ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Orientation.rightAngleRotation_neg_orientation
theorem rightAngleRotation_neg_orientation (x : E) :
    (-o).rightAngleRotation x = -o.rightAngleRotation x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: (-o).rightAngleRotation x = -o.rightAngleRotation x
-- Orientation.rightAngleRotation_trans_neg_orientation
@[simp]
theorem rightAngleRotation_trans_neg_orientation :
    (-o).rightAngleRotation = o.rightAngleRotation.trans (LinearIsometryEquiv.neg ℝ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: (-o).rightAngleRotation = o.rightAngleRotation.trans (LinearIsometryEquiv.neg ℝ)
-- Orientation.rightAngleRotation_map
theorem rightAngleRotation_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    [hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x : F) :
    (Orientation.map (Fin 2) φ.toLinearEquiv o).rightAngleRotation x =
      φ (o.rightAngleRotation (φ.symm x)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [hF : Fact (FiniteDimensional.finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x : F)
Goal: ((Orientation.map (Fin 2) φ.toLinearEquiv) o).rightAngleRotation x = φ (o.rightAngleRotation (φ.symm x))
-- Orientation.linearIsometryEquiv_comp_rightAngleRotation
theorem linearIsometryEquiv_comp_rightAngleRotation (φ : E ≃ₗᵢ[ℝ] E)
    (hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x : E) : φ (J x) = J (φ x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (φ : E ≃ₗᵢ[ℝ] E) (hφ : 0 < LinearMap.det ↑φ.toLinearEquiv) (x : E)
Goal: φ (o.rightAngleRotation x) = o.rightAngleRotation (φ x)
-- Orientation.rightAngleRotation_map'
theorem rightAngleRotation_map' {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    [Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) :
    (Orientation.map (Fin 2) φ.toLinearEquiv o).rightAngleRotation =
      (φ.symm.trans o.rightAngleRotation).trans φ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [Fact (FiniteDimensional.finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F)
Goal: ((Orientation.map (Fin 2) φ.toLinearEquiv) o).rightAngleRotation = (φ.symm.trans o.rightAngleRotation).trans φ
-- Orientation.linearIsometryEquiv_comp_rightAngleRotation'
theorem linearIsometryEquiv_comp_rightAngleRotation' (φ : E ≃ₗᵢ[ℝ] E)
    (hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) :
    LinearIsometryEquiv.trans J φ = φ.trans J 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (φ : E ≃ₗᵢ[ℝ] E) (hφ : 0 < LinearMap.det ↑φ.toLinearEquiv)
Goal: o.rightAngleRotation.trans φ = φ.trans o.rightAngleRotation
-- Orientation.coe_basisRightAngleRotation
@[simp]
theorem coe_basisRightAngleRotation (x : E) (hx : x ≠ 0) :
    ⇑(o.basisRightAngleRotation x hx) = ![x, J x] 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E) (hx : x ≠ 0)
Goal: ⇑(o.basisRightAngleRotation x hx) = ![x, o.rightAngleRotation x]
-- Orientation.inner_mul_inner_add_areaForm_mul_areaForm'
theorem inner_mul_inner_add_areaForm_mul_areaForm' (a x : E) :
    ⟪a, x⟫ • innerₛₗ ℝ a + ω a x • ω a = ‖a‖ ^ 2 • innerₛₗ ℝ x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a x : E)
Goal: ⟪a, x⟫_ℝ • (innerₛₗ ℝ) a + (o.areaForm a) x • o.areaForm a = ‖a‖ ^ 2 • (innerₛₗ ℝ) x
-- Orientation.inner_mul_inner_add_areaForm_mul_areaForm
theorem inner_mul_inner_add_areaForm_mul_areaForm (a x y : E) :
    ⟪a, x⟫ * ⟪a, y⟫ + ω a x * ω a y = ‖a‖ ^ 2 * ⟪x, y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a x y : E)
Goal: ⟪a, x⟫_ℝ * ⟪a, y⟫_ℝ + (o.areaForm a) x * (o.areaForm a) y = ‖a‖ ^ 2 * ⟪x, y⟫_ℝ
-- Orientation.inner_sq_add_areaForm_sq
theorem inner_sq_add_areaForm_sq (a b : E) : ⟪a, b⟫ ^ 2 + ω a b ^ 2 = ‖a‖ ^ 2 * ‖b‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a b : E)
Goal: ⟪a, b⟫_ℝ ^ 2 + (o.areaForm a) b ^ 2 = ‖a‖ ^ 2 * ‖b‖ ^ 2
-- Orientation.inner_mul_areaForm_sub'
theorem inner_mul_areaForm_sub' (a x : E) : ⟪a, x⟫ • ω a - ω a x • innerₛₗ ℝ a = ‖a‖ ^ 2 • ω x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a x : E)
Goal: ⟪a, x⟫_ℝ • o.areaForm a - (o.areaForm a) x • (innerₛₗ ℝ) a = ‖a‖ ^ 2 • o.areaForm x
-- Orientation.inner_mul_areaForm_sub
theorem inner_mul_areaForm_sub (a x y : E) : ⟪a, x⟫ * ω a y - ω a x * ⟪a, y⟫ = ‖a‖ ^ 2 * ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a x y : E)
Goal: ⟪a, x⟫_ℝ * (o.areaForm a) y - (o.areaForm a) x * ⟪a, y⟫_ℝ = ‖a‖ ^ 2 * (o.areaForm x) y
-- Orientation.nonneg_inner_and_areaForm_eq_zero_iff_sameRay
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: 0 ≤ ⟪x, y⟫_ℝ ∧ (o.areaForm x) y = 0 ↔ SameRay ℝ x y
-- Orientation.kahler_apply_apply
theorem kahler_apply_apply (x y : E) : o.kahler x y = ⟪x, y⟫ + ω x y • Complex.I 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler x) y = ↑⟪x, y⟫_ℝ + (o.areaForm x) y • Complex.I
-- Orientation.kahler_swap
theorem kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler x) y = (starRingEnd ℂ) ((o.kahler y) x)
-- Orientation.kahler_apply_self
@[simp]
theorem kahler_apply_self (x : E) : o.kahler x x = ‖x‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: (o.kahler x) x = ↑‖x‖ ^ 2
-- Orientation.kahler_rightAngleRotation_left
@[simp]
theorem kahler_rightAngleRotation_left (x y : E) :
    o.kahler (J x) y = -Complex.I * o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler (o.rightAngleRotation x)) y = -Complex.I * (o.kahler x) y
-- Orientation.kahler_rightAngleRotation_right
@[simp]
theorem kahler_rightAngleRotation_right (x y : E) :
    o.kahler x (J y) = Complex.I * o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler x) (o.rightAngleRotation y) = Complex.I * (o.kahler x) y
-- Orientation.kahler_comp_rightAngleRotation
theorem kahler_comp_rightAngleRotation (x y : E) : o.kahler (J x) (J y) = o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler (o.rightAngleRotation x)) (o.rightAngleRotation y) = (o.kahler x) y
-- Orientation.kahler_comp_rightAngleRotation'
theorem kahler_comp_rightAngleRotation' (x y : E) :
    -(Complex.I * (Complex.I * o.kahler x y)) = o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: -(Complex.I * (Complex.I * (o.kahler x) y)) = (o.kahler x) y
-- Orientation.kahler_neg_orientation
@[simp]
theorem kahler_neg_orientation (x y : E) : (-o).kahler x y = conj (o.kahler x y) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ((-o).kahler x) y = (starRingEnd ℂ) ((o.kahler x) y)
-- Orientation.kahler_mul
theorem kahler_mul (a x y : E) : o.kahler x a * o.kahler a y = ‖a‖ ^ 2 * o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (a x y : E)
Goal: (o.kahler x) a * (o.kahler a) y = ↑‖a‖ ^ 2 * (o.kahler x) y
-- Orientation.normSq_kahler
theorem normSq_kahler (x y : E) : Complex.normSq (o.kahler x y) = ‖x‖ ^ 2 * ‖y‖ ^ 2 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: Complex.normSq ((o.kahler x) y) = ‖x‖ ^ 2 * ‖y‖ ^ 2
-- Orientation.abs_kahler
theorem abs_kahler (x y : E) : Complex.abs (o.kahler x y) = ‖x‖ * ‖y‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: Complex.abs ((o.kahler x) y) = ‖x‖ * ‖y‖
-- Orientation.norm_kahler
theorem norm_kahler (x y : E) : ‖o.kahler x y‖ = ‖x‖ * ‖y‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ‖(o.kahler x) y‖ = ‖x‖ * ‖y‖
-- Orientation.eq_zero_or_eq_zero_of_kahler_eq_zero
theorem eq_zero_or_eq_zero_of_kahler_eq_zero {x y : E} (hx : o.kahler x y = 0) : x = 0 ∨ y = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {x y : E} (hx : (o.kahler x) y = 0)
Goal: x = 0 ∨ y = 0
-- Orientation.kahler_eq_zero_iff
theorem kahler_eq_zero_iff (x y : E) : o.kahler x y = 0 ↔ x = 0 ∨ y = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler x) y = 0 ↔ x = 0 ∨ y = 0
-- Orientation.kahler_ne_zero
theorem kahler_ne_zero {x y : E} (hx : x ≠ 0) (hy : y ≠ 0) : o.kahler x y ≠ 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {x y : E} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (o.kahler x) y ≠ 0
-- Orientation.kahler_ne_zero_iff
theorem kahler_ne_zero_iff (x y : E) : o.kahler x y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.kahler x) y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0
-- Orientation.kahler_map
theorem kahler_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    [hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
    (Orientation.map (Fin 2) φ.toLinearEquiv o).kahler x y = o.kahler (φ.symm x) (φ.symm y) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [hF : Fact (FiniteDimensional.finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F)
Goal: (((Orientation.map (Fin 2) φ.toLinearEquiv) o).kahler x) y = (o.kahler (φ.symm x)) (φ.symm y)
-- Orientation.kahler_comp_linearIsometryEquiv
theorem kahler_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
    (hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
    o.kahler (φ x) (φ y) = o.kahler x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (φ : E ≃ₗᵢ[ℝ] E) (hφ : 0 < LinearMap.det ↑φ.toLinearEquiv) (x y : E)
Goal: (o.kahler (φ x)) (φ y) = (o.kahler x) y
-- Complex.areaForm
@[simp]
protected theorem areaForm (w z : ℂ) : Complex.orientation.areaForm w z = (conj w * z).im 
Arguments: (w z : ℂ)
Goal: (Complex.orientation.areaForm w) z = ((starRingEnd ℂ) w * z).im
-- Complex.rightAngleRotation
@[simp]
protected theorem rightAngleRotation (z : ℂ) :
    Complex.orientation.rightAngleRotation z = I * z 
Arguments: (z : ℂ)
Goal: Complex.orientation.rightAngleRotation z = Complex.I * z
-- Complex.kahler
@[simp]
protected theorem kahler (w z : ℂ) : Complex.orientation.kahler w z = conj w * z 
Arguments: (w z : ℂ)
Goal: (Complex.orientation.kahler w) z = (starRingEnd ℂ) w * z
-- Orientation.areaForm_map_complex
theorem areaForm_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x y : E) :
    ω x y = (conj (f x) * f y).im 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (f : E ≃ₗᵢ[ℝ] ℂ) (hf : (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x y : E)
Goal: (o.areaForm x) y = ((starRingEnd ℂ) (f x) * f y).im
-- Orientation.rightAngleRotation_map_complex
theorem rightAngleRotation_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x : E) :
    f (J x) = I * f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (f : E ≃ₗᵢ[ℝ] ℂ) (hf : (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x : E)
Goal: f (o.rightAngleRotation x) = Complex.I * f x
-- Orientation.kahler_map_complex
theorem kahler_map_complex (f : E ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x y : E) :
    o.kahler x y = conj (f x) * f y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (f : E ≃ₗᵢ[ℝ] ℂ) (hf : (Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x y : E)
Goal: (o.kahler x) y = (starRingEnd ℂ) (f x) * f y
-- unitary.spectrum_subset_circle
theorem unitary.spectrum_subset_circle (u : unitary E) :
    spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [NormedAlgebra 𝕜 E] [CompleteSpace E] (u : ↥(unitary E))
Goal: spectrum 𝕜 ↑u ⊆ Metric.sphere 0 1
-- spectrum.subset_circle_of_unitary
theorem spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :
    spectrum 𝕜 u ⊆ Metric.sphere 0 1 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [NormedAlgebra 𝕜 E] [CompleteSpace E] {u : E} (h : u ∈ unitary E)
Goal: spectrum 𝕜 u ⊆ Metric.sphere 0 1
-- IsSelfAdjoint.spectralRadius_eq_nnnorm
theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
    spectralRadius ℂ a = ‖a‖₊ 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] {a : A} (ha : IsSelfAdjoint a)
Goal: spectralRadius ℂ a = ↑‖a‖₊
-- IsSelfAdjoint.toReal_spectralRadius_complex_eq_norm
lemma IsSelfAdjoint.toReal_spectralRadius_complex_eq_norm {a : A} (ha : IsSelfAdjoint a) :
    (spectralRadius ℂ a).toReal = ‖a‖ 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] {a : A} (ha : IsSelfAdjoint a)
Goal: (spectralRadius ℂ a).toReal = ‖a‖
-- IsStarNormal.spectralRadius_eq_nnnorm
theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
    spectralRadius ℂ a = ‖a‖₊ 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] (a : A) [IsStarNormal a]
Goal: spectralRadius ℂ a = ↑‖a‖₊
-- IsSelfAdjoint.mem_spectrum_eq_re
theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ}
    (hz : z ∈ spectrum ℂ a) : z = z.re 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ} (hz : z ∈ spectrum ℂ a)
Goal: z = ↑z.re
-- selfAdjoint.mem_spectrum_eq_re
theorem selfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] (a : selfAdjoint A) {z : ℂ}
    (hz : z ∈ spectrum ℂ (a : A)) : z = z.re 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] (a : ↥(selfAdjoint A)) {z : ℂ} (hz : z ∈ spectrum ℂ ↑a)
Goal: z = ↑z.re
-- IsSelfAdjoint.val_re_map_spectrum
theorem IsSelfAdjoint.val_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) :
    spectrum ℂ a = ((↑) ∘ re '' spectrum ℂ a : Set ℂ) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a)
Goal: spectrum ℂ a = Complex.ofReal' ∘ Complex.re '' spectrum ℂ a
-- selfAdjoint.val_re_map_spectrum
theorem selfAdjoint.val_re_map_spectrum [StarModule ℂ A] (a : selfAdjoint A) :
    spectrum ℂ (a : A) = ((↑) ∘ re '' spectrum ℂ (a : A) : Set ℂ) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] (a : ↥(selfAdjoint A))
Goal: spectrum ℂ ↑a = Complex.ofReal' ∘ Complex.re '' spectrum ℂ ↑a
-- StarAlgHom.nnnorm_apply_le
theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ 
Arguments: {F : Type u_1} {A : Type u_2} {B : Type u_3} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [FunLike F A B] [AlgHomClass F ℂ A B] [StarAlgHomClass F ℂ A B] (φ : F) (a : A)
Goal: ‖φ a‖₊ ≤ ‖a‖₊
-- StarAlgHom.norm_apply_le
theorem norm_apply_le (a : A) : ‖(φ a : B)‖ ≤ ‖a‖ 
Arguments: {F : Type u_1} {A : Type u_2} {B : Type u_3} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [FunLike F A B] [AlgHomClass F ℂ A B] [StarAlgHomClass F ℂ A B] (φ : F) (a : A)
Goal: ‖φ a‖ ≤ ‖a‖
-- StarAlgEquiv.nnnorm_map
lemma nnnorm_map (φ : F) (a : A) : ‖φ a‖₊ = ‖a‖₊ 
Arguments: {F : Type u_1} {A : Type u_2} {B : Type u_3} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [EquivLike F A B] [NonUnitalAlgEquivClass F ℂ A B] [StarAlgEquivClass F ℂ A B] (φ : F) (a : A)
Goal: ‖φ a‖₊ = ‖a‖₊
-- StarAlgEquiv.norm_map
lemma norm_map (φ : F) (a : A) : ‖φ a‖ = ‖a‖ 
Arguments: {F : Type u_1} {A : Type u_2} {B : Type u_3} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [EquivLike F A B] [NonUnitalAlgEquivClass F ℂ A B] [StarAlgEquivClass F ℂ A B] (φ : F) (a : A)
Goal: ‖φ a‖ = ‖a‖
-- StarAlgEquiv.isometry
lemma isometry (φ : F) : Isometry φ 
Arguments: {F : Type u_1} {A : Type u_2} {B : Type u_3} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [EquivLike F A B] [NonUnitalAlgEquivClass F ℂ A B] [StarAlgEquivClass F ℂ A B] (φ : F)
Goal: Isometry ⇑φ
-- AlgHomClass.instStarAlgHomClass
lemma _root_.AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ 
Arguments: {F : Type u_1} {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] [FunLike F A ℂ] [hF : AlgHomClass F ℂ A ℂ]
Goal: StarAlgHomClass F ℂ A ℂ
-- Matrix.norm_def
theorem norm_def (A : Matrix m n α) : ‖A‖ = ‖fun i j => A i j‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖ = ‖fun i j => A i j‖
-- Matrix.norm_eq_sup_sup_nnnorm
lemma norm_eq_sup_sup_nnnorm (A : Matrix m n α) :
    ‖A‖ = Finset.sup Finset.univ fun i ↦ Finset.sup Finset.univ fun j ↦ ‖A i j‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)
-- Matrix.nnnorm_def
theorem nnnorm_def (A : Matrix m n α) : ‖A‖₊ = ‖fun i j => A i j‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖₊ = ‖fun i j => A i j‖₊
-- Matrix.norm_le_iff
theorem norm_le_iff {r : ℝ} (hr : 0 ≤ r) {A : Matrix m n α} : ‖A‖ ≤ r ↔ ∀ i j, ‖A i j‖ ≤ r 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ} (hr : 0 ≤ r) {A : Matrix m n α}
Goal: ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r
-- Matrix.nnnorm_le_iff
theorem nnnorm_le_iff {r : ℝ≥0} {A : Matrix m n α} : ‖A‖₊ ≤ r ↔ ∀ i j, ‖A i j‖₊ ≤ r 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ≥0} {A : Matrix m n α}
Goal: ‖A‖₊ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ ≤ r
-- Matrix.norm_lt_iff
theorem norm_lt_iff {r : ℝ} (hr : 0 < r) {A : Matrix m n α} : ‖A‖ < r ↔ ∀ i j, ‖A i j‖ < r 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ} (hr : 0 < r) {A : Matrix m n α}
Goal: ‖A‖ < r ↔ ∀ (i : m) (j : n), ‖A i j‖ < r
-- Matrix.nnnorm_lt_iff
theorem nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α} :
    ‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α}
Goal: ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r
-- Matrix.norm_entry_le_entrywise_sup_norm
theorem norm_entry_le_entrywise_sup_norm (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖ ≤ ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) {i : m} {j : n}
Goal: ‖A i j‖ ≤ ‖A‖
-- Matrix.nnnorm_entry_le_entrywise_sup_nnnorm
theorem nnnorm_entry_le_entrywise_sup_nnnorm (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖₊ ≤ ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) {i : m} {j : n}
Goal: ‖A i j‖₊ ≤ ‖A‖₊
-- Matrix.nnnorm_map_eq
@[simp]
theorem nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
    ‖A.map f‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊)
Goal: ‖A.map f‖₊ = ‖A‖₊
-- Matrix.norm_map_eq
@[simp]
theorem norm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖ = ‖a‖) : ‖A.map f‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖ = ‖a‖)
Goal: ‖A.map f‖ = ‖A‖
-- Matrix.nnnorm_transpose
@[simp]
theorem nnnorm_transpose (A : Matrix m n α) : ‖Aᵀ‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖Aᵀ‖₊ = ‖A‖₊
-- Matrix.norm_transpose
@[simp]
theorem norm_transpose (A : Matrix m n α) : ‖Aᵀ‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖Aᵀ‖ = ‖A‖
-- Matrix.nnnorm_conjTranspose
@[simp]
theorem nnnorm_conjTranspose [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) :
    ‖Aᴴ‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α)
Goal: ‖Aᴴ‖₊ = ‖A‖₊