ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top'
theorem rnDeriv_smul_right_of_ne_top' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
    {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ⊤)
Goal: ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_add'
lemma rnDeriv_add' (ν₁ ν₂ μ : Measure α) [SigmaFinite ν₁] [SigmaFinite ν₂] [SigmaFinite μ] :
    (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν₁ ν₂ μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SigmaFinite μ]
Goal: (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ + ν₂.rnDeriv μ
-- MeasureTheory.Measure.rnDeriv_add_of_mutuallySingular
lemma rnDeriv_add_of_mutuallySingular (ν₁ ν₂ μ : Measure α)
    [SigmaFinite ν₁] [SigmaFinite ν₂] [SigmaFinite μ] (h : ν₂ ⟂ₘ μ) :
    (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (ν₁ ν₂ μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SigmaFinite μ] (h : ν₂ ⟂ₘ μ)
Goal: (ν₁ + ν₂).rnDeriv μ =ᵐ[μ] ν₁.rnDeriv μ
-- spectrum_star_mul_self_of_isStarNormal
theorem spectrum_star_mul_self_of_isStarNormal :
    spectrum ℂ (star a * a) ⊆ Set.Icc (0 : ℂ) ‖star a * a‖ 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a]
Goal: spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
-- elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal
theorem elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal (h : IsUnit a) :
    IsUnit (⟨a, self_mem ℂ a⟩ : elementalStarAlgebra ℂ a) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {a : A} [IsStarNormal a] (h : IsUnit a)
Goal: IsUnit ⟨a, ⋯⟩
-- StarSubalgebra.isUnit_coe_inv_mem
theorem StarSubalgebra.isUnit_coe_inv_mem {S : StarSubalgebra ℂ A} (hS : IsClosed (S : Set A))
    {x : A} (h : IsUnit x) (hxS : x ∈ S) : ↑h.unit⁻¹ ∈ S 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : A} (h : IsUnit x) (hxS : x ∈ S)
Goal: ↑h.unit⁻¹ ∈ S
-- StarSubalgebra.coe_isUnit
theorem StarSubalgebra.coe_isUnit {S : StarSubalgebra ℂ A} (hS : IsClosed (S : Set A)) {x : S} :
    IsUnit (x : A) ↔ IsUnit x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : ↥S}
Goal: IsUnit ↑x ↔ IsUnit x
-- StarSubalgebra.mem_spectrum_iff
theorem StarSubalgebra.mem_spectrum_iff {S : StarSubalgebra ℂ A} (hS : IsClosed (S : Set A)) {x : S}
    {z : ℂ} : z ∈ spectrum ℂ x ↔ z ∈ spectrum ℂ (x : A) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : ↥S} {z : ℂ}
Goal: z ∈ spectrum ℂ x ↔ z ∈ spectrum ℂ ↑x
-- StarSubalgebra.spectrum_eq
theorem StarSubalgebra.spectrum_eq {S : StarSubalgebra ℂ A} (hS : IsClosed (S : Set A)) (x : S) :
    spectrum ℂ x = spectrum ℂ (x : A) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) (x : ↥S)
Goal: spectrum ℂ x = spectrum ℂ ↑x
-- elementalStarAlgebra.characterSpaceToSpectrum_coe
@[simps]
noncomputable def elementalStarAlgebra.characterSpaceToSpectrum (x : A)
    (φ : characterSpace ℂ (elementalStarAlgebra ℂ x)) : spectrum ℂ x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (x : A) (φ : ↑(WeakDual.characterSpace ℂ ↥(elementalStarAlgebra ℂ x)))
Goal: ↑(elementalStarAlgebra.characterSpaceToSpectrum x φ) = φ ⟨x, ⋯⟩
-- elementalStarAlgebra.continuous_characterSpaceToSpectrum
theorem elementalStarAlgebra.continuous_characterSpaceToSpectrum (x : A) :
    Continuous (elementalStarAlgebra.characterSpaceToSpectrum x) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (x : A)
Goal: Continuous (elementalStarAlgebra.characterSpaceToSpectrum x)
-- elementalStarAlgebra.bijective_characterSpaceToSpectrum
theorem elementalStarAlgebra.bijective_characterSpaceToSpectrum :
    Function.Bijective (elementalStarAlgebra.characterSpaceToSpectrum a) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a]
Goal: Function.Bijective (elementalStarAlgebra.characterSpaceToSpectrum a)
-- continuousFunctionalCalculus_map_id
theorem continuousFunctionalCalculus_map_id :
    continuousFunctionalCalculus a ((ContinuousMap.id ℂ).restrict (spectrum ℂ a)) =
      ⟨a, self_mem ℂ a⟩ 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a]
Goal: (continuousFunctionalCalculus a) (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) = ⟨a, ⋯⟩
-- convexHull_isClosed
@[simps! isClosed]
def convexHull : ClosureOperator (Set E) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E)
Goal: (convexHull 𝕜).IsClosed s = Convex 𝕜 s
-- subset_convexHull
theorem subset_convexHull : s ⊆ convexHull 𝕜 s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E)
Goal: s ⊆ (convexHull 𝕜) s
-- convex_convexHull
theorem convex_convexHull : Convex 𝕜 (convexHull 𝕜 s) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E)
Goal: Convex 𝕜 ((convexHull 𝕜) s)
-- convexHull_eq_iInter
theorem convexHull_eq_iInter : convexHull 𝕜 s = ⋂ (t : Set E) (_ : s ⊆ t) (_ : Convex 𝕜 t), t 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E)
Goal: (convexHull 𝕜) s = ⋂ t, ⋂ (_ : s ⊆ t), ⋂ (_ : Convex 𝕜 t), t
-- mem_convexHull_iff
theorem mem_convexHull_iff : x ∈ convexHull 𝕜 s ↔ ∀ t, s ⊆ t → Convex 𝕜 t → x ∈ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} {x : E}
Goal: x ∈ (convexHull 𝕜) s ↔ ∀ (t : Set E), s ⊆ t → Convex 𝕜 t → x ∈ t
-- convexHull_min
theorem convexHull_min : s ⊆ t → Convex 𝕜 t → convexHull 𝕜 s ⊆ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E}
Goal: s ⊆ t → Convex 𝕜 t → (convexHull 𝕜) s ⊆ t
-- Convex.convexHull_subset_iff
theorem Convex.convexHull_subset_iff (ht : Convex 𝕜 t) : convexHull 𝕜 s ⊆ t ↔ s ⊆ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} (ht : Convex 𝕜 t)
Goal: (convexHull 𝕜) s ⊆ t ↔ s ⊆ t
-- convexHull_mono
@[mono, gcongr]
theorem convexHull_mono (hst : s ⊆ t) : convexHull 𝕜 s ⊆ convexHull 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} (hst : s ⊆ t)
Goal: (convexHull 𝕜) s ⊆ (convexHull 𝕜) t
-- convexHull_eq_self
lemma convexHull_eq_self : convexHull 𝕜 s = s ↔ Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E}
Goal: (convexHull 𝕜) s = s ↔ Convex 𝕜 s
-- convexHull_univ
@[simp]
theorem convexHull_univ : convexHull 𝕜 (univ : Set E) = univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: (convexHull 𝕜) Set.univ = Set.univ
-- convexHull_empty
@[simp]
theorem convexHull_empty : convexHull 𝕜 (∅ : Set E) = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: (convexHull 𝕜) ∅ = ∅
-- convexHull_empty_iff
@[simp]
theorem convexHull_empty_iff : convexHull 𝕜 s = ∅ ↔ s = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E}
Goal: (convexHull 𝕜) s = ∅ ↔ s = ∅
-- convexHull_nonempty_iff
@[simp]
theorem convexHull_nonempty_iff : (convexHull 𝕜 s).Nonempty ↔ s.Nonempty 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E}
Goal: ((convexHull 𝕜) s).Nonempty ↔ s.Nonempty
-- segment_subset_convexHull
theorem segment_subset_convexHull (hx : x ∈ s) (hy : y ∈ s) : segment 𝕜 x y ⊆ convexHull 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} {x y : E} (hx : x ∈ s) (hy : y ∈ s)
Goal: segment 𝕜 x y ⊆ (convexHull 𝕜) s
-- convexHull_singleton
@[simp]
theorem convexHull_singleton (x : E) : convexHull 𝕜 ({x} : Set E) = {x} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x : E)
Goal: (convexHull 𝕜) {x} = {x}
-- convexHull_zero
@[simp]
theorem convexHull_zero : convexHull 𝕜 (0 : Set E) = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
Goal: (convexHull 𝕜) 0 = 0
-- convexHull_pair
@[simp]
theorem convexHull_pair (x y : E) : convexHull 𝕜 {x, y} = segment 𝕜 x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x y : E)
Goal: (convexHull 𝕜) {x, y} = segment 𝕜 x y
-- convexHull_convexHull_union_left
theorem convexHull_convexHull_union_left (s t : Set E) :
    convexHull 𝕜 (convexHull 𝕜 s ∪ t) = convexHull 𝕜 (s ∪ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s t : Set E)
Goal: (convexHull 𝕜) ((convexHull 𝕜) s ∪ t) = (convexHull 𝕜) (s ∪ t)
-- convexHull_convexHull_union_right
theorem convexHull_convexHull_union_right (s t : Set E) :
    convexHull 𝕜 (s ∪ convexHull 𝕜 t) = convexHull 𝕜 (s ∪ t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s t : Set E)
Goal: (convexHull 𝕜) (s ∪ (convexHull 𝕜) t) = (convexHull 𝕜) (s ∪ t)
-- Convex.convex_remove_iff_not_mem_convexHull_remove
theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
    Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x}) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (x : E)
Goal: Convex 𝕜 (s \ {x}) ↔ x ∉ (convexHull 𝕜) (s \ {x})
-- IsLinearMap.image_convexHull
theorem IsLinearMap.image_convexHull {f : E → F} (hf : IsLinearMap 𝕜 f) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] {f : E → F} (hf : IsLinearMap 𝕜 f) (s : Set E)
Goal: f '' (convexHull 𝕜) s = (convexHull 𝕜) (f '' s)
-- LinearMap.image_convexHull
theorem LinearMap.image_convexHull (f : E →ₗ[𝕜] F) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (f : E →ₗ[𝕜] F) (s : Set E)
Goal: ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)
-- convexHull_smul
theorem convexHull_smul (a : 𝕜) (s : Set E) : convexHull 𝕜 (a • s) = a • convexHull 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (a : 𝕜) (s : Set E)
Goal: (convexHull 𝕜) (a • s) = a • (convexHull 𝕜) s
-- AffineMap.image_convexHull
theorem AffineMap.image_convexHull (f : E →ᵃ[𝕜] F) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] (f : E →ᵃ[𝕜] F) (s : Set E)
Goal: ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)
-- convexHull_subset_affineSpan
theorem convexHull_subset_affineSpan (s : Set E) : convexHull 𝕜 s ⊆ (affineSpan 𝕜 s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: (convexHull 𝕜) s ⊆ ↑(affineSpan 𝕜 s)
-- affineSpan_convexHull
@[simp]
theorem affineSpan_convexHull (s : Set E) : affineSpan 𝕜 (convexHull 𝕜 s) = affineSpan 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: affineSpan 𝕜 ((convexHull 𝕜) s) = affineSpan 𝕜 s
-- convexHull_neg
theorem convexHull_neg (s : Set E) : convexHull 𝕜 (-s) = -convexHull 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (s : Set E)
Goal: (convexHull 𝕜) (-s) = -(convexHull 𝕜) s
-- convexHull_vadd
lemma convexHull_vadd (x : E) (s : Set E) : convexHull 𝕜 (x +ᵥ s) = x +ᵥ convexHull 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x : E) (s : Set E)
Goal: (convexHull 𝕜) (x +ᵥ s) = x +ᵥ (convexHull 𝕜) s
-- Unitization.inr_le_iff
lemma inr_le_iff (a b : A) (ha : IsSelfAdjoint a := by cfc_tac)
    (hb : IsSelfAdjoint b := by cfc_tac) :
    (a : Unitization ℂ A) ≤ (b : Unitization ℂ A) ↔ a ≤ b 
Arguments: {A : Type u_1} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [CstarRing A] [NormedSpace ℂ A] [StarModule ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] (a b : A) (ha : IsSelfAdjoint a := by cfc_tac) (hb : IsSelfAdjoint b := by cfc_tac)
Goal: ↑a ≤ ↑b ↔ a ≤ b
-- Unitization.inr_nonneg_iff
@[simp, norm_cast]
lemma inr_nonneg_iff {a : A} : 0 ≤ (a : Unitization ℂ A) ↔ 0 ≤ a 
Arguments: {A : Type u_1} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [CstarRing A] [NormedSpace ℂ A] [StarModule ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] {a : A}
Goal: 0 ≤ ↑a ↔ 0 ≤ a
-- IsSelfAdjoint.le_algebraMap_norm_self
lemma IsSelfAdjoint.le_algebraMap_norm_self {a : A} (ha : IsSelfAdjoint a := by cfc_tac) :
    a ≤ algebraMap ℝ A ‖a‖ 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] {a : A} (ha : IsSelfAdjoint a := by cfc_tac)
Goal: a ≤ (algebraMap ℝ A) ‖a‖
-- IsSelfAdjoint.neg_algebraMap_norm_le_self
lemma IsSelfAdjoint.neg_algebraMap_norm_le_self {a : A} (ha : IsSelfAdjoint a := by cfc_tac) :
    - (algebraMap ℝ A ‖a‖) ≤ a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] {a : A} (ha : IsSelfAdjoint a := by cfc_tac)
Goal: -(algebraMap ℝ A) ‖a‖ ≤ a
-- CstarRing.mul_star_le_algebraMap_norm_sq
lemma CstarRing.mul_star_le_algebraMap_norm_sq {a : A} : a * star a ≤ algebraMap ℝ A (‖a‖ ^ 2) 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] {a : A}
Goal: a * star a ≤ (algebraMap ℝ A) (‖a‖ ^ 2)
-- CstarRing.star_mul_le_algebraMap_norm_sq
lemma CstarRing.star_mul_le_algebraMap_norm_sq {a : A} : star a * a ≤ algebraMap ℝ A (‖a‖ ^ 2) 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] {a : A}
Goal: star a * a ≤ (algebraMap ℝ A) (‖a‖ ^ 2)
-- IsSelfAdjoint.toReal_spectralRadius_eq_norm
lemma IsSelfAdjoint.toReal_spectralRadius_eq_norm {a : A} (ha : IsSelfAdjoint a) :
    (spectralRadius ℝ a).toReal = ‖a‖ 
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: (spectralRadius ℝ a).toReal = ‖a‖
-- CstarRing.norm_or_neg_norm_mem_spectrum
lemma CstarRing.norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
    (ha : IsSelfAdjoint a := by cfc_tac) : ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [Nontrivial A] {a : A} (ha : IsSelfAdjoint a := by cfc_tac)
Goal: ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a
-- CstarRing.nnnorm_mem_spectrum_of_nonneg
lemma CstarRing.nnnorm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
    ‖a‖₊ ∈ spectrum ℝ≥0 a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac)
Goal: ‖a‖₊ ∈ spectrum ℝ≥0 a
-- CstarRing.norm_mem_spectrum_of_nonneg
lemma CstarRing.norm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
    ‖a‖ ∈ spectrum ℝ a 
Arguments: {A : Type u_1} [NormedRing A] [StarRing A] [CstarRing A] [CompleteSpace A] [NormedAlgebra ℂ A] [StarModule ℂ A] [PartialOrder A] [StarOrderedRing A] [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac)
Goal: ‖a‖ ∈ spectrum ℝ a
-- CstarRing.norm_le_norm_of_nonneg_of_le
lemma CstarRing.norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b) :
    ‖a‖ ≤ ‖b‖ 
Arguments: {A : Type u_1} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [CstarRing A] [NormedSpace ℂ A] [StarModule ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b)
Goal: ‖a‖ ≤ ‖b‖
-- CstarRing.conjugate_le_norm_smul
lemma CstarRing.conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
    star a * b * a ≤ ‖b‖ • (star a * a) 
Arguments: {A : Type u_1} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [CstarRing A] [NormedSpace ℂ A] [StarModule ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] {a b : A} (hb : IsSelfAdjoint b := by cfc_tac)
Goal: star a * b * a ≤ ‖b‖ • (star a * a)
-- CstarRing.conjugate_le_norm_smul'
lemma CstarRing.conjugate_le_norm_smul' {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
    a * b * star a ≤ ‖b‖ • (a * star a) 
Arguments: {A : Type u_1} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [CstarRing A] [NormedSpace ℂ A] [StarModule ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] {a b : A} (hb : IsSelfAdjoint b := by cfc_tac)
Goal: a * b * star a ≤ ‖b‖ • (a * star a)
-- Subsingleton.convexIndependent
theorem Subsingleton.convexIndependent [Subsingleton ι] (p : ι → E) : ConvexIndependent 𝕜 p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] [Subsingleton ι] (p : ι → E)
Goal: ConvexIndependent 𝕜 p
-- ConvexIndependent.injective
protected theorem ConvexIndependent.injective {p : ι → E} (hc : ConvexIndependent 𝕜 p) :
    Function.Injective p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E} (hc : ConvexIndependent 𝕜 p)
Goal: Function.Injective p
-- ConvexIndependent.comp_embedding
theorem ConvexIndependent.comp_embedding {ι' : Type*} (f : ι' ↪ ι) {p : ι → E}
    (hc : ConvexIndependent 𝕜 p) : ConvexIndependent 𝕜 (p ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {ι' : Type u_4} (f : ι' ↪ ι) {p : ι → E} (hc : ConvexIndependent 𝕜 p)
Goal: ConvexIndependent 𝕜 (p ∘ ⇑f)
-- ConvexIndependent.subtype
protected theorem ConvexIndependent.subtype {p : ι → E} (hc : ConvexIndependent 𝕜 p) (s : Set ι) :
    ConvexIndependent 𝕜 fun i : s => p i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E} (hc : ConvexIndependent 𝕜 p) (s : Set ι)
Goal: ConvexIndependent 𝕜 fun i => p ↑i
-- ConvexIndependent.range
protected theorem ConvexIndependent.range {p : ι → E} (hc : ConvexIndependent 𝕜 p) :
    ConvexIndependent 𝕜 ((↑) : Set.range p → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E} (hc : ConvexIndependent 𝕜 p)
Goal: ConvexIndependent 𝕜 Subtype.val
-- ConvexIndependent.mono
protected theorem ConvexIndependent.mono {s t : Set E} (hc : ConvexIndependent 𝕜 ((↑) : t → E))
    (hs : s ⊆ t) : ConvexIndependent 𝕜 ((↑) : s → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hc : ConvexIndependent 𝕜 Subtype.val) (hs : s ⊆ t)
Goal: ConvexIndependent 𝕜 Subtype.val
-- Function.Injective.convexIndependent_iff_set
theorem Function.Injective.convexIndependent_iff_set {p : ι → E} (hi : Function.Injective p) :
    ConvexIndependent 𝕜 ((↑) : Set.range p → E) ↔ ConvexIndependent 𝕜 p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E} (hi : Function.Injective p)
Goal: ConvexIndependent 𝕜 Subtype.val ↔ ConvexIndependent 𝕜 p
-- ConvexIndependent.mem_convexHull_iff
@[simp]
protected theorem ConvexIndependent.mem_convexHull_iff {p : ι → E} (hc : ConvexIndependent 𝕜 p)
    (s : Set ι) (i : ι) : p i ∈ convexHull 𝕜 (p '' s) ↔ i ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E} (hc : ConvexIndependent 𝕜 p) (s : Set ι) (i : ι)
Goal: p i ∈ (convexHull 𝕜) (p '' s) ↔ i ∈ s
-- convexIndependent_iff_not_mem_convexHull_diff
theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
    ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E}
Goal: ConvexIndependent 𝕜 p ↔ ∀ (i : ι) (s : Set ι), p i ∉ (convexHull 𝕜) (p '' (s \ {i}))
-- convexIndependent_set_iff_inter_convexHull_subset
theorem convexIndependent_set_iff_inter_convexHull_subset {s : Set E} :
    ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ t, t ⊆ s → s ∩ convexHull 𝕜 t ⊆ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: ConvexIndependent 𝕜 Subtype.val ↔ ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t
-- convexIndependent_set_iff_not_mem_convexHull_diff
theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
    ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: ConvexIndependent 𝕜 Subtype.val ↔ ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
-- convexIndependent_iff_finset
theorem convexIndependent_iff_finset {p : ι → E} :
    ConvexIndependent 𝕜 p ↔
      ∀ (s : Finset ι) (x : ι), p x ∈ convexHull 𝕜 (s.image p : Set E) → x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : ι → E}
Goal: ConvexIndependent 𝕜 p ↔ ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(Finset.image p s) → x ∈ s
-- Convex.convexIndependent_extremePoints
theorem Convex.convexIndependent_extremePoints (hs : Convex 𝕜 s) :
    ConvexIndependent 𝕜 ((↑) : s.extremePoints 𝕜 → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: ConvexIndependent 𝕜 Subtype.val
-- ContinuousLinearMap.opNorm_mul_flip_apply
lemma opNorm_mul_flip_apply (a : E) : ‖(mul 𝕜 E).flip a‖ = ‖a‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing E] [StarRing E] [NormedStarGroup E] [NormedSpace 𝕜 E] [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] [RegularNormedAlgebra 𝕜 E] (a : E)
Goal: ‖(ContinuousLinearMap.mul 𝕜 E).flip a‖ = ‖a‖
-- ContinuousLinearMap.opNNNorm_mul_flip_apply
lemma opNNNorm_mul_flip_apply (a : E) : ‖(mul 𝕜 E).flip a‖₊ = ‖a‖₊ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing E] [StarRing E] [NormedStarGroup E] [NormedSpace 𝕜 E] [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] [RegularNormedAlgebra 𝕜 E] (a : E)
Goal: ‖(ContinuousLinearMap.mul 𝕜 E).flip a‖₊ = ‖a‖₊
-- ContinuousLinearMap.isometry_mul_flip
lemma isometry_mul_flip : Isometry (mul 𝕜 E).flip 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing E] [StarRing E] [NormedStarGroup E] [NormedSpace 𝕜 E] [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] [RegularNormedAlgebra 𝕜 E]
Goal: Isometry ⇑(ContinuousLinearMap.mul 𝕜 E).flip