ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- val_cfcUnits
@[simps]
noncomputable def cfcUnits (hf' : ∀ x ∈ spectrum R a, f x ≠ 0)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) : Aˣ 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: ↑(cfcUnits f a hf' hf ha) = cfc f a
-- val_inv_cfcUnits
@[simps]
noncomputable def cfcUnits (hf' : ∀ x ∈ spectrum R a, f x ≠ 0)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) : Aˣ 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: ↑(cfcUnits f a hf' hf ha)⁻¹ = cfc (fun x => (f x)⁻¹) a
-- cfcUnits_pow
lemma cfcUnits_pow (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℕ)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    (cfcUnits f a hf') ^ n =
      cfcUnits (forall₂_imp (fun _ _ ↦ pow_ne_zero n) hf') (hf := hf.pow n) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℕ) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfcUnits f a hf' hf ha ^ n = cfcUnits (fun x => f x ^ n) a ⋯ ⋯ ha
-- cfc_inv
lemma cfc_inv (hf' : ∀ x ∈ spectrum R a, f x ≠ 0)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc (fun x ↦ (f x) ⁻¹) a = Ring.inverse (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => (f x)⁻¹) a = Ring.inverse (cfc f a)
-- cfc_inv_id
lemma cfc_inv_id (a : Aˣ) (ha : p a := by cfc_tac) :
    cfc (fun x ↦ x⁻¹ : R → R) (a : A) = a⁻¹ 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : Aˣ) (ha : p ↑a := by cfc_tac)
Goal: cfc (fun x => x⁻¹) ↑a = ↑a⁻¹
-- cfc_map_div
lemma cfc_map_div (f g : R → R) (a : A) (hg' : ∀ x ∈ spectrum R a, g x ≠ 0)
    (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) :
    cfc (fun x ↦ f x / g x) a = cfc f a * Ring.inverse (cfc g a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f g : R → R) (a : A) (hg' : ∀ x ∈ spectrum R a, g x ≠ 0) (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: cfc (fun x => f x / g x) a = cfc f a * Ring.inverse (cfc g a)
-- Units.continuousOn_inv₀_spectrum
@[fun_prop]
lemma Units.continuousOn_inv₀_spectrum (a : Aˣ) : ContinuousOn (· ⁻¹) (spectrum R (a : A)) 
Arguments: {R : Type u_1} {A : Type u_2} [Semifield R] [MetricSpace R] [HasContinuousInv₀ R] [Ring A] [Algebra R A] (a : Aˣ)
Goal: ContinuousOn (fun x => x⁻¹) (spectrum R ↑a)
-- Units.continuousOn_zpow₀_spectrum
@[fun_prop]
lemma Units.continuousOn_zpow₀_spectrum (a : Aˣ) (n : ℤ) :
    ContinuousOn (· ^ n) (spectrum R (a : A)) 
Arguments: {R : Type u_1} {A : Type u_2} [Semifield R] [MetricSpace R] [TopologicalSemiring R] [HasContinuousInv₀ R] [Ring A] [Algebra R A] (a : Aˣ) (n : ℤ)
Goal: ContinuousOn (fun x => x ^ n) (spectrum R ↑a)
-- cfc_comp_inv
lemma cfc_comp_inv (f : R → R) (a : Aˣ)
    (hf : ContinuousOn f ((· ⁻¹) '' (spectrum R (a : A))) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) :
    cfc (fun x ↦ f x⁻¹) (a : A) = cfc f (↑a⁻¹ : A) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : R → R) (a : Aˣ) (hf : ContinuousOn f ((fun x => x⁻¹) '' spectrum R ↑a) := by cfc_cont_tac) (ha : p ↑a := by cfc_tac)
Goal: cfc (fun x => f x⁻¹) ↑a = cfc f ↑a⁻¹
-- cfcUnits_zpow
lemma cfcUnits_zpow (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℤ)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    (cfcUnits f a hf') ^ n =
      cfcUnits (f ^ n) a (forall₂_imp (fun _ _ ↦ zpow_ne_zero n) hf')
        (hf.zpow₀ n (forall₂_imp (fun _ _ ↦ Or.inl) hf')) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℤ) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfcUnits f a hf' hf ha ^ n = cfcUnits (f ^ n) a ⋯ ⋯ ha
-- cfc_zpow
lemma cfc_zpow (a : Aˣ) (n : ℤ) (ha : p a := by cfc_tac) :
    cfc (fun x : R ↦ x ^ n) (a : A) = ↑(a ^ n) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (a : Aˣ) (n : ℤ) (ha : p ↑a := by cfc_tac)
Goal: cfc (fun x => x ^ n) ↑a = ↑(a ^ n)
-- cfc_comp_zpow
lemma cfc_comp_zpow (f : R → R) (n : ℤ) (a : Aˣ)
    (hf : ContinuousOn f ((· ^ n) '' (spectrum R (a : A))) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) :
    cfc (fun x ↦ f (x ^ n)) (a : A) = cfc f (↑(a ^ n) : A) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [HasContinuousInv₀ R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : R → R) (n : ℤ) (a : Aˣ) (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_sub
lemma cfc_sub : 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} [CommRing R] [StarRing R] [MetricSpace R] [TopologicalRing 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_neg
lemma cfc_neg : cfc (fun x ↦ - (f x)) a = - (cfc f a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A)
Goal: cfc (fun x => -f x) a = -cfc f a
-- cfc_neg_id
lemma cfc_neg_id (ha : p a := by cfc_tac) : cfc (- · : R → R) a = -a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [StarRing R] [MetricSpace R] [TopologicalRing 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 => -x) a = -a
-- cfc_comp_neg
lemma cfc_comp_neg (hf : ContinuousOn f ((- ·) '' (spectrum R (a : A))) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) : cfc (f <| - ·) a = cfc f (-a) 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [CommRing R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) [UniqueContinuousFunctionalCalculus R A] (hf : ContinuousOn f ((fun x => -x) '' spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc (fun x => f (-x)) a = cfc f (-a)
-- cfcHom_mono
lemma cfcHom_mono {a : A} (ha : p a) {f g : C(spectrum R a, R)} (hfg : f ≤ g) :
    cfcHom ha f ≤ cfcHom ha g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {a : A} (ha : p a) {f g : C(↑(spectrum R a), R)} (hfg : f ≤ g)
Goal: (cfcHom ha) f ≤ (cfcHom ha) g
-- cfcHom_nonneg_iff
lemma cfcHom_nonneg_iff {a : A} (ha : p a) {f : C(spectrum R a, R)} :
    0 ≤ cfcHom ha f ↔ 0 ≤ f 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] {a : A} (ha : p a) {f : C(↑(spectrum R a), R)}
Goal: 0 ≤ (cfcHom ha) f ↔ 0 ≤ f
-- cfc_mono
lemma cfc_mono {f g : R → R} {a : A} (h : ∀ x ∈ spectrum R a, f x ≤ g x)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (hg : ContinuousOn g (spectrum R a) := by cfc_cont_tac) :
    cfc f a ≤ cfc g a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f g : R → R} {a : A} (h : ∀ x ∈ spectrum R a, f x ≤ g x) (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
-- cfc_nonneg_iff
lemma cfc_nonneg_iff (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) : 0 ≤ cfc f a ↔ ∀ x ∈ spectrum R a, 0 ≤ f x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: 0 ≤ cfc f a ↔ ∀ x ∈ spectrum R a, 0 ≤ f x
-- StarOrderedRing.nonneg_iff_spectrum_nonneg
lemma StarOrderedRing.nonneg_iff_spectrum_nonneg (a : A) (ha : p a := by cfc_tac) :
    0 ≤ a ↔ ∀ x ∈ spectrum R a, 0 ≤ x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (a : A) (ha : p a := by cfc_tac)
Goal: 0 ≤ a ↔ ∀ x ∈ spectrum R a, 0 ≤ x
-- cfc_nonneg
lemma cfc_nonneg {f : R → R} {a : A} (h : ∀ x ∈ spectrum R a, 0 ≤ f x) :
    0 ≤ cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {f : R → R} {a : A} (h : ∀ x ∈ spectrum R a, 0 ≤ f x)
Goal: 0 ≤ cfc f a
-- cfc_nonpos
lemma cfc_nonpos (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ 0) :
    cfc f a ≤ 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ 0)
Goal: cfc f a ≤ 0
-- cfc_le_algebraMap
lemma cfc_le_algebraMap (f : R → R) (r : R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ r)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    cfc f a ≤ algebraMap R A r 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (r : R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ r) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc f a ≤ (algebraMap R A) r
-- algebraMap_le_cfc
lemma algebraMap_le_cfc (f : R → R) (r : R) (a : A) (h : ∀ x ∈ spectrum R a, r ≤ f x)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    algebraMap R A r ≤ cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (r : R) (a : A) (h : ∀ x ∈ spectrum R a, r ≤ f x) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: (algebraMap R A) r ≤ cfc f a
-- le_algebraMap_of_spectrum_le
lemma le_algebraMap_of_spectrum_le {r : R} {a : A} (h : ∀ x ∈ spectrum R a, x ≤ r)
    (ha : p a := by cfc_tac) : a ≤ algebraMap R A r 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {r : R} {a : A} (h : ∀ x ∈ spectrum R a, x ≤ r) (ha : p a := by cfc_tac)
Goal: a ≤ (algebraMap R A) r
-- algebraMap_le_of_le_spectrum
lemma algebraMap_le_of_le_spectrum {r : R} {a : A} (h : ∀ x ∈ spectrum R a, r ≤ x)
    (ha : p a := by cfc_tac) : algebraMap R A r ≤ a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] {r : R} {a : A} (h : ∀ x ∈ spectrum R a, r ≤ x) (ha : p a := by cfc_tac)
Goal: (algebraMap R A) r ≤ a
-- cfc_le_one
lemma cfc_le_one (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ 1) : cfc f a ≤ 1 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ 1)
Goal: cfc f a ≤ 1
-- one_le_cfc
lemma one_le_cfc (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, 1 ≤ f x)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
    1 ≤ cfc f a 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, 1 ≤ f x) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: 1 ≤ cfc f a
-- cfcHom_le_iff
lemma cfcHom_le_iff {a : A} (ha : p a) {f g : C(spectrum R a, R)} :
    cfcHom ha f ≤ cfcHom ha g ↔ f ≤ g 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] {a : A} (ha : p a) {f g : C(↑(spectrum R a), R)}
Goal: (cfcHom ha) f ≤ (cfcHom ha) g ↔ f ≤ g
-- cfc_le_iff
lemma cfc_le_iff (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) (ha : p a := by cfc_tac) :
    cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum R a, f x ≤ g x 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (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) (ha : p a := by cfc_tac)
Goal: cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum R a, f x ≤ g x
-- cfc_nonpos_iff
lemma cfc_nonpos_iff (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
    (ha : p a := by cfc_tac) : cfc f a ≤ 0 ↔ ∀ x ∈ spectrum R a, f x ≤ 0 
Arguments: {R : Type u_1} {A : Type u_2} {p : A → Prop} [OrderedCommRing R] [StarRing R] [MetricSpace R] [TopologicalRing R] [ContinuousStar R] [∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)] [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Algebra R A] [ContinuousFunctionalCalculus R p] [NonnegSpectrumClass R A] (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac)
Goal: cfc f a ≤ 0 ↔ ∀ x ∈ spectrum R a, f x ≤ 0
-- mem_intrinsicInterior
@[simp]
theorem mem_intrinsicInterior :
    x ∈ intrinsicInterior 𝕜 s ↔ ∃ y, y ∈ interior ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} {x : P}
Goal: x ∈ intrinsicInterior 𝕜 s ↔ ∃ y ∈ interior (Subtype.val ⁻¹' s), ↑y = x
-- mem_intrinsicFrontier
@[simp]
theorem mem_intrinsicFrontier :
    x ∈ intrinsicFrontier 𝕜 s ↔ ∃ y, y ∈ frontier ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} {x : P}
Goal: x ∈ intrinsicFrontier 𝕜 s ↔ ∃ y ∈ frontier (Subtype.val ⁻¹' s), ↑y = x
-- mem_intrinsicClosure
@[simp]
theorem mem_intrinsicClosure :
    x ∈ intrinsicClosure 𝕜 s ↔ ∃ y, y ∈ closure ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} {x : P}
Goal: x ∈ intrinsicClosure 𝕜 s ↔ ∃ y ∈ closure (Subtype.val ⁻¹' s), ↑y = x
-- intrinsicInterior_subset
theorem intrinsicInterior_subset : intrinsicInterior 𝕜 s ⊆ s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: intrinsicInterior 𝕜 s ⊆ s
-- intrinsicFrontier_subset
theorem intrinsicFrontier_subset (hs : IsClosed s) : intrinsicFrontier 𝕜 s ⊆ s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P} (hs : IsClosed s)
Goal: intrinsicFrontier 𝕜 s ⊆ s
-- intrinsicFrontier_subset_intrinsicClosure
theorem intrinsicFrontier_subset_intrinsicClosure : intrinsicFrontier 𝕜 s ⊆ intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: intrinsicFrontier 𝕜 s ⊆ intrinsicClosure 𝕜 s
-- subset_intrinsicClosure
theorem subset_intrinsicClosure : s ⊆ intrinsicClosure 𝕜 s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: s ⊆ intrinsicClosure 𝕜 s
-- intrinsicInterior_empty
@[simp]
theorem intrinsicInterior_empty : intrinsicInterior 𝕜 (∅ : Set P) = ∅ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P]
Goal: intrinsicInterior 𝕜 ∅ = ∅
-- intrinsicFrontier_empty
@[simp]
theorem intrinsicFrontier_empty : intrinsicFrontier 𝕜 (∅ : Set P) = ∅ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P]
Goal: intrinsicFrontier 𝕜 ∅ = ∅
-- intrinsicClosure_empty
@[simp]
theorem intrinsicClosure_empty : intrinsicClosure 𝕜 (∅ : Set P) = ∅ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P]
Goal: intrinsicClosure 𝕜 ∅ = ∅
-- intrinsicClosure_nonempty
@[simp]
theorem intrinsicClosure_nonempty : (intrinsicClosure 𝕜 s).Nonempty ↔ s.Nonempty 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s : Set P}
Goal: (intrinsicClosure 𝕜 s).Nonempty ↔ s.Nonempty
-- intrinsicInterior_singleton
@[simp]
theorem intrinsicInterior_singleton (x : P) : intrinsicInterior 𝕜 ({x} : Set P) = {x} 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (x : P)
Goal: intrinsicInterior 𝕜 {x} = {x}
-- intrinsicFrontier_singleton
@[simp]
theorem intrinsicFrontier_singleton (x : P) : intrinsicFrontier 𝕜 ({x} : Set P) = ∅ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (x : P)
Goal: intrinsicFrontier 𝕜 {x} = ∅
-- intrinsicClosure_singleton
@[simp]
theorem intrinsicClosure_singleton (x : P) : intrinsicClosure 𝕜 ({x} : Set P) = {x} 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] (x : P)
Goal: intrinsicClosure 𝕜 {x} = {x}
-- intrinsicClosure_mono
theorem intrinsicClosure_mono (h : s ⊆ t) : intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_5} [Ring 𝕜] [AddCommGroup V] [Module 𝕜 V] [TopologicalSpace P] [AddTorsor V P] {s t : Set P} (h : s ⊆ t)
Goal: intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t