ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- cfcₙ_add
lemma cfcₙ_add : 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
-- cfcₙ_sum
lemma cfcₙ_sum {ι : Type*} (f : ι → R → R) (a : A) (s : Finset ι)
    (hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
    (hf0 : ∀ i ∈ s, f i 0 = 0 := by cfc_zero_tac) :
    cfcₙ (∑ i in s, f i) a = ∑ i in s, cfcₙ (f i) 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] {ι : Type u_3} (f : ι → R → R) (a : A) (s : Finset ι) (hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac) (hf0 : ∀ i ∈ s, f i 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a
-- cfcₙ_sum_univ
lemma cfcₙ_sum_univ {ι : Type*} [Fintype ι] (f : ι → R → R) (a : A)
    (hf : ∀ i, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
    (hf0 : ∀ i, f i 0 = 0 := by cfc_zero_tac) :
    cfcₙ (∑ i, f i) a = ∑ i, cfcₙ (f i) 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] {ι : Type u_3} [Fintype ι] (f : ι → R → R) (a : A) (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: cfcₙ (∑ i : ι, f i) a = ∑ i : ι, cfcₙ (f i) a
-- cfcₙ_smul
lemma cfcₙ_smul {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
    [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
    (s : S) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (h0 : f 0 = 0 := by cfc_zero_tac) :
    cfcₙ (fun x ↦ s • f x) a = s • 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] {S : Type u_3} [SMulZeroClass S R] [ContinuousConstSMul S R] [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)] (s : S) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (h0 : f 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ (fun x => s • f x) a = s • cfcₙ f a
-- cfcₙ_const_mul
lemma cfcₙ_const_mul (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (h0 : f 0 = 0 := by cfc_zero_tac) :
    cfcₙ (fun x ↦ r * f x) a = r • 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] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (h0 : f 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ (fun x => r * f x) a = r • cfcₙ f a
-- cfcₙ_star
lemma cfcₙ_star : cfcₙ (fun x ↦ star (f x)) a = star (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: cfcₙ (fun x => star (f x)) a = star (cfcₙ f a)
-- cfcₙ_smul_id
lemma cfcₙ_smul_id {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
    [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
    (s : S) (a : A) (ha : p a := by cfc_tac) : cfcₙ (s • · : R → R) a = s • 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] {S : Type u_3} [SMulZeroClass S R] [ContinuousConstSMul S R] [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)] (s : S) (a : A) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => s • x) a = s • a
-- cfcₙ_const_mul_id
lemma cfcₙ_const_mul_id (r : R) (a : A) (ha : p a := by cfc_tac) : cfcₙ (r * ·) a = 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] (r : R) (a : A) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => r * x) a = r • a
-- cfcₙ_star_id
lemma cfcₙ_star_id : cfcₙ (star · : R → R) a = star 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 => star x) a = star a
-- cfcₙ_comp
lemma cfcₙ_comp (g f : R → R) (a : A)
    (hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
    (ha : p a := by cfc_tac) :
    cfcₙ (g ∘ f) a = cfcₙ g (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] [UniqueNonUnitalContinuousFunctionalCalculus R A] (g f : R → R) (a : A) (hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) (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 (g ∘ f) a = cfcₙ g (cfcₙ f a)
-- cfcₙ_comp'
lemma cfcₙ_comp' (g f : R → R) (a : A)
    (hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
    (ha : p a := by cfc_tac) :
    cfcₙ (g <| f ·) a = cfcₙ g (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] [UniqueNonUnitalContinuousFunctionalCalculus R A] (g f : R → R) (a : A) (hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) (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 (fun x => g (f x)) a = cfcₙ g (cfcₙ f a)
-- cfcₙ_comp_smul
lemma cfcₙ_comp_smul {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
    [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
    (s : S) (f : R → R) (a : A) (hf : ContinuousOn f ((s • ·) '' (σₙ R a)) := by cfc_cont_tac)
    (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ (f <| s • ·) a = cfcₙ f (s • 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] [UniqueNonUnitalContinuousFunctionalCalculus R A] {S : Type u_3} [SMulZeroClass S R] [ContinuousConstSMul S R] [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)] (s : S) (f : R → R) (a : A) (hf : ContinuousOn f ((fun x => s • x) '' σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => f (s • x)) a = cfcₙ f (s • a)
-- cfcₙ_comp_const_mul
lemma cfcₙ_comp_const_mul (r : R) (f : R → R) (a : A)
    (hf : ContinuousOn f ((r * ·) '' (σₙ R a)) := by cfc_cont_tac)
    (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ (f <| r * ·) a = cfcₙ f (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] [UniqueNonUnitalContinuousFunctionalCalculus R A] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f ((fun x => r * x) '' σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => f (r * x)) a = cfcₙ f (r • a)
-- cfcₙ_comp_star
lemma cfcₙ_comp_star (hf : ContinuousOn f (star '' (σₙ R a)) := by cfc_cont_tac)
    (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ (f <| star ·) a = cfcₙ f (star 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) [UniqueNonUnitalContinuousFunctionalCalculus R A] (hf : ContinuousOn f (star '' σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => f (star x)) a = cfcₙ f (star a)
-- CFC.eq_zero_of_quasispectrum_eq_zero
lemma CFC.eq_zero_of_quasispectrum_eq_zero (h_spec : σₙ R a ⊆ {0}) (ha : p a := by cfc_tac) :
    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) (h_spec : σₙ R a ⊆ {0}) (ha : p a := by cfc_tac)
Goal: a = 0
-- CFC.quasispectrum_zero_eq
lemma CFC.quasispectrum_zero_eq : σₙ 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]
Goal: σₙ R 0 = {0}
-- cfcₙ_apply_zero
@[simp] lemma cfcₙ_apply_zero {f : R → R} : cfcₙ f (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] {f : R → R}
Goal: cfcₙ f 0 = 0
-- cfcₙ_sub
lemma cfcₙ_sub : 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} [CommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [NonUnitalRing A] [StarRing 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
-- cfcₙ_neg
lemma cfcₙ_neg : cfcₙ (fun x ↦ - (f x)) a = - (cfcₙ f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: cfcₙ (fun x => -f x) a = -cfcₙ f a
-- cfcₙ_neg_id
lemma cfcₙ_neg_id (ha : p a := by cfc_tac) :
    cfcₙ (- · : R → R) a = -a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [NonUnitalRing A] [StarRing 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_comp_neg
lemma cfcₙ_comp_neg (hf : ContinuousOn f ((- ·) '' (σₙ R a)) := by cfc_cont_tac)
    (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ (f <| - ·) a = cfcₙ f (-a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A) [UniqueNonUnitalContinuousFunctionalCalculus R A] (hf : ContinuousOn f ((fun x => -x) '' σₙ R a) := by cfc_cont_tac) (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ (fun x => f (-x)) a = cfcₙ f (-a)
-- cfcₙHom_mono
lemma cfcₙHom_mono {a : A} (ha : p a) {f g : C(σₙ R a, R)₀} (hfg : f ≤ g) :
    cfcₙHom ha f ≤ cfcₙHom ha g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {a : A} (ha : p a) {f g : C(↑(σₙ R a), R)₀} (hfg : f ≤ g)
Goal: (cfcₙHom ha) f ≤ (cfcₙHom ha) g
-- cfcₙHom_nonneg_iff
lemma cfcₙHom_nonneg_iff {a : A} (ha : p a) {f : C(σₙ R a, R)₀} :
    0 ≤ cfcₙHom ha f ↔ 0 ≤ f 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] {a : A} (ha : p a) {f : C(↑(σₙ R a), R)₀}
Goal: 0 ≤ (cfcₙHom ha) f ↔ 0 ≤ f
-- cfcₙ_mono
lemma cfcₙ_mono {f g : R → R} {a : A} (h : ∀ x ∈ σₙ R a, f x ≤ g x)
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac)
    (hf0 : f 0 = 0 := by cfc_zero_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
    cfcₙ f a ≤ cfcₙ g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (h : ∀ x ∈ σₙ R a, f x ≤ g x) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ f a ≤ cfcₙ g a
-- cfcₙ_nonneg_iff
lemma cfcₙ_nonneg_iff (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    0 ≤ cfcₙ f a ↔ ∀ x ∈ σₙ R a, 0 ≤ f x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: 0 ≤ cfcₙ f a ↔ ∀ x ∈ σₙ R a, 0 ≤ f x
-- StarOrderedRing.nonneg_iff_quasispectrum_nonneg
lemma StarOrderedRing.nonneg_iff_quasispectrum_nonneg (a : A) (ha : p a := by cfc_tac) :
    0 ≤ a ↔ ∀ x ∈ quasispectrum R a, 0 ≤ x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (a : A) (ha : p a := by cfc_tac)
Goal: 0 ≤ a ↔ ∀ x ∈ σₙ R a, 0 ≤ x
-- cfcₙ_nonneg
lemma cfcₙ_nonneg {f : R → R} {a : A} (h : ∀ x ∈ σₙ R a, 0 ≤ f x) :
    0 ≤ cfcₙ f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] {f : R → R} {a : A} (h : ∀ x ∈ σₙ R a, 0 ≤ f x)
Goal: 0 ≤ cfcₙ f a
-- cfcₙ_nonpos
lemma cfcₙ_nonpos (f : R → R) (a : A) (h : ∀ x ∈ σₙ R a, f x ≤ 0) :
    cfcₙ f a ≤ 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] (f : R → R) (a : A) (h : ∀ x ∈ σₙ R a, f x ≤ 0)
Goal: cfcₙ f a ≤ 0
-- cfcₙHom_le_iff
lemma cfcₙHom_le_iff {a : A} (ha : p a) {f g : C(σₙ R a, R)₀} :
    cfcₙHom ha f ≤ cfcₙHom ha g ↔ f ≤ g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] {a : A} (ha : p a) {f g : C(↑(σₙ R a), R)₀}
Goal: (cfcₙHom ha) f ≤ (cfcₙHom ha) g ↔ f ≤ g
-- cfcₙ_le_iff
lemma cfcₙ_le_iff (f g : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
    (hg0 : g 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ f a ≤ cfcₙ g a ↔ ∀ x ∈ σₙ R a, f x ≤ g x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (f g : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ f a ≤ cfcₙ g a ↔ ∀ x ∈ σₙ R a, f x ≤ g x
-- cfcₙ_nonpos_iff
lemma cfcₙ_nonpos_iff (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
    (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
    cfcₙ f a ≤ 0 ↔ ∀ x ∈ σₙ R a, f x ≤ 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac)
Goal: cfcₙ f a ≤ 0 ↔ ∀ x ∈ σₙ R a, f x ≤ 0
-- closedEmbedding_cfcₙHom_of_cfcHom
lemma closedEmbedding_cfcₙHom_of_cfcHom {a : A} (ha : p a) :
    ClosedEmbedding (cfcₙHom_of_cfcHom R ha) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Field R] [StarRing R] [MetricSpace R] [CompleteSpace R] [TopologicalRing R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [ContinuousFunctionalCalculus R p] [h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)] {a : A} (ha : p a)
Goal: ClosedEmbedding ⇑(cfcₙHom_of_cfcHom R ha)
-- cfcₙHom_of_cfcHom_map_quasispectrum
lemma cfcₙHom_of_cfcHom_map_quasispectrum {a : A} (ha : p a) :
    ∀ f : C(σₙ R a, R)₀, σₙ R (cfcₙHom_of_cfcHom R ha f) = range f 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Field R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [ContinuousFunctionalCalculus R p] {a : A} (ha : p a) (f : C(↑(σₙ R a), R)₀)
Goal: σₙ R ((cfcₙHom_of_cfcHom R ha) f) = Set.range ⇑f
-- cfcₙHom_eq_cfcₙHom_of_cfcHom
lemma cfcₙHom_eq_cfcₙHom_of_cfcHom [UniqueNonUnitalContinuousFunctionalCalculus R A] {a : A}
    (ha : p a) : cfcₙHom (R := R) ha = cfcₙHom_of_cfcHom R ha 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Field R] [StarRing R] [MetricSpace R] [CompleteSpace R] [TopologicalRing R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [ContinuousFunctionalCalculus R p] [h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)] [UniqueNonUnitalContinuousFunctionalCalculus R A] {a : A} (ha : p a)
Goal: cfcₙHom ha = cfcₙHom_of_cfcHom R ha
-- cfcₙ_eq_cfc
lemma cfcₙ_eq_cfc [UniqueNonUnitalContinuousFunctionalCalculus R A] {f : R → R} {a : A}
    (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) :
    cfcₙ f a = cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Field R] [StarRing R] [MetricSpace R] [CompleteSpace R] [TopologicalRing R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [ContinuousFunctionalCalculus R p] [h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)] [UniqueNonUnitalContinuousFunctionalCalculus R A] {f : R → R} {a : A} (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
Goal: cfcₙ f a = cfc f a