ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- cfc_predicate_zero
lemma cfc_predicate_zero : p 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p]
Goal: p 0
-- cfc_predicate
lemma cfc_predicate (f : R → R) (a : A) : p (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: p (cfc f a)
-- cfc_predicate_algebraMap
lemma cfc_predicate_algebraMap (r : R) : p (algebraMap R A r) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R)
Goal: p ((algebraMap R A) r)
-- cfc_predicate_one
lemma cfc_predicate_one : p 1 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p]
Goal: p 1
-- cfc_congr
lemma cfc_congr {f g : R → R} {a : A} (hfg : (spectrum R a).EqOn f g) :
    cfc f a = cfc g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (hfg : Set.EqOn f g (spectrum R a))
Goal: cfc f a = cfc g a
-- eqOn_of_cfc_eq_cfc
lemma eqOn_of_cfc_eq_cfc {f g : R → R} {a : A} (h : cfc f a = cfc g a)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    (spectrum R a).EqOn f g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (h : cfc f a = cfc g a) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: Set.EqOn f g (spectrum R a)
-- cfc_eq_cfc_iff_eqOn
lemma cfc_eq_cfc_iff_eqOn : cfc f a = cfc g a ↔ (spectrum R a).EqOn f g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (ha : p a := by cfc_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac)
Goal: cfc f a = cfc g a ↔ Set.EqOn f g (spectrum R a)
-- cfc_one
lemma cfc_one : cfc (1 : R → R) a = 1 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfc 1 a = 1
-- cfc_const_one
lemma cfc_const_one : cfc (fun _ : R ↦ 1) a = 1 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun x => 1) a = 1
-- cfc_zero
@[simp]
lemma cfc_zero : cfc (0 : R → R) a = 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A)
Goal: cfc 0 a = 0
-- cfc_const_zero
@[simp]
lemma cfc_const_zero : cfc (fun _ : R ↦ 0) a = 0 
Arguments: (R : Type u_1) {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A)
Goal: cfc (fun x => 0) a = 0
-- cfc_mul
lemma cfc_mul (f g : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ f x * g x) a = cfc f a * cfc g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f g : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac)
Goal: cfc (fun x => f x * g x) a = cfc f a * cfc g a
-- cfc_pow
lemma cfc_pow (f : R → R) (n : ℕ) (a : A)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (fun x ↦ (f x) ^ n) a = cfc f a ^ n 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (n : ℕ) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f x ^ n) a = cfc f a ^ n
-- cfc_add
lemma cfc_add (f g : R → R) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ f x + g x) a = cfc f a + cfc g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (f g : R → R) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac)
Goal: cfc (fun x => f x + g x) a = cfc f a + cfc g a
-- cfc_const_add
lemma cfc_const_add (r : R) (f : R → R) (a : A)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (fun x => r + f x) a = algebraMap R A r + cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => r + f x) a = (algebraMap R A) r + cfc f a
-- cfc_add_const
lemma cfc_add_const (r : R) (f : R → R) (a : A)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (fun x => f x + r) a = cfc f a + algebraMap R A r 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f x + r) a = cfc f a + (algebraMap R A) r
-- cfc_sum
lemma cfc_sum {ι : Type*} (f : ι → R → R) (a : A) (s : Finset ι)
    (hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac) :
    cfc (∑ i in s, f i)  a = ∑ i in s, cfc (f i) a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {ι : Type u_3} (f : ι → R → R) (a : A) (s : Finset ι) (hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac)
Goal: cfc (∑ i ∈ s, f i) a = ∑ i ∈ s, cfc (f i) a
-- cfc_sum_univ
lemma cfc_sum_univ {ι : Type*} [Fintype ι] (f : ι → R → R) (a : A)
    (hf : ∀ i, ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac) :
    cfc (∑ i, f i) a = ∑ i, cfc (f i) a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {ι : Type u_3} [Fintype ι] (f : ι → R → R) (a : A) (hf : ∀ (i : ι), ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac)
Goal: cfc (∑ i : ι, f i) a = ∑ i : ι, cfc (f i) a
-- cfc_smul
lemma cfc_smul {S : Type*} [SMul 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 (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ s • f x) a = s • cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {S : Type u_3} [SMul 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 (spectrum R a) := by cfc_cont_tac)
Goal: cfc (fun x => s • f x) a = s • cfc f a
-- cfc_const_mul
lemma cfc_const_mul (r : R) (f : R → R) (a : A)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ r * f x) a = r • cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
Goal: cfc (fun x => r * f x) a = r • cfc f a
-- cfc_star
lemma cfc_star (f : R → R) (a : A) : cfc (fun x ↦ star (f x)) a = star (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: cfc (fun x => star (f x)) a = star (cfc f a)
-- cfc_pow_id
lemma cfc_pow_id (a : A) (n : ℕ) (ha : p a := by cfc_tac) : cfc (· ^ n : R → R) a = a ^ n 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (n : ℕ) (ha : p a := by cfc_tac)
Goal: cfc (fun x => x ^ n) a = a ^ n
-- cfc_smul_id
lemma cfc_smul_id {S : Type*} [SMul 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) : cfc (s • · : R → R) a = s • a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {S : Type u_3} [SMul 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: cfc (fun x => s • x) a = s • a
-- cfc_const_mul_id
lemma cfc_const_mul_id (r : R) (a : A) (ha : p a := by cfc_tac) : cfc (r * ·) a = r • a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun x => r * x) a = r • a
-- cfc_star_id
lemma cfc_star_id : cfc (star · : R → R) a = star a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun x => star x) a = star a
-- cfc_eval_X
lemma cfc_eval_X (ha : p a := by cfc_tac) : cfc (X : R[X]).eval a = a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun a => Polynomial.eval a Polynomial.X) a = a
-- cfc_eval_C
lemma cfc_eval_C (r : R) (a : A) (ha : p a := by cfc_tac) :
    cfc (C r).eval a = algebraMap R A r 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun a => Polynomial.eval a (Polynomial.C r)) a = (algebraMap R A) r
-- cfc_map_polynomial
lemma cfc_map_polynomial (q : R[X]) (f : R → R) (a : A) (ha : p a := by cfc_tac)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ q.eval (f x)) a = aeval (cfc f a) q 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (q : Polynomial R) (f : R → R) (a : A) (ha : p a := by cfc_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
Goal: cfc (fun x => Polynomial.eval (f x) q) a = (Polynomial.aeval (cfc f a)) q
-- cfc_polynomial
lemma cfc_polynomial (q : R[X]) (a : A) (ha : p a := by cfc_tac) :
    cfc q.eval a = aeval a q 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (q : Polynomial R) (a : A) (ha : p a := by cfc_tac)
Goal: cfc (fun a => Polynomial.eval a q) a = (Polynomial.aeval a) q
-- cfc_comp
lemma cfc_comp (g f : R → R) (a : A) (ha : p a := by cfc_tac)
    (hg : ContinuousOn g (f '' spectrum R a) := by cfc_cont_tac)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) :
    cfc (g ∘ f) a = cfc g (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (g f : R → R) (a : A) (ha : p a := by cfc_tac) (hg : ContinuousOn g (f '' spectrum R a) := by cfc_cont_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
Goal: cfc (g ∘ f) a = cfc g (cfc f a)
-- cfc_comp'
lemma cfc_comp' (g f : R → R) (a : A) (hg : ContinuousOn g (f '' spectrum R a) := by cfc_cont_tac)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (g <| f ·) a = cfc g (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (g f : R → R) (a : A) (hg : ContinuousOn g (f '' spectrum R a) := by cfc_cont_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => g (f x)) a = cfc g (cfc f a)
-- cfc_comp_pow
lemma cfc_comp_pow (f : R → R) (n : ℕ) (a : A)
    (hf : ContinuousOn f ((· ^ n) '' (spectrum R a)) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (f <| · ^ n) a = cfc f (a ^ n) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : R → R) (n : ℕ) (a : A) (hf : ContinuousOn f ((fun x => x ^ n) '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (x ^ n)) a = cfc f (a ^ n)
-- cfc_comp_smul
lemma cfc_comp_smul {S : Type*} [SMul 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 • ·) '' (spectrum R a)) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (f <| s • ·) a = cfc f (s • a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] {S : Type u_3} [SMul 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) '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (s • x)) a = cfc f (s • a)
-- cfc_comp_const_mul
lemma cfc_comp_const_mul (r : R) (f : R → R) (a : A)
    (hf : ContinuousOn f ((r * ·) '' (spectrum R a)) := by cfc_cont_tac)  (ha : p a := by cfc_tac) :
    cfc (f <| r * ·) a = cfc f (r • a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (r : R) (f : R → R) (a : A) (hf : ContinuousOn f ((fun x => r * x) '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (r * x)) a = cfc f (r • a)
-- cfc_comp_star
lemma cfc_comp_star (f : R → R) (a : A)
    (hf : ContinuousOn f (star '' (spectrum R a)) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (f <| star ·) a = cfc f (star a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : R → R) (a : A) (hf : ContinuousOn f (star '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (star x)) a = cfc f (star a)
-- cfc_comp_polynomial
lemma cfc_comp_polynomial (q : R[X]) (f : R → R) (a : A)
    (hf : ContinuousOn f (q.eval '' (spectrum R a)) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (f <| q.eval ·) a = cfc f (aeval a q) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (q : Polynomial R) (f : R → R) (a : A) (hf : ContinuousOn f ((fun a => Polynomial.eval a q) '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (Polynomial.eval x q)) a = cfc f ((Polynomial.aeval a) q)
-- CFC.eq_algebraMap_of_spectrum_subset_singleton
lemma CFC.eq_algebraMap_of_spectrum_subset_singleton (r : R) (h_spec : spectrum R a ⊆ {r})
    (ha : p a := by cfc_tac) : a = algebraMap R A r 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (r : R) (h_spec : spectrum R a ⊆ {r}) (ha : p a := by cfc_tac)
Goal: a = (algebraMap R A) r
-- CFC.eq_zero_of_spectrum_subset_zero
lemma CFC.eq_zero_of_spectrum_subset_zero (h_spec : spectrum 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] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (h_spec : spectrum R a ⊆ {0}) (ha : p a := by cfc_tac)
Goal: a = 0
-- CFC.eq_one_of_spectrum_subset_one
lemma CFC.eq_one_of_spectrum_subset_one (h_spec : spectrum R a ⊆ {1}) (ha : p a := by cfc_tac) :
    a = 1 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : A) (h_spec : spectrum R a ⊆ {1}) (ha : p a := by cfc_tac)
Goal: a = 1
-- CFC.spectrum_algebraMap_subset
lemma CFC.spectrum_algebraMap_subset (r : R) : spectrum R (algebraMap R A r) ⊆ {r} 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R)
Goal: spectrum R ((algebraMap R A) r) ⊆ {r}
-- CFC.spectrum_algebraMap_eq
lemma CFC.spectrum_algebraMap_eq [Nontrivial A] (r : R) :
    spectrum R (algebraMap R A r) = {r} 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [Nontrivial A] (r : R)
Goal: spectrum R ((algebraMap R A) r) = {r}
-- CFC.spectrum_zero_eq
lemma CFC.spectrum_zero_eq [Nontrivial A] :
    spectrum R (0 : A) = {0} 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [Nontrivial A]
Goal: spectrum R 0 = {0}
-- CFC.spectrum_one_eq
lemma CFC.spectrum_one_eq [Nontrivial A] :
    spectrum R (1 : A) = {1} 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [Nontrivial A]
Goal: spectrum R 1 = {1}
-- cfc_algebraMap
@[simp]
lemma cfc_algebraMap (r : R) (f : R → R) : cfc f (algebraMap R A r) = algebraMap R A (f r) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (r : R) (f : R → R)
Goal: cfc f ((algebraMap R A) r) = (algebraMap R A) (f r)
-- cfc_apply_zero
@[simp] lemma cfc_apply_zero {f : R → R} : cfc f (0 : A) = algebraMap R A (f 0) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f : R → R}
Goal: cfc f 0 = (algebraMap R A) (f 0)
-- cfc_apply_one
@[simp] lemma cfc_apply_one {f : R → R} : cfc f (1 : A) = algebraMap R A (f 1) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f : R → R}
Goal: cfc f 1 = (algebraMap R A) (f 1)
-- isUnit_cfc_iff
lemma isUnit_cfc_iff (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) : IsUnit (cfc f a) ↔ ∀ x ∈ spectrum R a, f x ≠ 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: IsUnit (cfc f a) ↔ ∀ x ∈ spectrum R a, f x ≠ 0