ℝ := fun w ↦ π ^ (↑(finrank ℝ V) / 2) * rexp (-π ^ 2 * ‖w‖ ^ 2)
⊢ Tendsto (fun c ↦ ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop
    (𝓝 (f v))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
t : ℂ
⊢ ∫ (x : ℝ), cexp (I * t * ↑x) * cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))

Tactic:
conv => enter [1, 2, x]; rw [← Complex.exp_add, add_comm, ← add_zero (-b * x ^ 2 + I * t * x)]

Goal state after:
b : ℂ
hb : 0 < b.re
t : ℂ
⊢ ∫ (x : ℝ), cexp (-b * ↑x ^ 2 + I * t * ↑x + 0) = (↑π / b) ^ (1 / 2) * cexp (-t ^ 2 / (4 * b))
Module: Mathlib.Topology.Algebra.Module.Spaces.PointwiseConvergenceCLM

Goal state before:
pp: ι : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_7
F : Type u_8
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : Module 𝕜₁ E
inst✝ : Module 𝕜₂ F
p : Filter ι
a : ι → E →SLₚₜ[σ] F
a₀ : E →SLₚₜ[σ] F
⊢ Filter.Tendsto a p (𝓝 a₀) ↔ ∀ (x : E), Filter.Tendsto (fun x_1 ↦ (a x_1) x) p (𝓝 (a₀ x))

Tactic:
by
  simp [(PointwiseConvergenceCLM.isEmbedding_coeFn σ E F).tendsto_nhds_iff, tendsto_pi_nhds]

Goal state after: