ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.continuous_arctan
@[continuity]
theorem continuous_arctan : Continuous arctan 
Goal: Continuous Real.arctan
-- Real.continuousAt_arctan
theorem continuousAt_arctan {x : ℝ} : ContinuousAt arctan x 
Arguments: {x : ℝ}
Goal: ContinuousAt Real.arctan x
-- Real.coe_tanPartialHomeomorph
@[simp]
theorem coe_tanPartialHomeomorph : ⇑tanPartialHomeomorph = tan 
Goal: ↑Real.tanPartialHomeomorph = Real.tan
-- Real.coe_tanPartialHomeomorph_symm
@[simp]
theorem coe_tanPartialHomeomorph_symm : ⇑tanPartialHomeomorph.symm = arctan 
Goal: ↑Real.tanPartialHomeomorph.symm = Real.arctan
-- Real.sin_lt
theorem sin_lt (h : 0 < x) : sin x < x 
Arguments: {x : ℝ} (h : 0 < x)
Goal: Real.sin x < x
-- Real.sin_le
lemma sin_le (hx : 0 ≤ x) : sin x ≤ x 
Arguments: {x : ℝ} (hx : 0 ≤ x)
Goal: Real.sin x ≤ x
-- Real.lt_sin
lemma lt_sin (hx : x < 0) : x < sin x 
Arguments: {x : ℝ} (hx : x < 0)
Goal: x < Real.sin x
-- Real.le_sin
lemma le_sin (hx : x ≤ 0) : x ≤ sin x 
Arguments: {x : ℝ} (hx : x ≤ 0)
Goal: x ≤ Real.sin x
-- Real.one_sub_sq_div_two_le_cos
lemma one_sub_sq_div_two_le_cos : 1 - x ^ 2 / 2 ≤ cos x 
Arguments: {x : ℝ}
Goal: 1 - x ^ 2 / 2 ≤ Real.cos x
-- Real.two_div_pi_mul_le_sin
lemma two_div_pi_mul_le_sin (hx₀ : 0 ≤ x) (hx : x ≤ π / 2) : 2 / π * x ≤ sin x 
Arguments: {x : ℝ} (hx₀ : 0 ≤ x) (hx : x ≤ Real.pi / 2)
Goal: 2 / Real.pi * x ≤ Real.sin x
-- Real.sin_le_two_div_pi_mul
lemma sin_le_two_div_pi_mul (hx : -(π / 2) ≤ x) (hx₀ : x ≤ 0) : sin x ≤ 2 / π * x 
Arguments: {x : ℝ} (hx : -(Real.pi / 2) ≤ x) (hx₀ : x ≤ 0)
Goal: Real.sin x ≤ 2 / Real.pi * x
-- Real.one_sub_two_div_pi_mul_le_cos
lemma one_sub_two_div_pi_mul_le_cos (hx₀ : 0 ≤ x) (hx : x ≤ π / 2) : 1 - 2 / π * x ≤ cos x 
Arguments: {x : ℝ} (hx₀ : 0 ≤ x) (hx : x ≤ Real.pi / 2)
Goal: 1 - 2 / Real.pi * x ≤ Real.cos x
-- Real.cos_quadratic_upper_bound
lemma cos_quadratic_upper_bound (hx : |x| ≤ π) : cos x ≤ 1 - 2 / π ^ 2 * x ^ 2 
Arguments: {x : ℝ} (hx : |x| ≤ Real.pi)
Goal: Real.cos x ≤ 1 - 2 / Real.pi ^ 2 * x ^ 2
-- Real.sin_gt_sub_cube
theorem sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < sin x 
Arguments: {x : ℝ} (h : 0 < x) (h' : x ≤ 1)
Goal: x - x ^ 3 / 4 < Real.sin x
-- Real.deriv_tan_sub_id
theorem deriv_tan_sub_id (x : ℝ) (h : cos x ≠ 0) :
    deriv (fun y : ℝ => tan y - y) x = 1 / cos x ^ 2 - 1 
Arguments: (x : ℝ) (h : Real.cos x ≠ 0)
Goal: deriv (fun y => Real.tan y - y) x = 1 / Real.cos x ^ 2 - 1
-- Real.lt_tan
theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x 
Arguments: {x : ℝ} (h1 : 0 < x) (h2 : x < Real.pi / 2)
Goal: x < Real.tan x
-- Real.le_tan
theorem le_tan {x : ℝ} (h1 : 0 ≤ x) (h2 : x < π / 2) : x ≤ tan x 
Arguments: {x : ℝ} (h1 : 0 ≤ x) (h2 : x < Real.pi / 2)
Goal: x ≤ Real.tan x
-- Real.cos_lt_one_div_sqrt_sq_add_one
theorem cos_lt_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2)
    (hx3 : x ≠ 0) : cos x < (1 / √(x ^ 2 + 1) : ℝ) 
Arguments: {x : ℝ} (hx1 : -(3 * Real.pi / 2) ≤ x) (hx2 : x ≤ 3 * Real.pi / 2) (hx3 : x ≠ 0)
Goal: Real.cos x < 1 / √(x ^ 2 + 1)
-- Real.cos_le_one_div_sqrt_sq_add_one
theorem cos_le_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2) :
    cos x ≤ (1 : ℝ) / √(x ^ 2 + 1) 
Arguments: {x : ℝ} (hx1 : -(3 * Real.pi / 2) ≤ x) (hx2 : x ≤ 3 * Real.pi / 2)
Goal: Real.cos x ≤ 1 / √(x ^ 2 + 1)
-- NonUnitalContinuousFunctionalCalculus.predicate_zero
class NonUnitalContinuousFunctionalCalculus (R : Type*) {A : Type*} (p : outParam (A → Prop))
    [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R]
    [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A]
    [IsScalarTower R A A] [SMulCommClass R A A] : Prop where
  predicate_zero : p 0
  exists_cfc_of_predicate : ∀ a, p a → ∃ φ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A,
    ClosedEmbedding φ ∧ φ ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a ∧
      (∀ f, σₙ R (φ f) = Set.range f) ∧ ∀ f, p (φ f)
Arguments: (R : Type u_1) {A : Type u_2} {p : outParam (A → Prop)} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [self : NonUnitalContinuousFunctionalCalculus R p]
Goal: p 0
-- NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate
class NonUnitalContinuousFunctionalCalculus (R : Type*) {A : Type*} (p : outParam (A → Prop))
    [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R]
    [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A]
    [IsScalarTower R A A] [SMulCommClass R A A] : Prop where
  predicate_zero : p 0
  exists_cfc_of_predicate : ∀ a, p a → ∃ φ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A,
    ClosedEmbedding φ ∧ φ ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a ∧
      (∀ f, σₙ R (φ f) = Set.range f) ∧ ∀ f, p (φ f)
Arguments: {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [self : NonUnitalContinuousFunctionalCalculus R p] (a : A)
Goal: p a →
  ∃ φ,
    ClosedEmbedding ⇑φ ∧
      φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a ∧
        (∀ (f : C(↑(σₙ R a), R)₀), σₙ R (φ f) = Set.range ⇑f) ∧ ∀ (f : C(↑(σₙ R a), R)₀), p (φ f)
-- UniqueNonUnitalContinuousFunctionalCalculus.compactSpace_quasispectrum
class UniqueNonUnitalContinuousFunctionalCalculus (R A : Type*) [CommSemiring R] [StarRing R]
    [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A]
    [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] : Prop where
  eq_of_continuous_of_map_id (s : Set R) [CompactSpace s] [Zero s] (h0 : (0 : s) = (0 : R))
    (φ ψ : C(s, R)₀ →⋆ₙₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (⟨.restrict s <| .id R, h0⟩) = ψ (⟨.restrict s <| .id R, h0⟩)) :
    φ = ψ
  compactSpace_quasispectrum (a : A) : CompactSpace (σₙ R a)
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [self : UniqueNonUnitalContinuousFunctionalCalculus R A] (a : A)
Goal: CompactSpace ↑(σₙ R a)
-- UniqueNonUnitalContinuousFunctionalCalculus.eq_of_continuous_of_map_id
class UniqueNonUnitalContinuousFunctionalCalculus (R A : Type*) [CommSemiring R] [StarRing R]
    [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A]
    [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] : Prop where
  eq_of_continuous_of_map_id (s : Set R) [CompactSpace s] [Zero s] (h0 : (0 : s) = (0 : R))
    (φ ψ : C(s, R)₀ →⋆ₙₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (⟨.restrict s <| .id R, h0⟩) = ψ (⟨.restrict s <| .id R, h0⟩)) :
    φ = ψ
  compactSpace_quasispectrum (a : A) : CompactSpace (σₙ R a)
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [self : UniqueNonUnitalContinuousFunctionalCalculus R A] (s : Set R) [CompactSpace ↑s] [Zero ↑s] (h0 : ↑0 = 0) (φ ψ : C(↑s, R)₀ →⋆ₙₐ[R] A) (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h :
  φ { toContinuousMap := ContinuousMap.restrict s (ContinuousMap.id R), map_zero' := h0 } =
    ψ { toContinuousMap := ContinuousMap.restrict s (ContinuousMap.id R), map_zero' := h0 })
Goal: φ = ψ
-- NonUnitalStarAlgHom.ext_continuousMap
lemma NonUnitalStarAlgHom.ext_continuousMap [UniqueNonUnitalContinuousFunctionalCalculus R A]
    (a : A) (φ ψ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ ⟨.restrict (σₙ R a) <| .id R, rfl⟩ = ψ ⟨.restrict (σₙ R a) <| .id R, rfl⟩) :
    φ = ψ 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [UniqueNonUnitalContinuousFunctionalCalculus R A] (a : A) (φ ψ : C(↑(σₙ R a), R)₀ →⋆ₙₐ[R] A) (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h :
  φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
    ψ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ })
Goal: φ = ψ
-- cfcₙHom_closedEmbedding
lemma cfcₙHom_closedEmbedding :
    ClosedEmbedding <| (cfcₙHom ha : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a)
Goal: ClosedEmbedding ⇑(cfcₙHom ha)
-- cfcₙHom_id
lemma cfcₙHom_id :
    cfcₙHom ha ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a)
Goal: (cfcₙHom ha) { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a
-- cfcₙHom_map_quasispectrum
lemma cfcₙHom_map_quasispectrum (f : C(σₙ R a, R)₀) :
    σₙ R (cfcₙHom ha f) = Set.range f 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a) (f : C(↑(σₙ R a), R)₀)
Goal: σₙ R ((cfcₙHom ha) f) = Set.range ⇑f
-- cfcₙHom_predicate
lemma cfcₙHom_predicate (f : C(σₙ R a, R)₀) :
    p (cfcₙHom ha f) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a) (f : C(↑(σₙ R a), R)₀)
Goal: p ((cfcₙHom ha) f)
-- cfcₙHom_eq_of_continuous_of_map_id
lemma cfcₙHom_eq_of_continuous_of_map_id [UniqueNonUnitalContinuousFunctionalCalculus R A]
    (φ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) (hφ₁ : Continuous φ)
    (hφ₂ : φ ⟨.restrict (σₙ R a) <| .id R, rfl⟩ = a) : cfcₙHom ha = φ 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a) [UniqueNonUnitalContinuousFunctionalCalculus R A] (φ : C(↑(σₙ R a), R)₀ →⋆ₙₐ[R] A) (hφ₁ : Continuous ⇑φ) (hφ₂ : φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a)
Goal: cfcₙHom ha = φ
-- cfcₙHom_comp
theorem cfcₙHom_comp [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(σₙ R a, R)₀)
    (f' : C(σₙ R a, σₙ R (cfcₙHom ha f))₀)
    (hff' : ∀ x, f x = f' x) (g : C(σₙ R (cfcₙHom ha f), R)₀) :
    cfcₙHom ha (g.comp f') = cfcₙHom (cfcₙHom_predicate ha f) g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a) [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(↑(σₙ R a), R)₀) (f' : C(↑(σₙ R a), ↑(σₙ R ((cfcₙHom ha) f)))₀) (hff' : ∀ (x : ↑(σₙ R a)), f x = ↑(f' x)) (g : C(↑(σₙ R ((cfcₙHom ha) f)), R)₀)
Goal: (cfcₙHom ha) (g.comp f') = (cfcₙHom ⋯) g
-- cfcₙ_def
noncomputable irreducible_def cfcₙ (f : R → R) (a : A) : A 
Arguments: {R : Type u_3} {A : Type u_4} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: cfcₙ f a =
  if h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0 then
    (cfcₙHom ⋯) { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := ⋯ }
  else 0
-- cfcₙ_apply
lemma cfcₙ_apply : cfcₙ f a = cfcₙHom (a := a) ha ⟨⟨_, hf.restrict⟩, hf0⟩ 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ f a = (cfcₙHom ha) { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := hf0 }
-- cfcₙ_apply_pi
lemma cfcₙ_apply_pi {ι : Type*} (f : ι → R → R) (a : A) (ha := by cfc_tac)
    (hf : ∀ i, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
    (hf0 : ∀ i, f i 0 = 0 := by cfc_zero_tac) :
    (fun i => cfcₙ (f i) a) = (fun i => cfcₙHom (a := a) ha ⟨⟨_, (hf i).restrict⟩, hf0 i⟩) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {ι : Type u_3} (f : ι → R → R) (a : A) (ha : p a := by cfc_tac) (hf : ∀ (i : ι), ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac) (hf0 : ∀ (i : ι), f i 0 = 0 := by cfc_zero_tac)
Goal: (fun i => cfcₙ (f i) a) = fun i =>
  (cfcₙHom ha) { toFun := (σₙ R a).restrict (f i), continuous_toFun := ⋯, map_zero' := ⋯ }
-- cfcₙ_apply_of_not_and_and
lemma cfcₙ_apply_of_not_and_and {f : R → R} (a : A)
    (ha : ¬ (p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)) :
    cfcₙ f a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f : R → R} (a : A) (ha : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0))
Goal: cfcₙ f a = 0
-- cfcₙ_apply_of_not_predicate
lemma cfcₙ_apply_of_not_predicate {f : R → R} (a : A) (ha : ¬ p a) :
    cfcₙ f a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f : R → R} (a : A) (ha : ¬p a)
Goal: cfcₙ f a = 0
-- cfcₙ_apply_of_not_continuousOn
lemma cfcₙ_apply_of_not_continuousOn {f : R → R} (a : A) (hf : ¬ ContinuousOn f (σₙ R a)) :
    cfcₙ f a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f : R → R} (a : A) (hf : ¬ContinuousOn f (σₙ R a))
Goal: cfcₙ f a = 0
-- cfcₙ_apply_of_not_map_zero
lemma cfcₙ_apply_of_not_map_zero {f : R → R} (a : A) (hf : ¬ f 0 = 0) :
    cfcₙ f a = 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f : R → R} (a : A) (hf : ¬f 0 = 0)
Goal: cfcₙ f a = 0
-- cfcₙHom_eq_cfcₙ_extend
lemma cfcₙHom_eq_cfcₙ_extend {a : A} (g : R → R) (ha : p a) (f : C(σₙ R a, R)₀) :
    cfcₙHom ha f = cfcₙ (Function.extend Subtype.val f g) a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (g : R → R) (ha : p a) (f : C(↑(σₙ R a), R)₀)
Goal: (cfcₙHom ha) f = cfcₙ (Function.extend Subtype.val (⇑f) g) a
-- cfcₙ_cases
lemma cfcₙ_cases (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0)
    (haf : ∀ (hf : ContinuousOn f (σₙ R a)) h0 ha, P (cfcₙHom ha ⟨⟨_, hf.restrict⟩, h0⟩)) :
    P (cfcₙ f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0) (haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := h0 }))
Goal: P (cfcₙ f a)
-- cfcₙ_id
lemma cfcₙ_id : cfcₙ (id : R → R) a = a 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfcₙ id a = a
-- cfcₙ_id'
lemma cfcₙ_id' : cfcₙ (fun x : R ↦ x) a = a 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => x) a = a
-- cfcₙ_map_quasispectrum
lemma cfcₙ_map_quasispectrum : σₙ R (cfcₙ f a) = f '' σₙ R a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: σₙ R (cfcₙ f a) = f '' σₙ R a
-- cfcₙ_predicate_zero
lemma cfcₙ_predicate_zero : p 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p]
Goal: p 0
-- cfcₙ_predicate
lemma cfcₙ_predicate (f : R → R) (a : A) : p (cfcₙ f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: p (cfcₙ f a)
-- cfcₙ_congr
lemma cfcₙ_congr {f g : R → R} {a : A} (hfg : (σₙ R a).EqOn f g) :
    cfcₙ f a = cfcₙ g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (hfg : Set.EqOn f g (σₙ R a))
Goal: cfcₙ f a = cfcₙ g a
-- eqOn_of_cfcₙ_eq_cfcₙ
lemma eqOn_of_cfcₙ_eq_cfcₙ {f g : R → R} {a : A} (h : cfcₙ f a = cfcₙ g a) (ha : p a := by cfc_tac)
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
    (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
    (σₙ R a).EqOn f g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (h : cfcₙ f a = cfcₙ g a) (ha : p a := by cfc_tac) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
Goal: Set.EqOn f g (σₙ R a)
-- cfcₙ_eq_cfcₙ_iff_eqOn
lemma cfcₙ_eq_cfcₙ_iff_eqOn {f g : R → R} {a : A} (ha : p a := by cfc_tac)
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
    (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
    cfcₙ f a = cfcₙ g a ↔ (σₙ R a).EqOn f g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (ha : p a := by cfc_tac) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ f a = cfcₙ g a ↔ Set.EqOn f g (σₙ R a)
-- cfcₙ_zero
@[simp]
lemma cfcₙ_zero : cfcₙ (0 : R → R) a = 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (a : A)
Goal: cfcₙ 0 a = 0
-- cfcₙ_const_zero
@[simp]
lemma cfcₙ_const_zero : cfcₙ (fun _ : R ↦ 0) a = 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (a : A)
Goal: cfcₙ (fun x => 0) a = 0
-- cfcₙ_mul
lemma cfcₙ_mul : cfcₙ (fun x ↦ f x * g x) a = cfcₙ f a * cfcₙ g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f g : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ (fun x => f x * g x) a = cfcₙ f a * cfcₙ g a