mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →L[𝕜] E
x : E
⊢ (T + S).rayleighQuotient x = T.rayleighQuotient x + S.rayleighQuotient x

Tactic:
simp [rayleighQuotient, reApplyInnerSelf_apply, inner_add_left, add_div]

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ (Monotone (f ∘ id) ∧ ∀ (i j : Fin n), i < j → f (id i) = f (id j) → id i < id j) ↔ Monotone f

Tactic:
Function.comp_id

Goal state after:
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ (Monotone f ∧ ∀ (i j : Fin n), i < j → f (id i) = f (id j) → id i < id j) ↔ Monotone f
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
⊢ (star ↑hA.eigenvectorUnitary * ↑hA.eigenvectorUnitary) *ᵥ Pi.single j 1 = Pi.single j 1

Tactic:
Unitary.coe_star_mul_self,

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
j : n
⊢ 1 *ᵥ Pi.single j 1 = Pi.single j 1
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
⊢ A.PosSemidef ↔ 0 ≤ hA.eigenvalues

Tactic:
conv_lhs => rw [hA.spectral_theorem]
  simp [isUnit_coe.posSemidef_star_right_conjugate_iff, posSemidef_diagonal_iff, Pi.le_def]

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
f : ℝ → ℝ
hA' : IsSelfAdjoint A
⊢ cfc f A = hA.cfc f

Tactic:
have := cfcHom_eq_of_continuous_of_map_id hA' hA.cfcAux hA.isClosedEmbedding_cfcAux.continuous
    hA.cfcAux_id

Goal state after:
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
f : ℝ → ℝ
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ cfc f A = hA.cfc f
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 (⇑RCLike.re) T.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn (hT.eigenvalues hn)

Tactic:
hT.roots_charpoly_eq_eigenvalues,

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) (Multiset.map (RCLike.ofReal ∘ hT.eigenvalues ⋯) Finset.univ.val)).sort
      fun x1 x2 ↦ x1 ≥ x2) =
    List.ofFn (hT.eigenvalues hn)
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
ha : 0 ≤ a
⊢ sqrt (star a * a) = a

Tactic:
ha.star_eq,

Goal state after:
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
ha : 0 ≤ a
⊢ sqrt (a * a) = a
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
ha : IsSelfAdjoint a
| sqrt (star a * a)

Tactic:
ha.star_eq,

Goal state after:
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
ha : IsSelfAdjoint a
| sqrt (a * a)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module ℝ A
inst✝¹² : SMulCommClass ℝ A A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁹ : PartialOrder A
inst✝⁸ : StarOrderedRing A
inst✝⁷ : NonnegSpectrumClass ℝ A
inst✝⁶ : IsTopologicalRing A
inst✝⁵ : T2Space A
inst✝⁴ : StarModule ℝ A
R : Type u_3
inst✝³ : Semiring R
inst✝² : SMulWithZero R ℝ≥0
inst✝¹ : SMul R A
inst✝ : IsScalarTower R ℝ≥0 A
r : R
a : A
this : ∀ (r : ℝ≥0), abs (r • a) = r • abs a
⊢ abs (r • a) = r • abs a

Tactic:
by simpa using this (r • 1)

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹⁸ : RCLike 𝕜
inst✝¹⁷ : NonUnitalRing A
inst✝¹⁶ : TopologicalSpace A
inst✝¹⁵ : Module 𝕜 A
inst✝¹⁴ : StarRing A
inst✝¹³ : PartialOrder A
inst✝¹² : StarOrderedRing A
inst✝¹¹ : IsScalarTower 𝕜 A A
inst✝¹⁰ : SMulCommClass 𝕜 A A
inst✝⁹ : Module ℝ A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : IsScalarTower ℝ A A
inst✝⁶ : NonnegSpectrumClass ℝ A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : StarModule 𝕜 A
inst✝¹ : StarModule ℝ A
inst✝ : IsScalarTower ℝ 𝕜 A
r : 𝕜
a : A
⊢ abs (r • a) = abs (‖r‖ • a)

pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹⁸ : RCLike 𝕜
inst✝¹⁷ : NonUnitalRing A
inst✝¹⁶ : TopologicalSpace A
inst✝¹⁵ : Module 𝕜 A
inst✝¹⁴ : StarRing A
inst✝¹³ : PartialOrder A
inst✝¹² : StarOrderedRing A
inst✝¹¹ : IsScalarTower 𝕜 A A
inst✝¹⁰ : SMulCommClass 𝕜 A A
inst✝⁹ : Module ℝ A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : IsScalarTower ℝ A A
inst✝⁶ : NonnegSpectrumClass ℝ A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : StarModule 𝕜 A
inst✝¹ : StarModule ℝ A
inst✝ : IsScalarTower ℝ 𝕜 A
r : 𝕜
a : A
⊢ abs (‖r‖ • a) = ‖r‖ • abs a

Tactic:
· simp only [abs, mul_smul_comm, smul_mul_assoc, star_smul, ← smul_assoc,
      RCLike.real_smul_eq_coe_smul (K := 𝕜)]
    simp [-algebraMap_smul, ← smul_mul_assoc, ← mul_comm (starRingEnd _ _), RCLike.conj_mul, sq]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝¹⁸ : RCLike 𝕜
inst✝¹⁷ : NonUnitalRing A
inst✝¹⁶ : TopologicalSpace A
inst✝¹⁵ : Module 𝕜 A
inst✝¹⁴ : StarRing A
inst✝¹³ : PartialOrder A
inst✝¹² : StarOrderedRing A
inst✝¹¹ : IsScalarTower 𝕜 A A
inst✝¹⁰ : SMulCommClass 𝕜 A A
inst✝⁹ : Module ℝ A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : IsScalarTower ℝ A A
inst✝⁶ : NonnegSpectrumClass ℝ A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : StarModule 𝕜 A
inst✝¹ : StarModule ℝ A
inst✝ : IsScalarTower ℝ 𝕜 A
r : 𝕜
a : A
⊢ abs (‖r‖ • a) = ‖r‖ • abs a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 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
a : A
ha : p a
⊢ cfcₙ (fun x ↦ ↑‖x‖ * ↑‖x‖) a = star a * a

Tactic:
conv_rhs => rw [← cfcₙ_id' 𝕜 a, ← cfcₙ_star, ← cfcₙ_mul ..]

Goal state after:
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
a : A
ha : p a
⊢ cfcₙ (fun x ↦ ↑‖x‖ * ↑‖x‖) a = cfcₙ (fun x ↦ star x * x) a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : Ring A
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : StarRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Algebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' spectrum 𝕜 a)
⊢ cfc (fun x ↦ f ↑‖x‖) a = cfc f (abs a)

Tactic:
rw [abs_eq_cfcₙ_coe_norm 𝕜 a, cfcₙ_eq_cfc, ← cfc_comp' ..]

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

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ abs a = 0 ↔ a = 0

Tactic:
rw [CFC.abs, sqrt_eq_zero_iff _, CStarRing.star_mul_self_eq_zero_iff]

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
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ g,
    eLpNorm (f - g) p μ ≤ ε ∧
      Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ HasCompactSupport g

Tactic:
apply hf.induction_dense hp _ _ _ _ hε

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
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s →
      μ s < ∞ →
        ∀ {ε : ℝ≥0∞},
          ε ≠ 0 →
            ∃ g,
              eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧
                Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧
                  MemLp g p μ ∧ HasCompactSupport g

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
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ (f g : α → E),
    Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ MemLp f p μ ∧ HasCompactSupport f →
      Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ HasCompactSupport g →
        Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f + g) ∧
          MemLp (f + g) p μ ∧ HasCompactSupport (f + g)

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
p : ℝ≥0∞
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
hp : p ≠ ∞
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ (f : α → E),
    Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ MemLp f p μ ∧ HasCompactSupport f →
      AEStronglyMeasurable f μ
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
⊢ ∃ g,
    eLpNorm (f - g) p μ ≤ ε ∧
      Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g p μ ∧ IsBounded (range g)

Tactic:
apply hf.induction_dense hp _ _ _ _ hε

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
⊢ ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s →
      μ s < ∞ →
        ∀ {ε : ℝ≥0∞},
          ε ≠ 0 →
            ∃ g,
              eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧
                Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧
                  MemLp g p μ ∧ IsBounded (range g)

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))

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 : α → E),
    Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ MemLp f p μ ∧ IsBounded (range f) →
      AEStronglyMeasurable f μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: 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
⊢ ∫ (x : E), ∑ x_1, (L (f x)) ((∂_{(stdOrthonormalBasis ℝ E) x_1} (∂_{(stdOrthonormalBasis ℝ E) x_1} g)) x) ∂μ =
    ∫ (x : E), ∑ i, (L ((∂_{(stdOrthonormalBasis ℝ E) i} (∂_{(stdOrthonormalBasis ℝ E) i} f)) x)) (g x) ∂μ

Tactic:
MeasureTheory.integral_finsetSum,

Goal state after:
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, ∫ (a : E), (L (f a)) ((∂_{(stdOrthonormalBasis ℝ E) i} (∂_{(stdOrthonormalBasis ℝ E) i} g)) a) ∂μ =
    ∫ (x : E), ∑ i, (L ((∂_{(stdOrthonormalBasis ℝ E) i} (∂_{(stdOrthonormalBasis ℝ E) i} f)) x)) (g x) ∂μ

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
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))
⊢ ∃ g, ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ MemLp (⇑g) (ENNReal.ofReal p) μ

Tactic:
refine ⟨g, ?_, 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
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
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
f₁ f₂ g : ↥(lp G 2)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by
          simp only [inner_add_left, Pi.add_apply, coeFn_add]

Goal state after:
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
⊢ ∑ b_1 ∈ s, (fun i ↦ ↑(b.repr x) i • b i) b_1 ∈ ↑(span 𝕜 (Set.range ⇑b)).toAddSubmonoid

Tactic:
simp only [SetLike.mem_coe]

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
⊢ ∑ x_1 ∈ s, ↑(b.repr x) x_1 • b x_1 ∈ (span 𝕜 (Set.range ⇑b)).toAddSubmonoid
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
hx : x ∈ Ioc a b
⊢ (fourier (-n)) ↑x • liftIoc (b - a) a f ↑x = (fourier (-n)) ↑x • f x

Tactic:
liftIoc_coe_apply

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
hx : x ∈ Ioc a b
⊢ (fourier (-n)) ↑x • f x = (fourier (-n)) ↑x • f x

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
hx : x ∈ Ioc a b
⊢ x ∈ Ioc a (a + (b - a))
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
v : ι → E
hv : Orthonormal 𝕜 v
hsp : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure
⊢ ⇑(HilbertBasis.mk hv hsp) = v

Tactic:
apply funext <| Orthonormal.linearIsometryEquiv_symm_apply_single_one hv hsp

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

Goal state before:
pp: case e_a
T : ℝ
hT : Fact (0 < T)
a : ℝ
f : ℝ → ℂ
n : ℤ
x : ℝ
hx : x ∈ uIoc a (a + T)
⊢ (fourier (-n)) ↑x • liftIoc T a f ↑x = (fourier (-n)) ↑x • f x

Tactic:
liftIoc_coe_apply

Goal state after:
case e_a
T : ℝ
hT : Fact (0 < T)
a : ℝ
f : ℝ → ℂ
n : ℤ
x : ℝ
hx : x ∈ uIoc a (a + T)
⊢ (fourier (-n)) ↑x • f x = (fourier (-n)) ↑x • f x

case e_a
T : ℝ
hT : Fact (0 < T)
a : ℝ
f : ℝ → ℂ
n : ℤ
x : ℝ
hx : x ∈ uIoc a (a + T)
⊢ x ∈ Ioc a (a + T)
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
n : ℤ
hn : n ≠ 0
x : ℝ
⊢ HasDerivAt (fun y ↦ ↑T / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑y) ((fourier (-n)) ↑x) x

Tactic:
convert! (hasDerivAt_fourier_neg T n x).div_const (-2 * π * I * n / T) using 1
  · ext1 y; rw [div_div_eq_mul_div]; ring
  · simp [mul_div_cancel_left₀, hn, (Fact.out : 0 < T).ne', Real.pi_pos.ne']

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)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (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))

Tactic:
have s2 : (b : AddCircle (b - a)) = (a : AddCircle (b - a)) := by
    simpa using coe_add_period (b - a) a

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)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (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.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 : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
A : WithSeminorms (schwartzSeminormFamily ℝ E F)
⊢ WithSeminorms (schwartzSeminormFamily 𝕜 E F)

Tactic:
SeminormFamily.withSeminorms_iff_nhds_eq_iInf

Goal state after:
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
A : nhds 0 = ⨅ i, Filter.comap (⇑(schwartzSeminormFamily ℝ E F i)) (nhds 0)
⊢ nhds 0 = ⨅ i, Filter.comap (⇑(schwartzSeminormFamily 𝕜 E F i)) (nhds 0)