mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
F : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : ↥(Lp F 2 μ)
⊢ ENNReal.ofReal 2 = 2

Tactic:
by

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)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
⊢ 0 ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
F : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : ↥(Lp F 2 μ)
⊢ eLpNorm (fun x ↦ ‖↑↑f x‖ ^ 2) 1 μ < ∞

Tactic:
by

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
f g : ↥(Lp E 2 μ)
x : α
⊢ ‖↑↑f x‖ * ‖↑↑g x‖ ≤ 2 * ‖↑↑f x‖ * ‖↑↑g x‖

Tactic:
by

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
f g : ↥(Lp E 2 μ)
⊢ ∀ (x : α), ‖⟪↑↑f x, ↑↑g x⟫‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖

Tactic:
by

Goal state after:
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
⊢ abs a = cfcₙ (fun z ↦ ↑‖z‖) a

Tactic:
by

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
f g : ↥(Lp E 2 μ)
⊢ eLpNorm (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) 1 μ < ∞

Tactic:
by

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)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
hx : x ∈ K
⊢ ‖iteratedFDeriv ℝ i (⇑T' ∘ ⇑f) x‖ = ‖T'.compContinuousMultilinearMap (iteratedFDeriv ℝ i (⇑f) x)‖

Tactic:
by

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)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
hx : x ∈ K
⊢ ‖T'‖ * ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
by

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:
by

Goal state after:
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
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a)
hf0 : ¬f 0 = 0
h : f ↑‖0‖ = 0
⊢ f 0 = 0

Tactic:
by

Goal state after:
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
f : 𝕜 → 𝕜
a : A
ha : p a
hf : ContinuousOn f ((fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a)
⊢ cfcₙ (fun x ↦ f ↑‖x‖) a = cfcₙ f (abs a)

Tactic:
by

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
f : ↥(Lp E 2 μ)
x : α
⊢ 2 = ↑2

Tactic:
by

Goal state after:
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
⊢ quasispectrum 𝕜 (abs a) = (fun z ↦ ↑‖z‖) '' quasispectrum 𝕜 a

Tactic:
by

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
a : A
ha : IsSelfAdjoint a
⊢ abs a = cfc (fun x ↦ ‖x‖) a

Tactic:
by

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

Goal state before:
pp: case neg
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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
i : ℕ
f : 𝓓^{n₁}_{K₁}(E, F)
H : ¬(n₂ ≤ n₁ ∧ K₁ ≤ K₂)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F n₂ K₂ i) ((monoLM 𝕜) f) ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁ i) f

Tactic:
· simp [monoLM_eq_zero, H]

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

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
⊢ abs 1 = 1

Tactic:
by

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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
i : ℕ
f : 𝓓^{n₁}_{K₁}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F n₂ K₂ i) ((monoLM 𝕜) f) ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁ i) f

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
x : ℝ≥0
⊢ abs ((algebraMap ℝ≥0 A) x) = (algebraMap ℝ≥0 A) x

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℕ
⊢ abs ↑n = ↑n

Tactic:
by

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
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫ ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Algebra ℝ A
inst✝⁷ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : StarModule ℝ A
n : ℕ
inst✝ : n.AtLeastTwo
⊢ abs (OfNat.ofNat n) = OfNat.ofNat n

Tactic:
by

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
f : ↥(Lp E 2 μ)
⊢ ENNReal.toReal 2 = 2

Tactic:
by

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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
i : ℕ
h₁ : n₁ = n₂
h₂ : K₁ ≤ K₂
f : 𝓓^{n₁}_{K₁}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F n₂ K₂ i) ((monoLM 𝕜) f) = (ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁ i) f

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
n : ℤ
⊢ abs ↑n = ↑|n|

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹⁴ : RCLike 𝕜
inst✝¹³ : Ring A
inst✝¹² : TopologicalSpace A
inst✝¹¹ : StarRing A
inst✝¹⁰ : PartialOrder A
inst✝⁹ : StarOrderedRing A
inst✝⁸ : Algebra 𝕜 A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : NonnegSpectrumClass ℝ A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : StarModule 𝕜 A
inst✝¹ : StarModule ℝ A
inst✝ : IsScalarTower ℝ 𝕜 A
c : 𝕜
⊢ abs ((algebraMap 𝕜 A) c) = (algebraMap ℝ A) ‖c‖

Tactic:
by

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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
H : ¬(n₂ ≤ n₁ ∧ K₁ ≤ K₂)
⊢ monoCLM 𝕜 = 0

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ abs a ^ 2 = star a * a

Tactic:
by

Goal state after:
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
a : A
ha : p a
⊢ spectrum 𝕜 (abs a) = (fun z ↦ ↑‖z‖) '' spectrum 𝕜 a

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : CompleteSpace A
⊢ Continuous fun a ↦ star a * a

Tactic:
by

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
f : ↥(Lp E 2 μ)
⊢ ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : CompleteSpace A
⊢ ∀ (x : A), star x * x ∈ {a | 0 ≤ a}

Tactic:
by

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
f g : ↥(Lp E 2 μ)
⊢ AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯ ∈ Lp 𝕜 1 μ

Tactic:
by

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:
by

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
f f' g : ↥(Lp E 2 μ)
⊢ ⟪f + f', g⟫ = ⟪f, g⟫ + ⟪f', g⟫

Tactic:
by

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‖ = ‖a‖

Tactic:
by

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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E (E →L[ℝ] F) k K i) ((fderivLM 𝕜 n k) f) ≤
    (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
dist_f_f : ∀ (x : M), dist (f x) (f x) < ε x
t : M → Set F := fun x ↦ {y | dist y (f x) < ε x ∧ (x ∈ S → y = f x) ∧ (f x = 0 → y = 0)}
t_conv : ∀ (x : M), Convex ℝ (t x)
x : M
hx : x ∉ S
this : ∀ᶠ (y : M) in 𝓝 x, y ∉ S ∧ dist (f x) (f y) < ε y
⊢ ∀ᶠ (y : M) in 𝓝 x, (y ∉ S ∧ dist (f x) (f y) < ε y) ∧ (f y = 0 → f x = 0)

Tactic:
by

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
K : Compacts E
i : ℕ
f : 𝓓_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E (E →L[ℝ] F) ⊤ K i) ((fderivLM 𝕜 ⊤ ⊤) f) =
    (ContDiffMapSupportedIn.seminorm 𝕜 E F ⊤ K (i + 1)) f

Tactic:
by

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
f g : ↥(Lp E 2 μ)
r : 𝕜
x : α
hx : ↑↑(r • f) x = (r • ↑↑f) x
⊢ (fun a ↦ ⟪↑↑(r • f) a, ↑↑g a⟫) x = (fun a ↦ (starRingEnd 𝕜) r • ⟪↑↑f a, ↑↑g a⟫) x

Tactic:
simp only

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 g : ↥(Lp E 2 μ)
r : 𝕜
x : α
hx : ↑↑(r • f) x = (r • ↑↑f) x
⊢ ⟪↑↑(r • f) x, ↑↑g x⟫ = (starRingEnd 𝕜) r • ⟪↑↑f x, ↑↑g x⟫
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 g : ↥(Lp E 2 μ)
r : 𝕜
⊢ ⟪r • f, g⟫ = (starRingEnd 𝕜) r * ⟪f, g⟫

Tactic:
by

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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
f : 𝓓^{n}_{K}(E, F)
⊢ StronglyMeasurable ⇑f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
x✝¹ x✝ : ↥(Lp E 2 μ)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
μ_finite : IsFiniteMeasure (μ.restrict ↑K)
f : 𝓓^{n}_{K}(E, F)
⊢ Integrable (⇑f) μ

Tactic:
by

Goal state after: