awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : CompleteSpace A
inst_9 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : (x : (↑t : Type u_1)) → Set 𝕜
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) t
have refine_1 : ∀ x₀ ∈ t,
  ∀ᶠ (x : X) in 𝓝[t] x₀,
    quasispectrum 𝕜 (a x) ⊆
      (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set 𝕜)) x₀
Tactic: classical
  refine ha_cont.cfcₙ (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⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : CompleteSpace A
inst_9 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : (x : (↑t : Type u_1)) → Set 𝕜
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      quasispectrum 𝕜 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set 𝕜)) x₀
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) t
have 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 𝕜)) x)
Tactic: classical
  refine ha_cont.cfcₙ (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⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : CompleteSpace A
inst_9 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : (x : (↑t : Type u_1)) → Set 𝕜
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      quasispectrum 𝕜 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set 𝕜)) 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 𝕜)) x)
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) t
have refine_1₁ : ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum 𝕜 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ sorry)
Tactic: classical
  refine ha_cont.cfcₙ (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⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : CompleteSpace A
inst_9 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : (x : (↑t : Type u_1)) → Set 𝕜
hS₁ : ∀ (x : (↑t : Type u_1)), IsCompact (S x)
hS₂ : ∀ (x : (↑t : Type u_1)), ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : (↑t : Type u_1)), S x ⊆ s
refine_1 :
  ∀ x₀ ∈ t,
    ∀ᶠ (x : X) in 𝓝[t] x₀,
      quasispectrum 𝕜 (a x) ⊆
        (fun (x : X) ↦ (if hx : x ∈ t then S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x hx) else ∅ : Set 𝕜)) 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 𝕜)) x)
refine_1₁ :
  ∀ (x₀ : X) (a_1 : x₀ ∈ t),
    ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum 𝕜 (a x) ⊆ S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x₀ ⋯)
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) t
have refine_2₁ : ∀ (x : X) (a : x ∈ t), ContinuousOn f (S (Subtype.mk (p := fun (x : X) ↦ x ∈ t) x sorry))
Tactic: classical
  refine ha_cont.cfcₙ (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⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
ha_cont : Continuous[inst_8, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum 𝕜 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
hf0 : f 0 = 0
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst_8, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦ cfcₙ (p := p) f (a x)
have rw : ∀ (ha_cont : ContinuousOn a Set.univ) (hs : ∀ (x : X), IsCompact (s x))
  (ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum 𝕜 (a x) ⊆ s x₀) (hf : ∀ (x : X), ContinuousOn f (s x))
  (hf0 : f 0 = 0) (ha' : ∀ (x : X), p (a x)),
  ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) Set.univ
Tactic: rw [← continuousOn_univ] at ha_cont ⊢
Cleared hypotheses: ha_cont, hs, ha, hf, hf0, ha'
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
ha_cont : Continuous[inst_8, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
hf : ContinuousOn f s
hf0 : f 0 = 0
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst_8, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦ cfcₙ (p := p) f (a x)
have rw : ∀ (ha_cont : ContinuousOn a Set.univ) (ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s) (hf : ContinuousOn f s)
  (hf0 : f 0 = 0) (ha' : ∀ (x : X), p (a x)),
  ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) Set.univ
Tactic: rw [← continuousOn_univ] at ha_cont ⊢
Cleared hypotheses: ha_cont, ha, hf, hf0, ha'
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : ContinuousStar A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst_8 : CompleteSpace A
inst_9 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ (x : X), quasispectrum 𝕜 (a x))) s
ha_cont : Continuous[inst_9, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha' : ∀ (x : X), p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ Continuous[inst_9, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦ cfcₙ (p := p) f (a x)
have rw : ∀ (ha_cont : ContinuousOn a Set.univ) (ha' : ∀ (x : X), p (a x)) (hf : ContinuousOn f s) (hf0 : f 0 = 0),
  ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfcₙ (p := p) f (a x)) Set.univ
Tactic: rw [← continuousOn_univ] at ha_cont ⊢
Cleared hypotheses: ha_cont, ha', hf, hf0
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ ContinuousOn (Y := A) (fun (x : A) ↦ cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f x) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
have this : Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
  {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
Tactic: have : {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}.EqOn (cfcₙ f)
      (cfcₙ (fun x : ℝ ↦ f x.toNNReal)) :=
    fun a ha ↦ cfcₙ_nnreal_eq_real _ _ ha.1
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ ContinuousOn (Y := A) (fun (x : A) ↦ cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f x) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
have refine : ContinuousOn (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
Tactic: refine ContinuousOn.congr ?_ this
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
have apply : Set.MapsTo Real.toNNReal (NNReal.toReal '' s) s
Tactic: apply hf.ofReal_map_toNNReal
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ Set.MapsTo Real.toNNReal (NNReal.toReal '' s) s
have rw : Set.MapsTo (Real.toNNReal ∘ NNReal.toReal) s s
Tactic: rw [Set.mapsTo_image_iff]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ ContinuousOn (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
have hf₁ : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
Tactic: replace hf : ContinuousOn (fun x ↦ f x.toNNReal : ℝ → ℝ) (NNReal.toReal '' s) := by
    apply hf.ofReal_map_toNNReal
    rw [Set.mapsTo_image_iff]
    intro x hx
    simpa
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
⊢ ContinuousOn (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
have refine : ∀ (a : A) (ha : Membership.mem (γ := Set A) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s} a),
  Membership.mem (γ := Set A) {a : A | IsSelfAdjoint a ∧ quasispectrum ℝ a ⊆ NNReal.toReal '' s} a
Tactic: refine continuousOn_cfcₙ A (hs.image NNReal.continuous_coe) _ hf |>.mono fun a ha ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
a : A
ha : Membership.mem (γ := Set A) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s} a
⊢ Membership.mem (γ := Set A) {a : A | IsSelfAdjoint a ∧ quasispectrum ℝ a ⊆ NNReal.toReal '' s} a
have ha₁ : (IsSelfAdjoint a ∧ QuasispectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)) ∧
  quasispectrum ℝ≥0 a ⊆ s
Tactic: simp only [Set.mem_setOf_eq, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts] at ha ⊢
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
a : A
ha :
  (IsSelfAdjoint a ∧ QuasispectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)) ∧
    quasispectrum ℝ≥0 a ⊆ s
⊢ IsSelfAdjoint a ∧ quasispectrum ℝ a ⊆ NNReal.toReal '' s
have refine : quasispectrum ℝ a ⊆ NNReal.toReal '' s
Tactic: refine ⟨ha.1.1, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf0 : f 0 = 0
this :
  Set.EqOn (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f) (cfcₙ (p := IsSelfAdjoint) fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ))
    {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (NNReal.toReal '' s)
a : A
ha :
  (IsSelfAdjoint a ∧ QuasispectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)) ∧
    quasispectrum ℝ≥0 a ⊆ s
⊢ quasispectrum ℝ a ⊆ NNReal.toReal '' s
have rw : (⇑(algebraMap ℝ≥0 ℝ) : (a : ℝ≥0) → ℝ) '' quasispectrum ℝ≥0 a ⊆ NNReal.toReal '' s
Tactic: rw [← ha.1.2.algebraMap_image]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, quasispectrum ℝ≥0 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, 0 ≤ a x
ha₀ : quasispectrum ℝ≥0 a₀ ⊆ s
ha₀' : 0 ≤ a₀
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ Tendsto (fun (x : X) ↦ cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f (a x)) l (𝓝 (cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f a₀))
have apply : Tendsto a l (𝓝[{a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}] a₀)
Tactic: apply continuousOn_cfcₙ_nnreal A hs f |>.continuousWithinAt ⟨ha₀', ha₀⟩ |>.tendsto.comp
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, quasispectrum ℝ≥0 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, 0 ≤ a x
ha₀ : quasispectrum ℝ≥0 a₀ ⊆ s
ha₀' : 0 ≤ a₀
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ Tendsto a l (𝓝[{a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}] a₀)
have rw : Tendsto a l (𝓝 a₀) ∧ ∀ᶠ (n : X) in l, Membership.mem (γ := Set A) {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s} (a n)
Tactic: rw [tendsto_nhdsWithin_iff]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
X : Type u_1
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
inst_12 : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
hf0 : f 0 = 0
⊢ ContinuousOn.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f (a x)) t
have ha_cont₁ : ∀ x ∈ t, ContinuousWithinAt a t x
Tactic: rw [ContinuousOn] at ha_cont ⊢
Cleared hypotheses: ha_cont, ha, ha', hf, hf0
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
X : Type u_1
A : Type u_2
inst : NonUnitalNormedRing A
inst_1 : StarRing A
inst_2 : NormedSpace ℝ A
inst_3 : IsScalarTower ℝ A A
inst_4 : SMulCommClass ℝ A A
inst_5 : ContinuousStar A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_7 : PartialOrder A
inst_8 : StarOrderedRing A
inst_9 : NonnegSpectrumClass ℝ A
inst_10 : T2Space A
inst_11 : IsSemitopologicalRing A
inst_12 : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
hf0 : f 0 = 0
refine_1 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x_1 : X) in 𝓝[t] x, quasispectrum ℝ≥0 (a x_1) ⊆ s x
⊢ ∀ x ∈ t, ContinuousWithinAt.{u_1, u_2} (Y := A) (fun (x : X) ↦ cfcₙ (p := fun (x : A) ↦ 0 ≤ x) f (a x)) t x
have refine_2 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x : X) in 𝓝[t] x, 0 ≤ a x
Tactic: refine fun x hx ↦ (ha_cont x hx).cfcₙ_nnreal (hs x hx) f hx ?_ ?_ (hf x hx)