ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- spectrum.exists_nnnorm_eq_spectralRadius_of_nonempty
theorem exists_nnnorm_eq_spectralRadius_of_nonempty [ProperSpace 𝕜] {a : A} (ha : (σ a).Nonempty) :
    ∃ k ∈ σ a, (‖k‖₊ : ℝ≥0∞) = spectralRadius 𝕜 a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] {a : A} (ha : (spectrum 𝕜 a).Nonempty)
Goal: ∃ k ∈ spectrum 𝕜 a, ↑‖k‖₊ = spectralRadius 𝕜 a
-- spectrum.spectralRadius_lt_of_forall_lt_of_nonempty
theorem spectralRadius_lt_of_forall_lt_of_nonempty [ProperSpace 𝕜] {a : A} (ha : (σ a).Nonempty)
    {r : ℝ≥0} (hr : ∀ k ∈ σ a, ‖k‖₊ < r) : spectralRadius 𝕜 a < r 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] {a : A} (ha : (spectrum 𝕜 a).Nonempty) {r : ℝ≥0} (hr : ∀ k ∈ spectrum 𝕜 a, ‖k‖₊ < r)
Goal: spectralRadius 𝕜 a < ↑r
-- spectrum.spectralRadius_le_pow_nnnorm_pow_one_div
theorem spectralRadius_le_pow_nnnorm_pow_one_div (a : A) (n : ℕ) :
    spectralRadius 𝕜 a ≤ (‖a ^ (n + 1)‖₊ : ℝ≥0∞) ^ (1 / (n + 1) : ℝ) *
      (‖(1 : A)‖₊ : ℝ≥0∞) ^ (1 / (n + 1) : ℝ) 
Arguments: (𝕜 : Type u_1) {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) (n : ℕ)
Goal: spectralRadius 𝕜 a ≤ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1))
-- spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div
theorem spectralRadius_le_liminf_pow_nnnorm_pow_one_div (a : A) :
    spectralRadius 𝕜 a ≤ atTop.liminf fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) 
Arguments: (𝕜 : Type u_1) {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: spectralRadius 𝕜 a ≤ Filter.liminf (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop
-- spectrum.hasDerivAt_resolvent
theorem hasDerivAt_resolvent {a : A} {k : 𝕜} (hk : k ∈ ρ a) :
    HasDerivAt (resolvent a) (-resolvent a k ^ 2) k 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (hk : k ∈ resolventSet 𝕜 a)
Goal: HasDerivAt (resolvent a) (-resolvent a k ^ 2) k
-- spectrum.eventually_isUnit_resolvent
theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: ∀ᶠ (z : 𝕜) in Bornology.cobounded 𝕜, IsUnit (resolvent a z)
-- spectrum.resolvent_isBigO_inv
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: resolvent a =O[Bornology.cobounded 𝕜] Inv.inv
-- spectrum.resolvent_tendsto_cobounded
theorem resolvent_tendsto_cobounded (a : A) : Tendsto (resolvent a) (cobounded 𝕜) (𝓝 0) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: Filter.Tendsto (resolvent a) (Bornology.cobounded 𝕜) (nhds 0)
-- spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ 
Arguments: (𝕜 : Type u_1) {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: HasFPowerSeriesOnBall (fun z => Ring.inverse (1 - z • a)) (fun n => ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n))
  0 (↑‖a‖₊)⁻¹
-- spectrum.isUnit_one_sub_smul_of_lt_inv_radius
theorem isUnit_one_sub_smul_of_lt_inv_radius {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹) :
    IsUnit (1 - z • a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹)
Goal: IsUnit (1 - z • a)
-- spectrum.differentiableOn_inverse_one_sub_smul
theorem differentiableOn_inverse_one_sub_smul [CompleteSpace A] {a : A} {r : ℝ≥0}
    (hr : (r : ℝ≥0∞) < (spectralRadius 𝕜 a)⁻¹) :
    DifferentiableOn 𝕜 (fun z : 𝕜 => Ring.inverse (1 - z • a)) (Metric.closedBall 0 r) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {r : ℝ≥0} (hr : ↑r < (spectralRadius 𝕜 a)⁻¹)
Goal: DifferentiableOn 𝕜 (fun z => Ring.inverse (1 - z • a)) (Metric.closedBall 0 ↑r)
-- spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius
theorem limsup_pow_nnnorm_pow_one_div_le_spectralRadius (a : A) :
    limsup (fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) atTop ≤ spectralRadius ℂ a 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A)
Goal: Filter.limsup (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop ≤ spectralRadius ℂ a
-- spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius
theorem pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A) :
    Tendsto (fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) atTop (𝓝 (spectralRadius ℂ a)) 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A)
Goal: Filter.Tendsto (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop (nhds (spectralRadius ℂ a))
-- spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius
theorem pow_norm_pow_one_div_tendsto_nhds_spectralRadius (a : A) :
    Tendsto (fun n : ℕ => ENNReal.ofReal (‖a ^ n‖ ^ (1 / n : ℝ))) atTop
      (𝓝 (spectralRadius ℂ a)) 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A)
Goal: Filter.Tendsto (fun n => ENNReal.ofReal (‖a ^ n‖ ^ (1 / ↑n))) Filter.atTop (nhds (spectralRadius ℂ a))
-- spectrum.nonempty
protected theorem nonempty : (spectrum ℂ a).Nonempty 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A)
Goal: (spectrum ℂ a).Nonempty
-- spectrum.exists_nnnorm_eq_spectralRadius
theorem exists_nnnorm_eq_spectralRadius : ∃ z ∈ spectrum ℂ a, (‖z‖₊ : ℝ≥0∞) = spectralRadius ℂ a 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A)
Goal: ∃ z ∈ spectrum ℂ a, ↑‖z‖₊ = spectralRadius ℂ a
-- spectrum.spectralRadius_lt_of_forall_lt
theorem spectralRadius_lt_of_forall_lt {r : ℝ≥0} (hr : ∀ z ∈ spectrum ℂ a, ‖z‖₊ < r) :
    spectralRadius ℂ a < r 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) {r : ℝ≥0} (hr : ∀ z ∈ spectrum ℂ a, ‖z‖₊ < r)
Goal: spectralRadius ℂ a < ↑r
-- spectrum.map_polynomial_aeval
theorem map_polynomial_aeval (p : ℂ[X]) :
    spectrum ℂ (aeval a p) = (fun k => eval k p) '' spectrum ℂ a 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) (p : Polynomial ℂ)
Goal: spectrum ℂ ((Polynomial.aeval a) p) = (fun k => Polynomial.eval k p) '' spectrum ℂ a
-- spectrum.map_pow
protected theorem map_pow (n : ℕ) :
    spectrum ℂ (a ^ n) = (· ^ n) '' spectrum ℂ a 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) (n : ℕ)
Goal: spectrum ℂ (a ^ n) = (fun x => x ^ n) '' spectrum ℂ a
-- spectrum.algebraMap_eq_of_mem
theorem algebraMap_eq_of_mem {a : A} {z : ℂ} (h : z ∈ σ a) : algebraMap ℂ A z = a 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) {a : A} {z : ℂ} (h : z ∈ spectrum ℂ a)
Goal: (algebraMap ℂ A) z = a
-- NormedRing.algEquivComplexOfComplete_apply
@[simps]
noncomputable def _root_.NormedRing.algEquivComplexOfComplete [CompleteSpace A] : ℂ ≃ₐ[ℂ] A 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] (a : ℂ)
Goal: (NormedRing.algEquivComplexOfComplete hA) a = (algebraMap ℂ A) a
-- NormedRing.algEquivComplexOfComplete_symm_apply
@[simps]
noncomputable def _root_.NormedRing.algEquivComplexOfComplete [CompleteSpace A] : ℂ ≃ₐ[ℂ] A 
Arguments: {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] (a : A)
Goal: (NormedRing.algEquivComplexOfComplete hA).symm a = ⋯.some
-- spectrum.exp_mem_exp
theorem exp_mem_exp [RCLike 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
    {z : 𝕜} (hz : z ∈ spectrum 𝕜 a) : exp 𝕜 z ∈ spectrum 𝕜 (exp 𝕜 a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) {z : 𝕜} (hz : z ∈ spectrum 𝕜 a)
Goal: NormedSpace.exp 𝕜 z ∈ spectrum 𝕜 (NormedSpace.exp 𝕜 a)
-- AlgHom.coe_toContinuousLinearMap
@[simp]
theorem coe_toContinuousLinearMap (φ : A →ₐ[𝕜] 𝕜) : ⇑φ.toContinuousLinearMap = φ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (φ : A →ₐ[𝕜] 𝕜)
Goal: ⇑φ.toContinuousLinearMap = ⇑φ
-- AlgHom.norm_apply_le_self_mul_norm_one
theorem norm_apply_le_self_mul_norm_one [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A) :
    ‖f a‖ ≤ ‖a‖ * ‖(1 : A)‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A)
Goal: ‖f a‖ ≤ ‖a‖ * ‖1‖
-- AlgHom.norm_apply_le_self
theorem norm_apply_le_self [NormOneClass A] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜]
    (f : F) (a : A) : ‖f a‖ ≤ ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A)
Goal: ‖f a‖ ≤ ‖a‖
-- AlgHom.toContinuousLinearMap_norm
@[simp]
theorem toContinuousLinearMap_norm [NormOneClass A] (φ : A →ₐ[𝕜] 𝕜) :
    ‖φ.toContinuousLinearMap‖ = 1 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (φ : A →ₐ[𝕜] 𝕜)
Goal: ‖φ.toContinuousLinearMap‖ = 1
-- WeakDual.CharacterSpace.equivAlgHom_coe
@[simp]
theorem equivAlgHom_coe (f : characterSpace 𝕜 A) : ⇑(equivAlgHom f) = f 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] (f : ↑(WeakDual.characterSpace 𝕜 A))
Goal: ⇑(WeakDual.CharacterSpace.equivAlgHom f) = ⇑f
-- WeakDual.CharacterSpace.equivAlgHom_symm_coe
@[simp]
theorem equivAlgHom_symm_coe (f : A →ₐ[𝕜] 𝕜) : ⇑(equivAlgHom.symm f) = f 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] (f : A →ₐ[𝕜] 𝕜)
Goal: ⇑(WeakDual.CharacterSpace.equivAlgHom.symm f) = ⇑f
-- SpectrumRestricts.spectralRadius_eq
lemma spectralRadius_eq {𝕜₁ 𝕜₂ A : Type*} [NormedField 𝕜₁] [NormedField 𝕜₂]
    [NormedRing A] [NormedAlgebra 𝕜₁ A] [NormedAlgebra 𝕜₂ A] [NormedAlgebra 𝕜₁ 𝕜₂]
    [IsScalarTower 𝕜₁ 𝕜₂ A] {f : 𝕜₂ → 𝕜₁} {a : A} (h : SpectrumRestricts a f) :
    spectralRadius 𝕜₁ a = spectralRadius 𝕜₂ a 
Arguments: {𝕜₁ : Type u_3} {𝕜₂ : Type u_4} {A : Type u_5} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedRing A] [NormedAlgebra 𝕜₁ A] [NormedAlgebra 𝕜₂ A] [NormedAlgebra 𝕜₁ 𝕜₂] [IsScalarTower 𝕜₁ 𝕜₂ A] {f : 𝕜₂ → 𝕜₁} {a : A} (h : SpectrumRestricts a f)
Goal: spectralRadius 𝕜₁ a = spectralRadius 𝕜₂ a
-- SpectrumRestricts.nnreal_iff
lemma nnreal_iff [Algebra ℝ A] {a : A} :
    SpectrumRestricts a ContinuousMap.realToNNReal ↔ ∀ x ∈ spectrum ℝ a, 0 ≤ x 
Arguments: {A : Type u_3} [Ring A] [Algebra ℝ A] {a : A}
Goal: SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ∀ x ∈ spectrum ℝ a, 0 ≤ x
-- SpectrumRestricts.nnreal_of_nonneg
lemma nnreal_of_nonneg {A : Type*} [Ring A] [PartialOrder A] [Algebra ℝ A]
    [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a) :
    SpectrumRestricts a ContinuousMap.realToNNReal 
Arguments: {A : Type u_4} [Ring A] [PartialOrder A] [Algebra ℝ A] [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a)
Goal: SpectrumRestricts a ⇑ContinuousMap.realToNNReal
-- SpectrumRestricts.real_iff
lemma real_iff [Algebra ℂ A] {a : A} :
    SpectrumRestricts a Complex.reCLM ↔ ∀ x ∈ spectrum ℂ a, x = x.re 
Arguments: {A : Type u_3} [Ring A] [Algebra ℂ A] {a : A}
Goal: SpectrumRestricts a ⇑Complex.reCLM ↔ ∀ x ∈ spectrum ℂ a, x = ↑x.re
-- SpectrumRestricts.nnreal_iff_spectralRadius_le
lemma nnreal_iff_spectralRadius_le [Algebra ℝ A] {a : A} {t : ℝ≥0} (ht : spectralRadius ℝ a ≤ t) :
    SpectrumRestricts a ContinuousMap.realToNNReal ↔
      spectralRadius ℝ (algebraMap ℝ A t - a) ≤ t 
Arguments: {A : Type u_3} [Ring A] [Algebra ℝ A] {a : A} {t : ℝ≥0} (ht : spectralRadius ℝ a ≤ ↑t)
Goal: SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
-- NNReal.spectralRadius_mem_spectrum
lemma _root_.NNReal.spectralRadius_mem_spectrum {A : Type*} [NormedRing A] [NormedAlgebra ℝ A]
    [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty)
    (ha' : SpectrumRestricts a ContinuousMap.realToNNReal) :
    (spectralRadius ℝ a).toNNReal ∈ spectrum ℝ≥0 a 
Arguments: {A : Type u_4} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty) (ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal)
Goal: (spectralRadius ℝ a).toNNReal ∈ spectrum ℝ≥0 a
-- Real.spectralRadius_mem_spectrum
lemma _root_.Real.spectralRadius_mem_spectrum {A : Type*} [NormedRing A] [NormedAlgebra ℝ A]
    [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty)
    (ha' : SpectrumRestricts a ContinuousMap.realToNNReal) :
    (spectralRadius ℝ a).toReal ∈ spectrum ℝ a 
Arguments: {A : Type u_4} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty) (ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal)
Goal: (spectralRadius ℝ a).toReal ∈ spectrum ℝ a
-- Real.spectralRadius_mem_spectrum_or
lemma _root_.Real.spectralRadius_mem_spectrum_or {A : Type*} [NormedRing A] [NormedAlgebra ℝ A]
    [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty) :
    (spectralRadius ℝ a).toReal ∈ spectrum ℝ a ∨ -(spectralRadius ℝ a).toReal ∈ spectrum ℝ a 
Arguments: {A : Type u_4} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {a : A} (ha : (spectrum ℝ a).Nonempty)
Goal: (spectralRadius ℝ a).toReal ∈ spectrum ℝ a ∨ -(spectralRadius ℝ a).toReal ∈ spectrum ℝ a
-- QuasispectrumRestricts.compactSpace
lemma compactSpace {R S A : Type*} [Semifield R] [Field S] [NonUnitalRing A]
    [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A]
    [IsScalarTower R S A] [TopologicalSpace R] [TopologicalSpace S] {a : A} (f : C(S, R))
    (h : QuasispectrumRestricts a f) [h_cpct : CompactSpace (σₙ S a)] :
    CompactSpace (σₙ R a) 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] [TopologicalSpace R] [TopologicalSpace S] {a : A} (f : C(S, R)) (h : QuasispectrumRestricts a ⇑f) [h_cpct : CompactSpace ↑(quasispectrum S a)]
Goal: CompactSpace ↑(quasispectrum R a)
-- QuasispectrumRestricts.nnreal_iff
lemma nnreal_iff [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] {a : A} :
    QuasispectrumRestricts a ContinuousMap.realToNNReal ↔ ∀ x ∈ σₙ ℝ a, 0 ≤ x 
Arguments: {A : Type u_3} [NonUnitalRing A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] {a : A}
Goal: QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ∀ x ∈ quasispectrum ℝ a, 0 ≤ x
-- QuasispectrumRestricts.nnreal_of_nonneg
lemma nnreal_of_nonneg [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [PartialOrder A]
    [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a) :
    QuasispectrumRestricts a ContinuousMap.realToNNReal 
Arguments: {A : Type u_3} [NonUnitalRing A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [PartialOrder A] [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a)
Goal: QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
-- QuasispectrumRestricts.real_iff
lemma real_iff [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] {a : A} :
    QuasispectrumRestricts a Complex.reCLM ↔ ∀ x ∈ σₙ ℂ a, x = x.re 
Arguments: {A : Type u_3} [NonUnitalRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] {a : A}
Goal: QuasispectrumRestricts a ⇑Complex.reCLM ↔ ∀ x ∈ quasispectrum ℂ a, x = ↑x.re
-- coe_mem_spectrum_real_of_nonneg
lemma coe_mem_spectrum_real_of_nonneg [Algebra ℝ A] [NonnegSpectrumClass ℝ A] {a : A} {x : ℝ≥0}
    (ha : 0 ≤ a := by cfc_tac) :
    (x : ℝ) ∈ spectrum ℝ a ↔ x ∈ spectrum ℝ≥0 a 
Arguments: {A : Type u_3} [Ring A] [PartialOrder A] [Algebra ℝ A] [NonnegSpectrumClass ℝ A] {a : A} {x : ℝ≥0} (ha : 0 ≤ a := by cfc_tac)
Goal: ↑x ∈ spectrum ℝ a ↔ x ∈ spectrum ℝ≥0 a
-- conformalAt_id
theorem conformalAt_id (x : X) : ConformalAt _root_.id x 
Arguments: {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] (x : X)
Goal: ConformalAt id x
-- conformalAt_const_smul
theorem conformalAt_const_smul {c : ℝ} (h : c ≠ 0) (x : X) : ConformalAt (fun x' : X => c • x') x 
Arguments: {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {c : ℝ} (h : c ≠ 0) (x : X)
Goal: ConformalAt (fun x' => c • x') x
-- Subsingleton.conformalAt
@[nontriviality]
theorem Subsingleton.conformalAt [Subsingleton X] (f : X → Y) (x : X) : ConformalAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] [Subsingleton X] (f : X → Y) (x : X)
Goal: ConformalAt f x
-- conformalAt_iff_isConformalMap_fderiv
theorem conformalAt_iff_isConformalMap_fderiv {f : X → Y} {x : X} :
    ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} {x : X}
Goal: ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x)
-- ConformalAt.differentiableAt
theorem differentiableAt {f : X → Y} {x : X} (h : ConformalAt f x) : DifferentiableAt ℝ f x 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} {x : X} (h : ConformalAt f x)
Goal: DifferentiableAt ℝ f x
-- ConformalAt.congr
theorem congr {f g : X → Y} {x : X} {u : Set X} (hx : x ∈ u) (hu : IsOpen u) (hf : ConformalAt f x)
    (h : ∀ x : X, x ∈ u → g x = f x) : ConformalAt g x 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f g : X → Y} {x : X} {u : Set X} (hx : x ∈ u) (hu : IsOpen u) (hf : ConformalAt f x) (h : ∀ x ∈ u, g x = f x)
Goal: ConformalAt g x
-- ConformalAt.comp
theorem comp {f : X → Y} {g : Y → Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
    ConformalAt (g ∘ f) x 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedAddCommGroup Z] [NormedSpace ℝ X] [NormedSpace ℝ Y] [NormedSpace ℝ Z] {f : X → Y} {g : Y → Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x)
Goal: ConformalAt (g ∘ f) x
-- ConformalAt.const_smul
theorem const_smul {f : X → Y} {x : X} {c : ℝ} (hc : c ≠ 0) (hf : ConformalAt f x) :
    ConformalAt (c • f) x 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} {x : X} {c : ℝ} (hc : c ≠ 0) (hf : ConformalAt f x)
Goal: ConformalAt (c • f) x
-- conformal_id
theorem conformal_id : Conformal (id : X → X) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X]
Goal: Conformal id
-- conformal_const_smul
theorem conformal_const_smul {c : ℝ} (h : c ≠ 0) : Conformal fun x : X => c • x 
Arguments: {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {c : ℝ} (h : c ≠ 0)
Goal: Conformal fun x => c • x
-- Conformal.conformalAt
theorem conformalAt {f : X → Y} (h : Conformal f) (x : X) : ConformalAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} (h : Conformal f) (x : X)
Goal: ConformalAt f x
-- Conformal.differentiable
theorem differentiable {f : X → Y} (h : Conformal f) : Differentiable ℝ f 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} (h : Conformal f)
Goal: Differentiable ℝ f
-- Conformal.comp
theorem comp {f : X → Y} {g : Y → Z} (hf : Conformal f) (hg : Conformal g) : Conformal (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedAddCommGroup Z] [NormedSpace ℝ X] [NormedSpace ℝ Y] [NormedSpace ℝ Z] {f : X → Y} {g : Y → Z} (hf : Conformal f) (hg : Conformal g)
Goal: Conformal (g ∘ f)
-- Conformal.const_smul
theorem const_smul {f : X → Y} (hf : Conformal f) {c : ℝ} (hc : c ≠ 0) : Conformal (c • f) 
Arguments: {X : Type u_1} {Y : Type u_2} [NormedAddCommGroup X] [NormedAddCommGroup Y] [NormedSpace ℝ X] [NormedSpace ℝ Y] {f : X → Y} (hf : Conformal f) {c : ℝ} (hc : c ≠ 0)
Goal: Conformal (c • f)
-- IsExtrFilter.hasLineDerivAt_eq_zero
theorem IsExtrFilter.hasLineDerivAt_eq_zero {l : Filter E} (h : IsExtrFilter f l a)
    (hd : HasLineDerivAt ℝ f f' a b) (h' : Tendsto (fun t : ℝ ↦ a + t • b) (𝓝 0) l) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} {l : Filter E} (h : IsExtrFilter f l a) (hd : HasLineDerivAt ℝ f f' a b) (h' : Filter.Tendsto (fun t => a + t • b) (𝓝 0) l)
Goal: f' = 0
-- IsExtrFilter.lineDeriv_eq_zero
theorem IsExtrFilter.lineDeriv_eq_zero {l : Filter E} (h : IsExtrFilter f l a)
    (h' : Tendsto (fun t : ℝ ↦ a + t • b) (𝓝 0) l) : lineDeriv ℝ f a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {a b : E} {l : Filter E} (h : IsExtrFilter f l a) (h' : Filter.Tendsto (fun t => a + t • b) (𝓝 0) l)
Goal: lineDeriv ℝ f a b = 0
-- IsExtrOn.hasLineDerivAt_eq_zero
theorem IsExtrOn.hasLineDerivAt_eq_zero (h : IsExtrOn f s a) (hd : HasLineDerivAt ℝ f f' a b)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsExtrOn f s a) (hd : HasLineDerivAt ℝ f f' a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0
-- IsExtrOn.lineDeriv_eq_zero
theorem IsExtrOn.lineDeriv_eq_zero (h : IsExtrOn f s a) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) :
    lineDeriv ℝ f a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsExtrOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDeriv ℝ f a b = 0
-- IsMinOn.hasLineDerivAt_eq_zero
theorem IsMinOn.hasLineDerivAt_eq_zero (h : IsMinOn f s a) (hd : HasLineDerivAt ℝ f f' a b)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsMinOn f s a) (hd : HasLineDerivAt ℝ f f' a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0
-- IsMinOn.lineDeriv_eq_zero
theorem IsMinOn.lineDeriv_eq_zero (h : IsMinOn f s a) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) :
    lineDeriv ℝ f a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsMinOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDeriv ℝ f a b = 0
-- IsMaxOn.hasLineDerivAt_eq_zero
theorem IsMaxOn.hasLineDerivAt_eq_zero (h : IsMaxOn f s a) (hd : HasLineDerivAt ℝ f f' a b)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsMaxOn f s a) (hd : HasLineDerivAt ℝ f f' a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0
-- IsMaxOn.lineDeriv_eq_zero
theorem IsMaxOn.lineDeriv_eq_zero (h : IsMaxOn f s a) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) :
    lineDeriv ℝ f a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsMaxOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDeriv ℝ f a b = 0
-- IsExtrOn.hasLineDerivWithinAt_eq_zero
theorem IsExtrOn.hasLineDerivWithinAt_eq_zero (h : IsExtrOn f s a)
    (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsExtrOn f s a) (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0