mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ((Multiset.map (fun x ↦ RCLike.re x) (Multiset.map (RCLike.ofReal ∘ hA.eigenvalues₀) Finset.univ.val)).sort
      fun x1 x2 ↦ x1 ≥ x2) =
    List.ofFn hA.eigenvalues₀

Tactic:
Fin.univ_val_map,

Goal state after:
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
⊢ ((Multiset.map (fun x ↦ RCLike.re x) ↑(List.ofFn (RCLike.ofReal ∘ hA.eigenvalues₀))).sort fun x1 x2 ↦ x1 ≥ x2) =
    List.ofFn hA.eigenvalues₀
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: case inl
n : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
inst✝ : Nontrivial (Matrix n n 𝕜)
a : Matrix n n 𝕜
ha : IsSelfAdjoint a
h : IsEmpty n
x y : Matrix n n 𝕜
hxy : x ≠ y
⊢ (spectrum ℝ a).Nonempty

Tactic:
exact False.elim <| Matrix.of.symm.injective.ne hxy <| Subsingleton.elim _ _

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
⊢ ((Multiset.map (fun x ↦ RCLike.re x) (Multiset.map (RCLike.ofReal ∘ hT.eigenvalues ⋯) Finset.univ.val)).sort
      fun x1 x2 ↦ x1 ≥ x2) =
    List.ofFn (hT.eigenvalues hn)

Tactic:
Fin.univ_val_map,

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
⊢ ((Multiset.map (fun x ↦ RCLike.re x) ↑(List.ofFn (RCLike.ofReal ∘ hT.eigenvalues ⋯))).sort fun x1 x2 ↦ x1 ≥ x2) =
    List.ofFn (hT.eigenvalues hn)
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
i : ℕ
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hi : k + ↑i ≤ n
hi' : ↑i ≤ ↑n
a✝ : E
x✝ : Fin i → E
⊢ ((ContDiffMapSupportedIn.of_support_subset ⋯ ⋯) a✝) x✝ =
    (((RingHom.id 𝕜) c • ContDiffMapSupportedIn.of_support_subset ⋯ ⋯) a✝) x✝

Tactic:
simp [iteratedFDeriv_const_smul_apply (f.contDiff.of_le hi').contDiffAt, FunLike.coe_smul]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
⊢ ⨅ i, comap (⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) (comap (eval i) PseudoMetricSpace.toUniformSpace) =
    ⨅ i, comap (⇑(structureMapLM ℝ n i)) inferInstance

Tactic:
← comap_comap

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
⊢ ⨅ i, comap (eval i ∘ ⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) PseudoMetricSpace.toUniformSpace =
    ⨅ i, comap (⇑(structureMapLM ℝ n i)) inferInstance
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : IsSelfAdjoint A
⊢ Isometry ⇑(cfcHom hA)

Tactic:
by
    rw [← isHermitian_iff_isSelfAdjoint] at hA
    rw [IsHermitian.cfcHom_eq_cfcAux hA, AddMonoidHomClass.isometry_iff_norm]
    intro f
    simp only [IsHermitian.cfcAux_apply, Unitary.conjStarAlgAut_apply, ← Unitary.coe_star,
      CStarRing.norm_mul_coe_unitary, CStarRing.norm_coe_unitary_mul, l2_opNorm_diagonal]
    rw [((algebraMap_isometry ℝ 𝕜).postcomp_pi).norm_map_of_map_zero (by ext; simp)]
    let : Fintype (spectrum ℝ A) := .ofFinite _
    rw [ContinuousMap.norm_eq_norm_coeFn]
    refine Function.Surjective.pi_norm_comp ?_ _
    rw [← Function.Surjective.of_comp_iff'
      (Equiv.setCongr hA.spectrum_real_eq_range_eigenvalues).bijective]
    exact Set.codRestrict_range_surjective hA.eigenvalues

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F' n K i) ((postcompLM T) f) ≤
    ‖T‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F n K i) f

Tactic:
set T' := T.restrictScalars ℝ

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F' n K i) ((postcompLM T) f) ≤
    ‖T‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F n K i) f
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫ ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal

Tactic:
simp_rw [inner_self_eq_norm_sq_to_K]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module 𝕜 A
inst✝¹² : StarRing A
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : StarOrderedRing A
inst✝⁹ : IsScalarTower 𝕜 A A
inst✝⁸ : SMulCommClass 𝕜 A A
inst✝⁷ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a)
hf0 : ¬f 0 = 0
⊢ cfcₙ (fun x ↦ f ↑‖x‖) a = cfcₙ f (abs a)

Tactic:
rw [cfcₙ_apply_of_not_map_zero _ hf0,
      cfcₙ_apply_of_not_map_zero _ (fun h ↦ (hf0 <| by simpa using h).elim)]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ∞
⊢ ⟪indicatorConstLp 2 hs hμs c, f⟫ = ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ

Tactic:
rw [inner_def, ← integral_indicator hs]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ∞
⊢ ∫ (a : α), ⟪↑↑(indicatorConstLp 2 hs hμs c) a, ↑↑f a⟫ ∂μ = ∫ (x : α), s.indicator (fun x ↦ ⟪c, ↑↑f x⟫) x ∂μ
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace ℝ E
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace ℝ F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : SMulCommClass ℝ 𝕜 F
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace ℝ F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k✝ : ℕ∞
K K₁ K₂ : Compacts E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
G : Type u_8
inst✝⁹ : NormedAlgebra ℝ 𝕜
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace 𝕜 F₁
inst✝⁶ : NormedSpace ℝ F₁
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
inst✝³ : NormedSpace ℝ F₂
inst✝² : NormedAddCommGroup F₃
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℝ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
g : E → F₂
hg : ContDiff ℝ (↑n) g
k : ℕ
hk : ↑k ≤ n
hcont :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
    (Finset.range (k + 1)).sup' ⋯ fun i ↦ ‖iteratedFDeriv ℝ i g x‖
C₀ : ℝ
hC₀ : ∀ x ∈ ↑K, ‖(Finset.range (k + 1)).sup' ⋯ fun i ↦ ‖iteratedFDeriv ℝ i g x‖‖ ≤ C₀
hgC₀ : ∀ i ≤ k, ∀ x ∈ K, ‖iteratedFDeriv ℝ i g x‖ ≤ ‖C₀‖
φ : 𝓓^{n}_{K}(E, F₁)
x : E
hx : x ∈ K
⊢ ‖B‖ *
      ∑ i ∈ Finset.range (k + 1),
        ↑(k.choose i) * ((Finset.Iic k).sup fun m ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K m) φ * ‖C₀‖ =
    ‖B‖ * 2 ^ k * ‖C₀‖ * ((Finset.Iic k).sup fun m ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K m) φ

Tactic:
simp_rw [← Finset.sum_mul, ← Nat.cast_sum, Nat.sum_range_choose]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace ℝ E
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace ℝ F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : SMulCommClass ℝ 𝕜 F
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace ℝ F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k✝ : ℕ∞
K K₁ K₂ : Compacts E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
G : Type u_8
inst✝⁹ : NormedAlgebra ℝ 𝕜
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace 𝕜 F₁
inst✝⁶ : NormedSpace ℝ F₁
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
inst✝³ : NormedSpace ℝ F₂
inst✝² : NormedAddCommGroup F₃
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℝ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
g : E → F₂
hg : ContDiff ℝ (↑n) g
k : ℕ
hk : ↑k ≤ n
hcont :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
    (Finset.range (k + 1)).sup' ⋯ fun i ↦ ‖iteratedFDeriv ℝ i g x‖
C₀ : ℝ
hC₀ : ∀ x ∈ ↑K, ‖(Finset.range (k + 1)).sup' ⋯ fun i ↦ ‖iteratedFDeriv ℝ i g x‖‖ ≤ C₀
hgC₀ : ∀ i ≤ k, ∀ x ∈ K, ‖iteratedFDeriv ℝ i g x‖ ≤ ‖C₀‖
φ : 𝓓^{n}_{K}(E, F₁)
x : E
hx : x ∈ K
⊢ ‖B‖ * (↑(2 ^ k) * ((Finset.Iic k).sup fun m ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K m) φ * ‖C₀‖) =
    ‖B‖ * 2 ^ k * ‖C₀‖ * ((Finset.Iic k).sup fun m ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K m) φ
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
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
⊢ ENNReal.ofReal p ≠ 0

Tactic:
simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp

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

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
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 → E
g_support : HasCompactSupport g
g_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g (↑p.toNNReal) μ
hg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε

Tactic:
(hf.sub g_mem).eLpNorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,

Goal state after:
α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
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 → E
g_support : HasCompactSupport g
g_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g (↑p.toNNReal) μ
hg :
  ENNReal.ofReal ((∫ (a : α), ‖(f - g) a‖ ^ (ENNReal.ofReal p).toReal ∂μ) ^ (ENNReal.ofReal p).toReal⁻¹) ≤
    ENNReal.ofReal (ε ^ (1 / p))
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: case hf
E : Type u_5
F₁ : Type u_9
F₂ : Type u_10
F₃ : Type u_11
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : FiniteDimensional ℝ E
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace ℝ F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace ℝ F₃
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : 𝓢(E, F₁)
g : 𝓢(E, F₂)
L : F₁ →L[ℝ] F₂ →L[ℝ] F₃
⊢ ∀ i ∈ Finset.univ,
    Integrable (fun x ↦ (L (f x)) ((∂_{(stdOrthonormalBasis ℝ E) i} (∂_{(stdOrthonormalBasis ℝ E) i} g)) x)) μ

Tactic:
exact fun _ _ ↦ (pairing L f (∂_{_} <| ∂_{_} g)).integrable

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
⊢ ENNReal.ofReal p ≠ 0

Tactic:
simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp

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 : α →ᵇ E
g_mem : MemLp (⇑g) (↑p.toNNReal) μ
hg : eLpNorm (f - ⇑g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε

Tactic:
(hf.sub g_mem).eLpNorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,

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
g_mem : MemLp (⇑g) (↑p.toNNReal) μ
hg :
  ENNReal.ofReal ((∫ (a : α), ‖(f - ⇑g) a‖ ^ (ENNReal.ofReal p).toReal ∂μ) ^ (ENNReal.ofReal p).toReal⁻¹) ≤
    ENNReal.ofReal (ε ^ (1 / p))
⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 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 : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
i j : ℕ
⊢ (fun x ↦ x • D.e i)
      (({ toFun := coeff i, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖succSub D.P i‖ / ‖D.e i‖) ⋯) (D.e j)) =
    (fun x ↦ x • D.e i) (Pi.single j 1 i)

Tactic:
obtain ⟨x, hx⟩ : ∃ x, (succSub D.P j) x = D.e j := D.e_mem_range j

Goal state after:
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 : SchauderBasis 𝕜 X
D : RankOneDecomposition 𝕜 X
coeff : ℕ → X → 𝕜 := D.basisCoeff
hcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coeff n x • D.e n
i j : ℕ
x : X
hx : (succSub D.P j) x = D.e j
⊢ (fun x ↦ x • D.e i)
      (({ toFun := coeff i, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖succSub D.P i‖ / ‖D.e i‖) ⋯) (D.e j)) =
    (fun x ↦ x • D.e i) (Pi.single j 1 i)
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: case inr
T : ℝ
hT : Fact (0 < T)
n m : ℤ
hmn : m ≠ n
⊢ inner ℂ (fourierLp 2 m) (fourierLp 2 n) = Pi.single n 1 m

Tactic:
rw [(orthonormal_fourier (hT := hT)).2 hmn]

Goal state after:
case inr
T : ℝ
hT : Fact (0 < T)
n m : ℤ
hmn : m ≠ n
⊢ 0 = Pi.single n 1 m
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w

Tactic:
have h4 : (∀ᵐ v ∂μ, ∀ (w' : W), w' ∈ Metric.ball w 1 → ‖F' w' v‖ ≤ B v) := by
    filter_upwards with v w' _
    rw [Circle.norm_smul _ (fourierSMulRight L f v)]
    exact norm_fourierSMulRight_le L f v

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
h4 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, ‖F' w' v‖ ≤ B v
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
h4 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, ‖F' w' v‖ ≤ B v
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w

Tactic:
have h5 : Integrable B μ := by simpa only [← mul_assoc] using hf'.const_mul (2 * π * ‖L‖)

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
h4 : ∀ᵐ (v : V) ∂μ, ∀ w' ∈ Metric.ball w 1, ‖F' w' v‖ ≤ B v
h5 : Integrable B μ
⊢ HasFDerivAt (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
    (fourierIntegral 𝐞 μ (ContinuousLinearMap.toLinearMap₁₂ L) (fourierSMulRight L f) w) w
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 := ⋯
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
⊢ IntegrableOn (fun x ↦ c x ^ n) (v ∩ s) μ

Tactic:
exact (I n).mono inter_subset_right le_rfl

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : CommRing 𝕜
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module 𝕜 V
inst✝⁶ : MeasurableSpace V
W : Type u_3
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module 𝕜 W
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : MeasurableAdd V
e : AddChar 𝕜 𝕊
μ : Measure V
inst✝ : μ.IsAddRightInvariant
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
f : V → E
v₀ : V
⊢ fourierIntegral e μ L (f ∘ fun v ↦ v + v₀) = fun w ↦ e ((L v₀) w) • fourierIntegral e μ L f w

Tactic:
by
  ext1 w
  dsimp only [fourierIntegral, Function.comp_apply, Circle.smul_def]
  conv in L _ => rw [← add_sub_cancel_right v v₀]
  rw [integral_add_right_eq_self fun v : V ↦ (e (-L (v - v₀) w) : ℂ) • f v, ← integral_smul]
  congr 1 with v
  rw [← smul_assoc, smul_eq_mul, ← Circle.coe_mul, ← e.map_add_eq_mul, ← LinearMap.neg_apply,
    ← sub_eq_add_neg, ← LinearMap.sub_apply, map_sub, neg_sub]

Goal state after: