ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineSubspace.setOf_wSameSide_eq_image2
theorem setOf_wSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
    { y | s.WSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s)
Goal: {y | s.WSameSide x y} = Set.image2 (fun t q => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) ↑s
-- AffineSubspace.setOf_sSameSide_eq_image2
theorem setOf_sSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
    { y | s.SSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ioi 0) s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s)
Goal: {y | s.SSameSide x y} = Set.image2 (fun t q => t • (x -ᵥ p) +ᵥ q) (Set.Ioi 0) ↑s
-- AffineSubspace.setOf_wOppSide_eq_image2
theorem setOf_wOppSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
    { y | s.WOppSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Iic 0) s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s)
Goal: {y | s.WOppSide x y} = Set.image2 (fun t q => t • (x -ᵥ p) +ᵥ q) (Set.Iic 0) ↑s
-- AffineSubspace.setOf_sOppSide_eq_image2
theorem setOf_sOppSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
    { y | s.SOppSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Iio 0) s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s)
Goal: {y | s.SOppSide x y} = Set.image2 (fun t q => t • (x -ᵥ p) +ᵥ q) (Set.Iio 0) ↑s
-- AffineSubspace.wOppSide_pointReflection
theorem wOppSide_pointReflection {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) :
    s.WOppSide y (pointReflection R x y) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s)
Goal: s.WOppSide y ((AffineEquiv.pointReflection R x) y)
-- AffineSubspace.sOppSide_pointReflection
theorem sOppSide_pointReflection {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) :
    s.SOppSide y (pointReflection R x y) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s)
Goal: s.SOppSide y ((AffineEquiv.pointReflection R x) y)
-- AffineSubspace.isConnected_setOf_wSameSide
theorem isConnected_setOf_wSameSide {s : AffineSubspace ℝ P} (x : P) (h : (s : Set P).Nonempty) :
    IsConnected { y | s.WSameSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} (x : P) (h : (↑s).Nonempty)
Goal: IsConnected {y | s.WSameSide x y}
-- AffineSubspace.isPreconnected_setOf_wSameSide
theorem isPreconnected_setOf_wSameSide (s : AffineSubspace ℝ P) (x : P) :
    IsPreconnected { y | s.WSameSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P)
Goal: IsPreconnected {y | s.WSameSide x y}
-- AffineSubspace.isConnected_setOf_sSameSide
theorem isConnected_setOf_sSameSide {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s)
    (h : (s : Set P).Nonempty) : IsConnected { y | s.SSameSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s) (h : (↑s).Nonempty)
Goal: IsConnected {y | s.SSameSide x y}
-- AffineSubspace.isPreconnected_setOf_sSameSide
theorem isPreconnected_setOf_sSameSide (s : AffineSubspace ℝ P) (x : P) :
    IsPreconnected { y | s.SSameSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P)
Goal: IsPreconnected {y | s.SSameSide x y}
-- AffineSubspace.isConnected_setOf_wOppSide
theorem isConnected_setOf_wOppSide {s : AffineSubspace ℝ P} (x : P) (h : (s : Set P).Nonempty) :
    IsConnected { y | s.WOppSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} (x : P) (h : (↑s).Nonempty)
Goal: IsConnected {y | s.WOppSide x y}
-- AffineSubspace.isPreconnected_setOf_wOppSide
theorem isPreconnected_setOf_wOppSide (s : AffineSubspace ℝ P) (x : P) :
    IsPreconnected { y | s.WOppSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P)
Goal: IsPreconnected {y | s.WOppSide x y}
-- AffineSubspace.isConnected_setOf_sOppSide
theorem isConnected_setOf_sOppSide {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s)
    (h : (s : Set P).Nonempty) : IsConnected { y | s.SOppSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s) (h : (↑s).Nonempty)
Goal: IsConnected {y | s.SOppSide x y}
-- AffineSubspace.isPreconnected_setOf_sOppSide
theorem isPreconnected_setOf_sOppSide (s : AffineSubspace ℝ P) (x : P) :
    IsPreconnected { y | s.SOppSide x y } 
Arguments: {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P)
Goal: IsPreconnected {y | s.SOppSide x y}
-- cfcₙAux_id
lemma cfcₙAux_id : cfcₙAux hp₁ a ha (ContinuousMapZero.id rfl) = a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁]
Goal: (cfcₙAux ⋯ a ha) (ContinuousMapZero.id ⋯) = ↑a
-- closedEmbedding_cfcₙAux
lemma closedEmbedding_cfcₙAux : ClosedEmbedding (cfcₙAux hp₁ a ha) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁]
Goal: ClosedEmbedding ⇑(cfcₙAux ⋯ a ha)
-- spec_cfcₙAux
lemma spec_cfcₙAux (f : C(σₙ 𝕜 a, 𝕜)₀) : σ 𝕜 (cfcₙAux hp₁ a ha f) = Set.range f 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] (f : ContinuousMapZero (↑(quasispectrum 𝕜 a)) 𝕜)
Goal: spectrum 𝕜 ((cfcₙAux ⋯ a ha) f) = Set.range ⇑f
-- cfcₙAux_mem_range_inr
lemma cfcₙAux_mem_range_inr (f : C(σₙ 𝕜 a, 𝕜)₀) :
    cfcₙAux hp₁ a ha f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [CompleteSpace A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] (f : ContinuousMapZero (↑(quasispectrum 𝕜 a)) 𝕜)
Goal: (cfcₙAux ⋯ a ha) f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)
-- RCLike.nonUnitalContinuousFunctionalCalculus
theorem RCLike.nonUnitalContinuousFunctionalCalculus :
    NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) [ContinuousFunctionalCalculus 𝕜 p₁]
Goal: NonUnitalContinuousFunctionalCalculus 𝕜 p
-- isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts
lemma isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts {a : A} :
    IsSelfAdjoint a ↔ IsStarNormal a ∧ QuasispectrumRestricts a Complex.reCLM 
Arguments: {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] {a : A}
Goal: IsSelfAdjoint a ↔ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM
-- QuasispectrumRestricts.isSelfAdjoint
lemma QuasispectrumRestricts.isSelfAdjoint (a : A) (ha : QuasispectrumRestricts a Complex.reCLM)
    [IsStarNormal a] : IsSelfAdjoint a 
Arguments: {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] (a : A) (ha : QuasispectrumRestricts a ⇑Complex.reCLM) [IsStarNormal a]
Goal: IsSelfAdjoint a
-- isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts
lemma isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts {a : A} :
    IsSelfAdjoint a ↔ IsStarNormal a ∧ SpectrumRestricts a Complex.reCLM 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {a : A}
Goal: IsSelfAdjoint a ↔ IsStarNormal a ∧ SpectrumRestricts a ⇑Complex.reCLM
-- IsSelfAdjoint.spectrumRestricts
lemma IsSelfAdjoint.spectrumRestricts {a : A} (ha : IsSelfAdjoint a) :
    SpectrumRestricts a Complex.reCLM 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} (ha : IsSelfAdjoint a)
Goal: SpectrumRestricts a ⇑Complex.reCLM
-- SpectrumRestricts.isSelfAdjoint
lemma SpectrumRestricts.isSelfAdjoint (a : A) (ha : SpectrumRestricts a Complex.reCLM)
    [IsStarNormal a] : IsSelfAdjoint a 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] (a : A) (ha : SpectrumRestricts a ⇑Complex.reCLM) [IsStarNormal a]
Goal: IsSelfAdjoint a
-- CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts
lemma CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts {A : Type*} [NonUnitalRing A]
    [StarRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A ]
    [NonUnitalContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
    {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : QuasispectrumRestricts a ContinuousMap.realToNNReal) :
    ∃ x : A, IsSelfAdjoint x ∧ QuasispectrumRestricts x ContinuousMap.realToNNReal ∧ x * x = a 
Arguments: {A : Type u_1} [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal)
Goal: ∃ x, IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x * x = a
-- nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts
lemma nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts {a : A} :
    0 ≤ a ↔ IsSelfAdjoint a ∧ QuasispectrumRestricts a ContinuousMap.realToNNReal 
Arguments: {A : Type u_1} [NonUnitalRing A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A}
Goal: 0 ≤ a ↔ IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
-- CFC.exists_sqrt_of_isSelfAdjoint_of_spectrumRestricts
lemma CFC.exists_sqrt_of_isSelfAdjoint_of_spectrumRestricts {A : Type*} [Ring A] [StarRing A]
    [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
    {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : SpectrumRestricts a ContinuousMap.realToNNReal) :
    ∃ x : A, IsSelfAdjoint x ∧ SpectrumRestricts x ContinuousMap.realToNNReal ∧ x ^ 2 = a 
Arguments: {A : Type u_1} [Ring A] [StarRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal)
Goal: ∃ x, IsSelfAdjoint x ∧ SpectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x ^ 2 = a
-- nonneg_iff_isSelfAdjoint_and_spectrumRestricts
lemma nonneg_iff_isSelfAdjoint_and_spectrumRestricts {a : A} :
    0 ≤ a ↔ IsSelfAdjoint a ∧ SpectrumRestricts a ContinuousMap.realToNNReal 
Arguments: {A : Type u_1} [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A}
Goal: 0 ≤ a ↔ IsSelfAdjoint a ∧ SpectrumRestricts a ⇑ContinuousMap.realToNNReal
-- SpectrumRestricts.nnreal_iff_nnnorm
lemma SpectrumRestricts.nnreal_iff_nnnorm {a : A} {t : ℝ≥0} (ha : IsSelfAdjoint a) (ht : ‖a‖₊ ≤ t) :
    SpectrumRestricts a ContinuousMap.realToNNReal ↔ ‖algebraMap ℝ A t - a‖₊ ≤ t 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] {a : A} {t : NNReal} (ha : IsSelfAdjoint a) (ht : ‖a‖₊ ≤ t)
Goal: SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ‖(algebraMap ℝ A) ↑t - a‖₊ ≤ t
-- SpectrumRestricts.nnreal_add
lemma SpectrumRestricts.nnreal_add {a b : A} (ha₁ : IsSelfAdjoint a)
    (hb₁ : IsSelfAdjoint b) (ha₂ : SpectrumRestricts a ContinuousMap.realToNNReal)
    (hb₂ : SpectrumRestricts b ContinuousMap.realToNNReal) :
    SpectrumRestricts (a + b) ContinuousMap.realToNNReal 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] {a b : A} (ha₁ : IsSelfAdjoint a) (hb₁ : IsSelfAdjoint b) (ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal) (hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal)
Goal: SpectrumRestricts (a + b) ⇑ContinuousMap.realToNNReal
-- IsSelfAdjoint.sq_spectrumRestricts
lemma IsSelfAdjoint.sq_spectrumRestricts {a : A} (ha : IsSelfAdjoint a) :
    SpectrumRestricts (a ^ 2) ContinuousMap.realToNNReal 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a)
Goal: SpectrumRestricts (a ^ 2) ⇑ContinuousMap.realToNNReal
-- SpectrumRestricts.eq_zero_of_neg
lemma SpectrumRestricts.eq_zero_of_neg {a : A} (ha : IsSelfAdjoint a)
    (ha₁ : SpectrumRestricts a ContinuousMap.realToNNReal)
    (ha₂ : SpectrumRestricts (-a) ContinuousMap.realToNNReal) :
    a = 0 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) (ha₁ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal) (ha₂ : SpectrumRestricts (-a) ⇑ContinuousMap.realToNNReal)
Goal: a = 0
-- SpectrumRestricts.smul_of_nonneg
lemma SpectrumRestricts.smul_of_nonneg {A : Type*} [Ring A] [Algebra ℝ A] {a : A}
    (ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ} (hr : 0 ≤ r) :
    SpectrumRestricts (r • a) ContinuousMap.realToNNReal 
Arguments: {A : Type u_2} [Ring A] [Algebra ℝ A] {a : A} (ha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal) {r : ℝ} (hr : 0 ≤ r)
Goal: SpectrumRestricts (r • a) ⇑ContinuousMap.realToNNReal
-- spectrum_star_mul_self_nonneg
lemma spectrum_star_mul_self_nonneg {b : A} : ∀ x ∈ spectrum ℝ (star b * b), 0 ≤ x 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] {b : A} (x : ℝ)
Goal: x ∈ spectrum ℝ (star b * b) → 0 ≤ x
-- IsSelfAdjoint.coe_mem_spectrum_complex
lemma IsSelfAdjoint.coe_mem_spectrum_complex {A : Type*} [TopologicalSpace A] [Ring A]
    [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ (IsStarNormal : A → Prop)]
    {a : A} {x : ℝ} (ha : IsSelfAdjoint a := by cfc_tac) :
    (x : ℂ) ∈ spectrum ℂ a ↔ x ∈ spectrum ℝ a 
Arguments: {A : Type u_2} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} {x : ℝ} (ha : IsSelfAdjoint a := by cfc_tac)
Goal: ↑x ∈ spectrum ℂ a ↔ x ∈ spectrum ℝ a
-- CstarRing.spectralOrderedRing
lemma CstarRing.spectralOrderedRing : @StarOrderedRing A _ (CstarRing.spectralOrder A) _ 
Arguments: (A : Type u_1) [NormedRing A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedAlgebra ℂ A] [StarModule ℂ A]
Goal: StarOrderedRing A
-- cfcHom_real_eq_restrict
lemma cfcHom_real_eq_restrict {a : A} (ha : IsSelfAdjoint a) :
    cfcHom ha = ha.spectrumRestricts.starAlgHom (cfcHom ha.isStarNormal) (f := Complex.reCLM) 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] [UniqueContinuousFunctionalCalculus ℂ A] {a : A} (ha : IsSelfAdjoint a)
Goal: cfcHom ha = SpectrumRestricts.starAlgHom (cfcHom ⋯) ⋯
-- cfc_real_eq_complex
lemma cfc_real_eq_complex {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac)  :
    cfc f a = cfc (fun x ↦ f x.re : ℂ → ℂ) a 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [UniqueContinuousFunctionalCalculus ℝ A] [UniqueContinuousFunctionalCalculus ℂ A] {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac)
Goal: cfc f a = cfc (fun x => ↑(f x.re)) a
-- cfcHom_nnreal_eq_restrict
lemma cfcHom_nnreal_eq_restrict {a : A} (ha : 0 ≤ a) :
    cfcHom ha = (SpectrumRestricts.nnreal_of_nonneg ha).starAlgHom
      (cfcHom (IsSelfAdjoint.of_nonneg ha)) 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Algebra ℝ A] [TopologicalRing A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [ContinuousFunctionalCalculus NNReal fun x => 0 ≤ x] [UniqueContinuousFunctionalCalculus ℝ A] [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a)
Goal: cfcHom ha = SpectrumRestricts.starAlgHom (cfcHom ⋯) ⋯
-- cfc_nnreal_eq_real
lemma cfc_nnreal_eq_real {a : A} (f : ℝ≥0 → ℝ≥0) (ha : 0 ≤ a := by cfc_tac)  :
    cfc f a = cfc (fun x ↦ f x.toNNReal : ℝ → ℝ) a 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Algebra ℝ A] [TopologicalRing A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [ContinuousFunctionalCalculus NNReal fun x => 0 ≤ x] [UniqueContinuousFunctionalCalculus ℝ A] [NonnegSpectrumClass ℝ A] {a : A} (f : NNReal → NNReal) (ha : 0 ≤ a := by cfc_tac)
Goal: cfc f a = cfc (fun x => ↑(f x.toNNReal)) a
-- ENNReal.log_zero
@[simp] lemma log_zero : log 0 = ⊥ 
Goal: ENNReal.log 0 = ⊥
-- ENNReal.log_one
@[simp] lemma log_one : log 1 = 0 
Goal: ENNReal.log 1 = 0
-- ENNReal.log_top
@[simp] lemma log_top : log ⊤ = ⊤ 
Goal: ⊤.log = ⊤
-- ENNReal.log_ofReal
@[simp]
lemma log_ofReal (x : ℝ) : log (ENNReal.ofReal x) = if x ≤ 0 then ⊥ else ↑(Real.log x) 
Arguments: (x : ℝ)
Goal: (ENNReal.ofReal x).log = if x ≤ 0 then ⊥ else ↑(Real.log x)
-- ENNReal.log_ofReal_of_pos
lemma log_ofReal_of_pos {x : ℝ} (hx : 0 < x) : log (ENNReal.ofReal x) = Real.log x 
Arguments: {x : ℝ} (hx : 0 < x)
Goal: (ENNReal.ofReal x).log = ↑(Real.log x)
-- ENNReal.log_pos_real
theorem log_pos_real {x : ℝ≥0∞} (h : x ≠ 0) (h' : x ≠ ⊤) :
    log x = Real.log (ENNReal.toReal x) 
Arguments: {x : ENNReal} (h : x ≠ 0) (h' : x ≠ ⊤)
Goal: x.log = ↑(Real.log x.toReal)
-- ENNReal.log_pos_real'
theorem log_pos_real' {x : ℝ≥0∞} (h : 0 < x.toReal) :
    log x = Real.log (ENNReal.toReal x) 
Arguments: {x : ENNReal} (h : 0 < x.toReal)
Goal: x.log = ↑(Real.log x.toReal)
-- ENNReal.log_of_nnreal
theorem log_of_nnreal {x : ℝ≥0} (h : x ≠ 0) :
    log (x : ℝ≥0∞) = Real.log x 
Arguments: {x : NNReal} (h : x ≠ 0)
Goal: (↑x).log = ↑(Real.log ↑x)
-- ENNReal.log_strictMono
theorem log_strictMono : StrictMono log 
Goal: StrictMono ENNReal.log
-- ENNReal.log_monotone
theorem log_monotone : Monotone log 
Goal: Monotone ENNReal.log
-- ENNReal.log_injective
theorem log_injective : Function.Injective log 
Goal: Function.Injective ENNReal.log
-- ENNReal.log_surjective
theorem log_surjective : Function.Surjective log 
Goal: Function.Surjective ENNReal.log
-- ENNReal.log_bijective
theorem log_bijective : Function.Bijective log 
Goal: Function.Bijective ENNReal.log
-- ENNReal.log_eq_iff
@[simp]
theorem log_eq_iff {x y : ℝ≥0∞} : log x = log y ↔ x = y 
Arguments: {x y : ENNReal}
Goal: x.log = y.log ↔ x = y
-- ENNReal.log_eq_bot_iff
@[simp] theorem log_eq_bot_iff {x : ℝ≥0∞} : log x = ⊥ ↔ x = 0 
Arguments: {x : ENNReal}
Goal: x.log = ⊥ ↔ x = 0
-- ENNReal.log_eq_one_iff
@[simp] theorem log_eq_one_iff {x : ℝ≥0∞} : log x = 0 ↔ x = 1 
Arguments: {x : ENNReal}
Goal: x.log = 0 ↔ x = 1
-- ENNReal.log_eq_top_iff
@[simp] theorem log_eq_top_iff {x : ℝ≥0∞} : log x = ⊤ ↔ x = ⊤ 
Arguments: {x : ENNReal}
Goal: x.log = ⊤ ↔ x = ⊤
-- ENNReal.log_lt_log_iff
@[simp] lemma log_lt_log_iff {x y : ℝ≥0∞} : log x < log y ↔ x < y 
Arguments: {x y : ENNReal}
Goal: x.log < y.log ↔ x < y
-- ENNReal.bot_lt_log_iff
@[simp] lemma bot_lt_log_iff {x : ℝ≥0∞} : ⊥ < log x ↔ 0 < x 
Arguments: {x : ENNReal}
Goal: ⊥ < x.log ↔ 0 < x
-- ENNReal.log_lt_top_iff
@[simp] lemma log_lt_top_iff {x : ℝ≥0∞} : log x < ⊤ ↔ x < ⊤ 
Arguments: {x : ENNReal}
Goal: x.log < ⊤ ↔ x < ⊤
-- ENNReal.log_lt_zero_iff
@[simp] lemma log_lt_zero_iff {x : ℝ≥0∞} : log x < 0 ↔ x < 1 
Arguments: {x : ENNReal}
Goal: x.log < 0 ↔ x < 1
-- ENNReal.zero_lt_log_iff
@[simp] lemma zero_lt_log_iff {x : ℝ≥0∞} : 0 < log x ↔ 1 < x 
Arguments: {x : ENNReal}
Goal: 0 < x.log ↔ 1 < x
-- ENNReal.log_le_log_iff
@[simp] lemma log_le_log_iff {x y : ℝ≥0∞} : log x ≤ log y ↔ x ≤ y 
Arguments: {x y : ENNReal}
Goal: x.log ≤ y.log ↔ x ≤ y
-- ENNReal.log_le_zero_iff
@[simp] lemma log_le_zero_iff {x : ℝ≥0∞} : log x ≤ 0 ↔ x ≤ 1 
Arguments: {x : ENNReal}
Goal: x.log ≤ 0 ↔ x ≤ 1
-- ENNReal.zero_le_log_iff
@[simp] lemma zero_le_log_iff {x : ℝ≥0∞} : 0 ≤ log x ↔ 1 ≤ x 
Arguments: {x : ENNReal}
Goal: 0 ≤ x.log ↔ 1 ≤ x
-- ENNReal.log_mul_add
theorem log_mul_add {x y : ℝ≥0∞} : log (x * y) = log x + log y 
Arguments: {x y : ENNReal}
Goal: (x * y).log = x.log + y.log
-- ENNReal.log_pow
theorem log_pow {x : ℝ≥0∞} {n : ℕ} : log (x ^ n) = n * log x 
Arguments: {x : ENNReal} {n : ℕ}
Goal: (x ^ n).log = ↑n * x.log
-- ENNReal.log_rpow
theorem log_rpow {x : ℝ≥0∞} {y : ℝ} : log (x ^ y) = y * log x 
Arguments: {x : ENNReal} {y : ℝ}
Goal: (x ^ y).log = ↑y * x.log
-- ENNReal.log_inv
lemma log_inv {x : ℝ≥0∞} : log x⁻¹ = - log x 
Arguments: {x : ENNReal}
Goal: x⁻¹.log = -x.log
-- Real.log_mul_self_monotoneOn
theorem log_mul_self_monotoneOn : MonotoneOn (fun x : ℝ => log x * x) { x | 1 ≤ x } 
Goal: MonotoneOn (fun x => Real.log x * x) {x | 1 ≤ x}
-- Real.log_div_self_antitoneOn
theorem log_div_self_antitoneOn : AntitoneOn (fun x : ℝ => log x / x) { x | exp 1 ≤ x } 
Goal: AntitoneOn (fun x => Real.log x / x) {x | Real.exp 1 ≤ x}
-- Real.log_div_self_rpow_antitoneOn
theorem log_div_self_rpow_antitoneOn {a : ℝ} (ha : 0 < a) :
    AntitoneOn (fun x : ℝ => log x / x ^ a) { x | exp (1 / a) ≤ x } 
Arguments: {a : ℝ} (ha : 0 < a)
Goal: AntitoneOn (fun x => Real.log x / x ^ a) {x | Real.exp (1 / a) ≤ x}
-- Real.log_div_sqrt_antitoneOn
theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } 
Goal: AntitoneOn (fun x => Real.log x / √x) {x | Real.exp 2 ≤ x}
-- Real.continuous_mul_log
lemma continuous_mul_log : Continuous fun x ↦ x * log x 
Goal: Continuous fun x => x * Real.log x
-- Real.differentiableOn_mul_log
lemma differentiableOn_mul_log : DifferentiableOn ℝ (fun x ↦ x * log x) {0}ᶜ 
Goal: DifferentiableOn ℝ (fun x => x * Real.log x) {0}ᶜ
-- Real.deriv_mul_log
lemma deriv_mul_log {x : ℝ} (hx : x ≠ 0) : deriv (fun x ↦ x * log x) x = log x + 1 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: deriv (fun x => x * Real.log x) x = Real.log x + 1