awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ∀ x ∈ t,
    IsCompact (X := ℝ≥0)
      ((fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x)

Tactic:
simpa +contextual using hS₁
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      spectrum ℝ≥0 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x₀
refine_2 :
  ∀ x ∈ t,
    ContinuousOn f
      ((fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x)
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) t

Tactic:
refine ha_cont.cfc_nnreal (s := fun x : X ↦ if hx : x ∈ t then S ⟨x, hx⟩ else ∅) f
    (by simpa +contextual using hS₁) ?_ ha' ?_

Premises used: ContinuousOn.cfc_nnreal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ ⋯)
⊢ ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      spectrum ℝ≥0 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x₀

Tactic:
simp +contextual only [↓reduceDIte]

Premises used: reduceDIte
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_2 : ∀ (x : X) (a : x ∈ t), ContinuousOn f (S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x ⋯))
⊢ ∀ x ∈ t,
    ContinuousOn f
      ((fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x)

Tactic:
simp +contextual only [↓reduceDIte]

Premises used: reduceDIte
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ ⋯)

Tactic:
fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

Premises used: Subtype.mk, Filter.Tendsto.eventually, ContinuousOn.continuousWithinAt
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ ⋯)

Tactic:
exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

Premises used: Subtype.mk, Filter.Tendsto.eventually, ContinuousOn.continuousWithinAt
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ∀ (x : X) (a : x ∈ t), ContinuousOn f (S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x ⋯))

Tactic:
fun x hx ↦ hf.mono <| hS₃ ⟨x, hx⟩

Premises used: Subtype.mk, ContinuousOn.mono
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ∀ (x : X) (a : x ∈ t), ContinuousOn f (S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x ⋯))

Tactic:
exact fun x hx ↦ hf.mono <| hS₃ ⟨x, hx⟩

Premises used: Subtype.mk, ContinuousOn.mono
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : (x : (↑t : Type u_1)) → Set ℝ≥0
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      spectrum ℝ≥0 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x₀
refine_2 :
  ∀ x ∈ t,
    ContinuousOn f
      ((fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set ℝ≥0)) x)
refine_1₁ :
  ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ ⋯)
refine_2₁ refine_2₂ : ∀ (x : X) (a : x ∈ t), ContinuousOn f (S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x ⋯))
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) t

Tactic:
classical
  refine ha_cont.cfc_nnreal (s := fun x : X ↦ if hx : x ∈ t then S ⟨x, hx⟩ else ∅) f
    (by simpa +contextual using hS₁) ?_ ha' ?_
  all_goals simp +contextual only [↓reduceDIte]
  · exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩
  · exact fun x hx ↦ hf.mono <| hS₃ ⟨x, hx⟩

Premises used: reduceDIte, Subtype.mk, Filter.Tendsto.eventually, ContinuousOn.mono, ContinuousOn.continuousWithinAt
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : Continuous[inst_10, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
rw :
  ∀ (ha_cont : ContinuousOn a Set.univ) (hs : ∀ (x : X), IsCompact (s x))
    (ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀) (hf : ∀ (x : X), ContinuousOn f (s x))
    (ha' : ∀ (x : X), 0 ≤ a x),
    ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ
⊢ Continuous[inst_10, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Premises used: continuousOn_univ
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
x : X
h : Membership.mem (γ := Set X) Set.univ x
⊢ ∀ᶠ (x_1 : X) in 𝓝[Set.univ] x, spectrum ℝ≥0 (a x_1) ⊆ s x

Tactic:
simpa using ha x
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ

Tactic:
exact ha_cont.cfc_nnreal f (fun x _ ↦ hs x) (fun x _ ↦ by simpa using ha x) (fun x _ ↦ ha' x)

Premises used: ContinuousOn.cfc_nnreal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : Continuous[inst_10, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
rw :
  ∀ (ha_cont : ContinuousOn a Set.univ) (ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s) (hf : ContinuousOn f s)
    (ha' : ∀ (x : X), 0 ≤ a x),
    ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ
⊢ Continuous[inst_10, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Premises used: continuousOn_univ
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ

Tactic:
ha_cont.cfc_nnreal' hs f (fun x _ ↦ ha x) (fun x _ ↦ ha' x)

Premises used: ContinuousOn.cfc_nnreal'
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : TopologicalSpace X
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ

Tactic:
exact ha_cont.cfc_nnreal' hs f (fun x _ ↦ ha x) (fun x _ ↦ ha' x)

Premises used: ContinuousOn.cfc_nnreal'
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ (x : X), spectrum ℝ≥0 (a x))) s
ha_cont : Continuous[inst_11, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
rw :
  ∀ (ha_cont : ContinuousOn a Set.univ) (ha' : ∀ (x : X), 0 ≤ a x) (hf : ContinuousOn f s),
    ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ
⊢ Continuous[inst_11, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Premises used: continuousOn_univ
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ (x : X), spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ (x : X), ⋃ (_ : Membership.mem (γ := Set X) Set.univ x), spectrum ℝ≥0 (a x)))
    s

Tactic:
simpa
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ (x : X), spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ ∀ (x : X), Membership.mem (γ := Set X) Set.univ x → 0 ≤ a x

Tactic:
simpa
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
A : Type u_2
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_4 : ContinuousStar A
inst_5 : PartialOrder A
inst_6 : StarOrderedRing A
inst_7 : NonnegSpectrumClass ℝ A
inst_8 : T2Space A
inst_9 : IsSemitopologicalRing A
inst_10 : CompleteSpace A
inst_11 : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : Membership.mem (γ := Filter ℝ≥0) (𝓝ˢ (⋃ (x : X), spectrum ℝ≥0 (a x))) s
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfc (p := fun (x : A) ↦ 0 ≤ x) f (a x)) Set.univ

Tactic:
exact ha_cont.cfc_nnreal_of_mem_nhdsSet f (by simpa) (by simpa)

Premises used: ContinuousOn.cfc_nnreal_of_mem_nhdsSet
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst : CommSemiring R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : Nontrivial R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : SMulCommClass R A A
inst_11 : IsScalarTower R A A
inst_12 : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
inl :
  ∀ (h_tendsto : TendstoUniformlyOn F f ⊥ (quasispectrum R a))
    (hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (quasispectrum R a)) (hF0 : ∀ᶠ (x : X) in ⊥, F x 0 = 0),
    Tendsto (fun (x : X) ↦ cfcₙ (p := p) (F x) a) ⊥ (𝓝 (cfcₙ (p := p) f a))
inr : ∀ (hl : l.NeBot), Tendsto (fun (x : X) ↦ cfcₙ (p := p) (F x) a) l (𝓝 (cfcₙ (p := p) f a))
⊢ Tendsto (fun (x : X) ↦ cfcₙ (p := p) (F x) a) l (𝓝 (cfcₙ (p := p) f a))

Tactic:
obtain (rfl | hl) := l.eq_or_neBot

Premises used: Filter.eq_or_neBot, rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst : CommSemiring R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : Nontrivial R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : SMulCommClass R A A
inst_11 : IsScalarTower R A A
inst_12 : NonUnitalContinuousFunctionalCalculus R A p
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f ⊥ (quasispectrum R a)
hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in ⊥, F x 0 = 0
⊢ Tendsto (fun (x : X) ↦ cfcₙ (p := p) (F x) a) ⊥ (𝓝 (cfcₙ (p := p) f a))

Tactic:
simp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst : CommSemiring R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : Nontrivial R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : SMulCommClass R A A
inst_11 : IsScalarTower R A A
inst_12 : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
⊢ ContinuousOn f (quasispectrum R a)

Tactic:
have hf := h_tendsto.continuousOn hF.frequently