ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- gauge_ball
theorem gauge_ball (hr : 0 ≤ r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 ≤ r) (x : E)
Goal: gauge (Metric.ball 0 r) x = ‖x‖ / r
-- gauge_ball'
@[deprecated gauge_ball (since := "2023-07-24")]
theorem gauge_ball' (hr : 0 < r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 < r) (x : E)
Goal: gauge (Metric.ball 0 r) x = ‖x‖ / r
-- gauge_closure_zero
@[simp]
theorem gauge_closure_zero : gauge (closure (0 : Set E)) = 0 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
Goal: gauge (closure 0) = 0
-- gauge_closedBall
@[simp]
theorem gauge_closedBall (hr : 0 ≤ r) (x : E) : gauge (closedBall (0 : E) r) x = ‖x‖ / r 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 ≤ r) (x : E)
Goal: gauge (Metric.closedBall 0 r) x = ‖x‖ / r
-- mul_gauge_le_norm
theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E} (hs : Metric.ball 0 r ⊆ s)
Goal: r * gauge s x ≤ ‖x‖
-- Convex.lipschitzWith_gauge
theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r)
    (hs : Metric.ball (0 : E) r ⊆ s) : LipschitzWith r⁻¹ (gauge s) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r) (hs : Metric.ball 0 ↑r ⊆ s)
Goal: LipschitzWith r⁻¹ (gauge s)
-- Convex.lipschitz_gauge
theorem Convex.lipschitz_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
    ∃ K, LipschitzWith K (gauge s) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 0)
Goal: ∃ K, LipschitzWith K (gauge s)
-- Convex.uniformContinuous_gauge
theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
    UniformContinuous (gauge s) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 0)
Goal: UniformContinuous (gauge s)
-- le_gauge_of_subset_closedBall
theorem le_gauge_of_subset_closedBall (hs : Absorbent ℝ s) (hr : 0 ≤ r) (hsr : s ⊆ closedBall 0 r) :
    ‖x‖ / r ≤ gauge s x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E} (hs : Absorbent ℝ s) (hr : 0 ≤ r) (hsr : s ⊆ Metric.closedBall 0 r)
Goal: ‖x‖ / r ≤ gauge s x
-- PartitionOfUnity.finsum_smul_mem_convex
theorem PartitionOfUnity.finsum_smul_mem_convex {s : Set X} (f : PartitionOfUnity ι X s)
    {g : ι → X → E} {t : Set E} {x : X} (hx : x ∈ s) (hg : ∀ i, f i x ≠ 0 → g i x ∈ t)
    (ht : Convex ℝ t) : (∑ᶠ i, f i x • g i x) ∈ t 
Arguments: {ι : Type u_1} {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) {g : ι → X → E} {t : Set E} {x : X} (hx : x ∈ s) (hg : ∀ (i : ι), (f i) x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t)
Goal: ∑ᶠ (i : ι), (f i) x • g i x ∈ t
-- exists_continuous_forall_mem_convex_of_local
theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
    (H : ∀ x : X, ∃ U ∈ 𝓝 x, ∃ g : X → E, ContinuousOn g U ∧ ∀ y ∈ U, g y ∈ t y) :
    ∃ g : C(X, E), ∀ x, g x ∈ t x 
Arguments: {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {t : X → Set E} (ht : ∀ (x : X), Convex ℝ (t x)) (H : ∀ (x : X), ∃ U ∈ 𝓝 x, ∃ g, ContinuousOn g U ∧ ∀ y ∈ U, g y ∈ t y)
Goal: ∃ g, ∀ (x : X), g x ∈ t x
-- exists_continuous_forall_mem_convex_of_local_const
theorem exists_continuous_forall_mem_convex_of_local_const (ht : ∀ x, Convex ℝ (t x))
    (H : ∀ x : X, ∃ c : E, ∀ᶠ y in 𝓝 x, c ∈ t y) : ∃ g : C(X, E), ∀ x, g x ∈ t x 
Arguments: {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {t : X → Set E} (ht : ∀ (x : X), Convex ℝ (t x)) (H : ∀ (x : X), ∃ c, ∀ᶠ (y : X) in 𝓝 x, c ∈ t y)
Goal: ∃ g, ∀ (x : X), g x ∈ t x
-- Sbtw.dist_lt_max_dist
theorem Sbtw.dist_lt_max_dist (p : P) {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) :
    dist p₂ p < max (dist p₁ p) (dist p₃ p) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃)
Goal: dist p₂ p < max (dist p₁ p) (dist p₃ p)
-- Wbtw.dist_le_max_dist
theorem Wbtw.dist_le_max_dist (p : P) {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) :
    dist p₂ p ≤ max (dist p₁ p) (dist p₃ p) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃)
Goal: dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)
-- Collinear.wbtw_of_dist_eq_of_dist_le
theorem Collinear.wbtw_of_dist_eq_of_dist_le {p p₁ p₂ p₃ : P} {r : ℝ}
    (h : Collinear ℝ ({p₁, p₂, p₃} : Set P)) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p ≤ r)
    (hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃) : Wbtw ℝ p₁ p₂ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {p p₁ p₂ p₃ : P} {r : ℝ} (h : Collinear ℝ {p₁, p₂, p₃}) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p ≤ r) (hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃)
Goal: Wbtw ℝ p₁ p₂ p₃
-- Collinear.sbtw_of_dist_eq_of_dist_lt
theorem Collinear.sbtw_of_dist_eq_of_dist_lt {p p₁ p₂ p₃ : P} {r : ℝ}
    (h : Collinear ℝ ({p₁, p₂, p₃} : Set P)) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p < r)
    (hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃) : Sbtw ℝ p₁ p₂ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {p p₁ p₂ p₃ : P} {r : ℝ} (h : Collinear ℝ {p₁, p₂, p₃}) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p < r) (hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃)
Goal: Sbtw ℝ p₁ p₂ p₃
-- dist_add_dist_eq_iff
lemma dist_add_dist_eq_iff : dist a b + dist b c = dist a c ↔ Wbtw ℝ a b c 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {a b c : P}
Goal: dist a b + dist b c = dist a c ↔ Wbtw ℝ a b c
-- eq_lineMap_of_dist_eq_mul_of_dist_eq_mul
lemma eq_lineMap_of_dist_eq_mul_of_dist_eq_mul (hxy : dist x y = r * dist x z)
    (hyz : dist y z = (1 - r) * dist x z) : y = AffineMap.lineMap x z r 
Arguments: {E : Type u_3} {PE : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] {r : ℝ} {x y z : PE} (hxy : dist x y = r * dist x z) (hyz : dist y z = (1 - r) * dist x z)
Goal: y = (AffineMap.lineMap x z) r
-- eq_midpoint_of_dist_eq_half
lemma eq_midpoint_of_dist_eq_half (hx : dist x y = dist x z / 2) (hy : dist y z = dist x z / 2) :
    y = midpoint ℝ x z 
Arguments: {E : Type u_3} {PE : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] {x y z : PE} (hx : dist x y = dist x z / 2) (hy : dist y z = dist x z / 2)
Goal: y = midpoint ℝ x z
-- Isometry.coe_affineIsometryOfStrictConvexSpace
@[simp] lemma coe_affineIsometryOfStrictConvexSpace (hi : Isometry f) :
    ⇑hi.affineIsometryOfStrictConvexSpace = f 
Arguments: {E : Type u_3} {F : Type u_4} {PE : Type u_5} {PF : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [StrictConvexSpace ℝ E] [MetricSpace PE] [MetricSpace PF] [NormedAddTorsor E PE] [NormedAddTorsor F PF] {f : PF → PE} (hi : Isometry f)
Goal: ⇑hi.affineIsometryOfStrictConvexSpace = f
-- Isometry.affineIsometryOfStrictConvexSpace_apply
@[simp] lemma affineIsometryOfStrictConvexSpace_apply (hi : Isometry f) (p : PF) :
    hi.affineIsometryOfStrictConvexSpace p = f p 
Arguments: {E : Type u_3} {F : Type u_4} {PE : Type u_5} {PF : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [StrictConvexSpace ℝ E] [MetricSpace PE] [MetricSpace PF] [NormedAddTorsor E PE] [NormedAddTorsor F PF] {f : PF → PE} (hi : Isometry f) (p : PF)
Goal: hi.affineIsometryOfStrictConvexSpace p = f p
-- ConvexCone.coe_closure
@[simp, norm_cast]
theorem coe_closure (K : ConvexCone 𝕜 E) : (K.closure : Set E) = closure K 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [SMul 𝕜 E] [ContinuousConstSMul 𝕜 E] (K : ConvexCone 𝕜 E)
Goal: ↑K.closure = closure ↑K
-- ConvexCone.mem_closure
@[simp]
protected theorem mem_closure {K : ConvexCone 𝕜 E} {a : E} :
    a ∈ K.closure ↔ a ∈ closure (K : Set E) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [SMul 𝕜 E] [ContinuousConstSMul 𝕜 E] {K : ConvexCone 𝕜 E} {a : E}
Goal: a ∈ K.closure ↔ a ∈ closure ↑K
-- ConvexCone.closure_eq
@[simp]
theorem closure_eq {K L : ConvexCone 𝕜 E} : K.closure = L ↔ closure (K : Set E) = L 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [SMul 𝕜 E] [ContinuousConstSMul 𝕜 E] {K L : ConvexCone 𝕜 E}
Goal: K.closure = L ↔ closure ↑K = ↑L
-- PointedCone.toConvexCone_closure_pointed
lemma toConvexCone_closure_pointed (K : PointedCone 𝕜 E) : (K : ConvexCone 𝕜 E).closure.Pointed 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousConstSMul 𝕜 E] (K : PointedCone 𝕜 E)
Goal: (↑K).closure.Pointed
-- PointedCone.coe_closure
@[simp, norm_cast]
theorem coe_closure (K : PointedCone 𝕜 E) : (K.closure : Set E) = closure K 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousConstSMul 𝕜 E] (K : PointedCone 𝕜 E)
Goal: ↑K.closure = closure ↑K
-- PointedCone.mem_closure
@[simp]
protected theorem mem_closure {K : PointedCone 𝕜 E} {a : E} :
    a ∈ K.closure ↔ a ∈ closure (K : Set E) 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousConstSMul 𝕜 E] {K : PointedCone 𝕜 E} {a : E}
Goal: a ∈ K.closure ↔ a ∈ closure ↑K
-- PointedCone.closure_eq
@[simp]
theorem closure_eq {K L : PointedCone 𝕜 E} : K.closure = L ↔ closure (K : Set E) = L 
Arguments: {𝕜 : Type u_1} [OrderedSemiring 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousConstSMul 𝕜 E] {K L : PointedCone 𝕜 E}
Goal: K.closure = L ↔ closure ↑K = ↑L
-- ContinuousFunctionalCalculus.exists_cfc_of_predicate
class ContinuousFunctionalCalculus (R : Type*) {A : Type*} (p : outParam (A → Prop))
    [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R]
    [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] : Prop where
  predicate_zero : p 0
  exists_cfc_of_predicate : ∀ a, p a → ∃ φ : C(spectrum R a, R) →⋆ₐ[R] A,
    ClosedEmbedding φ ∧ φ ((ContinuousMap.id R).restrict <| spectrum R a) = a ∧
      (∀ f, spectrum 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] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [self : ContinuousFunctionalCalculus R p] (a : A)
Goal: p a →
  ∃ φ,
    ClosedEmbedding ⇑φ ∧
      φ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a ∧
        (∀ (f : C(↑(spectrum R a), R)), spectrum R (φ f) = Set.range ⇑f) ∧ ∀ (f : C(↑(spectrum R a), R)), p (φ f)
-- ContinuousFunctionalCalculus.predicate_zero
class ContinuousFunctionalCalculus (R : Type*) {A : Type*} (p : outParam (A → Prop))
    [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R]
    [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] : Prop where
  predicate_zero : p 0
  exists_cfc_of_predicate : ∀ a, p a → ∃ φ : C(spectrum R a, R) →⋆ₐ[R] A,
    ClosedEmbedding φ ∧ φ ((ContinuousMap.id R).restrict <| spectrum R a) = a ∧
      (∀ f, spectrum 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] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [self : ContinuousFunctionalCalculus R p]
Goal: p 0
-- UniqueContinuousFunctionalCalculus.eq_of_continuous_of_map_id
class UniqueContinuousFunctionalCalculus (R A : Type*) [CommSemiring R] [StarRing R]
    [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A]
    [TopologicalSpace A] [Algebra R A] : Prop where
  eq_of_continuous_of_map_id (s : Set R) [CompactSpace s]
    (φ ψ : C(s, R) →⋆ₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (.restrict s <| .id R) = ψ (.restrict s <| .id R)) :
    φ = ψ
  compactSpace_spectrum (a : A) : CompactSpace (spectrum R a)
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [self : UniqueContinuousFunctionalCalculus R A] (s : Set R) [CompactSpace ↑s] (φ ψ : C(↑s, R) →⋆ₐ[R] A) (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : φ (ContinuousMap.restrict s (ContinuousMap.id R)) = ψ (ContinuousMap.restrict s (ContinuousMap.id R)))
Goal: φ = ψ
-- UniqueContinuousFunctionalCalculus.compactSpace_spectrum
class UniqueContinuousFunctionalCalculus (R A : Type*) [CommSemiring R] [StarRing R]
    [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A]
    [TopologicalSpace A] [Algebra R A] : Prop where
  eq_of_continuous_of_map_id (s : Set R) [CompactSpace s]
    (φ ψ : C(s, R) →⋆ₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (.restrict s <| .id R) = ψ (.restrict s <| .id R)) :
    φ = ψ
  compactSpace_spectrum (a : A) : CompactSpace (spectrum R a)
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [self : UniqueContinuousFunctionalCalculus R A] (a : A)
Goal: CompactSpace ↑(spectrum R a)
-- StarAlgHom.ext_continuousMap
lemma StarAlgHom.ext_continuousMap [UniqueContinuousFunctionalCalculus R A]
    (a : A) (φ ψ : C(spectrum R a, R) →⋆ₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (.restrict (spectrum R a) <| .id R) = ψ (.restrict (spectrum R a) <| .id R)) :
    φ = ψ 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [UniqueContinuousFunctionalCalculus R A] (a : A) (φ ψ : C(↑(spectrum R a), R) →⋆ₐ[R] A) (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h :
  φ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) =
    ψ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)))
Goal: φ = ψ
-- cfcHom_closedEmbedding
lemma cfcHom_closedEmbedding :
    ClosedEmbedding <| (cfcHom ha : C(spectrum R a, 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] {a : A} (ha : p a)
Goal: ClosedEmbedding ⇑(cfcHom ha)
-- cfcHom_id
lemma cfcHom_id :
    cfcHom ha ((ContinuousMap.id R).restrict <| spectrum R 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)
Goal: (cfcHom ha) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a
-- cfcHom_map_spectrum
lemma cfcHom_map_spectrum (f : C(spectrum R a, R)) :
    spectrum R (cfcHom ha f) = Set.range f 
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) (f : C(↑(spectrum R a), R))
Goal: spectrum R ((cfcHom ha) f) = Set.range ⇑f
-- cfcHom_predicate
lemma cfcHom_predicate (f : C(spectrum R a, R)) :
    p (cfcHom ha f) 
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) (f : C(↑(spectrum R a), R))
Goal: p ((cfcHom ha) f)
-- cfcHom_eq_of_continuous_of_map_id
lemma cfcHom_eq_of_continuous_of_map_id [UniqueContinuousFunctionalCalculus R A]
    (φ : C(spectrum R a, R) →⋆ₐ[R] A) (hφ₁ : Continuous φ)
    (hφ₂ : φ (.restrict (spectrum R a) <| .id R) = a) : cfcHom ha = φ 
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) [UniqueContinuousFunctionalCalculus R A] (φ : C(↑(spectrum R a), R) →⋆ₐ[R] A) (hφ₁ : Continuous ⇑φ) (hφ₂ : φ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a)
Goal: cfcHom ha = φ
-- cfcHom_comp
theorem cfcHom_comp [UniqueContinuousFunctionalCalculus R A] (f : C(spectrum R a, R))
    (f' : C(spectrum R a, spectrum R (cfcHom ha f)))
    (hff' : ∀ x, f x = f' x) (g : C(spectrum R (cfcHom ha f), R)) :
    cfcHom ha (g.comp f') = cfcHom (cfcHom_predicate ha 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] {a : A} (ha : p a) [UniqueContinuousFunctionalCalculus R A] (f : C(↑(spectrum R a), R)) (f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))) (hff' : ∀ (x : ↑(spectrum R a)), f x = ↑(f' x)) (g : C(↑(spectrum R ((cfcHom ha) f)), R))
Goal: (cfcHom ha) (g.comp f') = (cfcHom ⋯) g
-- cfc_def
noncomputable irreducible_def cfc (f : R → R) (a : A) : A 
Arguments: {R : Type u_3} {A : Type u_4} {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 f a =
  if h : p a ∧ ContinuousOn f (spectrum R a) then
    (cfcHom ⋯) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ }
  else 0
-- cfc_apply
lemma cfc_apply : cfc f a = cfcHom (a := a) ha ⟨_, hf.restrict⟩ 
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) (ha : p a := by cfc_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
Goal: cfc f a = (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ }
-- cfc_apply_pi
lemma cfc_apply_pi {ι : Type*} (f : ι → R → R) (a : A) (ha : p a := by cfc_tac)
    (hf : ∀ i, ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac) :
    (fun i => cfc (f i) a) = (fun i => cfcHom (a := a) ha ⟨_, (hf i).restrict⟩) 
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) (ha : p a := by cfc_tac) (hf : ∀ (i : ι), ContinuousOn (f i) (spectrum R a) := by cfc_cont_tac)
Goal: (fun i => cfc (f i) a) = fun i => (cfcHom ha) { toFun := (spectrum R a).restrict (f i), continuous_toFun := ⋯ }
-- cfc_apply_of_not_and
lemma cfc_apply_of_not_and {f : R → R} (a : A) (ha : ¬ (p a ∧ ContinuousOn f (spectrum R a))) :
    cfc f 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] {f : R → R} (a : A) (ha : ¬(p a ∧ ContinuousOn f (spectrum R a)))
Goal: cfc f a = 0
-- cfc_apply_of_not_predicate
lemma cfc_apply_of_not_predicate {f : R → R} (a : A) (ha : ¬ p a) :
    cfc f 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] {f : R → R} (a : A) (ha : ¬p a)
Goal: cfc f a = 0
-- cfc_apply_of_not_continuousOn
lemma cfc_apply_of_not_continuousOn {f : R → R} (a : A) (hf : ¬ ContinuousOn f (spectrum R a)) :
    cfc f 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] {f : R → R} (a : A) (hf : ¬ContinuousOn f (spectrum R a))
Goal: cfc f a = 0
-- cfcHom_eq_cfc_extend
lemma cfcHom_eq_cfc_extend {a : A} (g : R → R) (ha : p a) (f : C(spectrum R a, R)) :
    cfcHom ha f = cfc (Function.extend Subtype.val f 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} (g : R → R) (ha : p a) (f : C(↑(spectrum R a), R))
Goal: (cfcHom ha) f = cfc (Function.extend Subtype.val (⇑f) g) a
-- cfc_cases
lemma cfc_cases (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0)
    (haf : (hf : ContinuousOn f (spectrum R a)) → (ha : p a) → P (cfcHom ha ⟨_, hf.restrict⟩)) :
    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] (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0) (haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ }))
Goal: P (cfc f a)
-- cfc_id
lemma cfc_id : cfc (id : R → R) 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 id a = a
-- cfc_id'
lemma cfc_id' : cfc (fun x : R ↦ x) 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 x => x) a = a
-- cfc_map_spectrum
lemma cfc_map_spectrum : spectrum R (cfc f a) = f '' spectrum 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] (f : R → R) (a : A) (ha : p a := by cfc_tac) (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
Goal: spectrum R (cfc f a) = f '' spectrum R a
-- cfc_const
lemma cfc_const (r : R) (a : A) (ha : p a := by cfc_tac) :
    cfc (fun _ ↦ r) 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 x => r) a = (algebraMap R A) r