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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace.{u_2} (↑s : Type u_2)
const :
  ∀ (r : 𝕜),
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) (ContinuousMap.const (↑s : Type u_2) r)
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
have id : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
  (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A)
    (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
Tactic: induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop
Cleared hypotheses: f
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace.{u_2} (↑s : Type u_2)
const :
  ∀ (r : 𝕜),
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) (ContinuousMap.const (↑s : Type u_2) r)
id :
  Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
have star_id : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
  (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A)
    (star (ContinuousMap.restrict s (ContinuousMap.id 𝕜)))
Tactic: induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop
Cleared hypotheses: f
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace.{u_2} (↑s : Type u_2)
const :
  ∀ (r : 𝕜),
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) (ContinuousMap.const (↑s : Type u_2) r)
id :
  Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
star_id :
  Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      (star (ContinuousMap.restrict s (ContinuousMap.id 𝕜)))
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
have add : ∀ (f g : C((↑s : Type u_2), 𝕜))
  (hf :
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A) f)
  (hg :
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A) g),
  Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A) (f + g)
Tactic: induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop
Cleared hypotheses: f
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
have open : CompactSpace.{u_2} (↑s : Type u_2)
Tactic: open scoped ContinuousFunctionalCalculus in
  have : CompactSpace s := by rwa [isCompact_iff_compactSpace] at hs
  induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
open : CompactSpace.{u_2} (↑s : Type u_2)
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
have open₁ : ∀ (this : CompactSpace.{u_2} (↑s : Type u_2)),
  Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A) f
Tactic: open scoped ContinuousFunctionalCalculus in
  have : CompactSpace s := by rwa [isCompact_iff_compactSpace] at hs
  induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f))
have h : ∀ (x : (↑{a : A | p a ∧ spectrum 𝕜 a ⊆ s} : Type u_3)),
  Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f) x =
    (cfcHomSuperset (p := p) sorry (s := s) sorry : (a : C((↑s : Type u_2), 𝕜)) → A)
      { toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry : C((↑s : Type u_2), 𝕜) }
Tactic: convert!
      continuous_cfcHomSuperset_left hs ⟨_, hf.restrict⟩ ((↑) : {a | p a ∧ spectrum 𝕜 a ⊆ s} → A)
        continuous_subtype_val (fun x ↦ x.2.2) with
      x
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (goal)
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
x : (↑{a : A | p a ∧ spectrum 𝕜 a ⊆ s} : Type u_3)
⊢ Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f) x =
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      { toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }
have h : Eq.{u_3 + 1} (α := A)
  ((cfcHom (p := p) sorry : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
      { toFun := (spectrum 𝕜 (↑x : A)).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry :
        C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜) } :
    A)
  ((cfcHom (p := p) sorry : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
      ({ toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry : C((↑s : Type u_2), 𝕜) }.comp
        { toFun := Subtype.map (p := Membership.mem (spectrum 𝕜 (↑x : A))) (q := Membership.mem s) id sorry,
          continuous_toFun := sorry : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), (↑s : Type u_2)) }) :
    A)
Tactic: rw [cfcHomSuperset_apply, Set.restrict_apply, cfc_apply _ _ x.2.1 (hf.mono x.2.2)]
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
⊢ Tendsto (fun (x : X) ↦ cfc (p := p) f (a x)) l (𝓝 (cfc (p := p) f a₀))
have apply : Tendsto a l (𝓝[{a : A | p a ∧ spectrum 𝕜 a ⊆ s}] a₀)
Tactic: apply continuousOn_cfc A hs f |>.continuousWithinAt ⟨ha₀', ha₀⟩ |>.tendsto.comp
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
⊢ Tendsto a l (𝓝[{a : A | p a ∧ spectrum 𝕜 a ⊆ s}] a₀)
have rw : Tendsto a l (𝓝 a₀) ∧ ∀ᶠ (n : X) in l, Membership.mem (γ := Set A) {a : A | p a ∧ spectrum 𝕜 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
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
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, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) 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
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
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, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
refine_1 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x_1 : X) in 𝓝[t] x, spectrum 𝕜 (a x_1) ⊆ s x
⊢ ∀ x ∈ t, ContinuousWithinAt.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t x
have refine_2 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x : X) in 𝓝[t] x, p (a x)
Tactic: refine fun x hx ↦ (ha_cont x hx).cfc (hs x hx) f hx ?_ ?_ (hf 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
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, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x : X
hx : x ∈ t
⊢ ∀ᶠ (x_1 : X) in 𝓝[t] x, spectrum 𝕜 (a x_1) ⊆ s x
have h : ∀ (x_1 : X) (hx : spectrum 𝕜 (a x_1) ⊆ s x) (hxt : x_1 ∈ t), spectrum 𝕜 (a x_1) ⊆ s x
Tactic: filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
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, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x : X
hx : x ∈ t
⊢ ∀ᶠ (x : X) in 𝓝[t] x, p (a x)
have h : ∀ (x_1 : X) (hx : spectrum 𝕜 (a x_1) ⊆ s x) (hxt : x_1 ∈ t), p (a x_1)
Tactic: filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t
have refine : ∀ x ∈ t, ∀ᶠ (x : X) in 𝓝[t] x, spectrum 𝕜 (a x) ⊆ s
Tactic: refine ContinuousOn.cfc _ (fun _ _ ↦ hs) ha_cont (fun _ _ ↦ ?_) ha'
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : CompleteSpace A
inst_7 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))) s
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t
have hs'₁ : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
Tactic: simp only [nhdsSet_iUnion, mem_iSup] at hs'
Cleared hypotheses: hs'
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : CompleteSpace A
inst_7 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
x : (↑t : Type u_1)
⊢ ∃ (S : Set 𝕜), IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
have obtain : ∀ (S : Set 𝕜) (hS₃ : id S ⊆ s) (hS₁ : S ∈ 𝓝ˢ (spectrum 𝕜 (a (↑x : X)))) (hS₂ : IsCompact S),
  ∃ (S : Set 𝕜), IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
Tactic: obtain ⟨S, ⟨hS₁, hS₂⟩, hS₃⟩ :=
      spectrum.isCompact (𝕜 := 𝕜) (a x) |>.nhdsSet_basis_isCompact.mem_iff.mp (hs' x x.2)
-- 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 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : CompleteSpace A
inst_7 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
x : (↑t : Type u_1)
S : Set 𝕜
hS₃ : id S ⊆ s
hS₁ : S ∈ 𝓝ˢ (spectrum 𝕜 (a (↑x : X)))
hS₂ : IsCompact S
⊢ ∃ (S : Set 𝕜), IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
have refine : ∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum 𝕜 x' ⊆ S
Tactic: refine ⟨S, hS₂, ?_, hS₃⟩
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity (hypothesis)
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : CompleteSpace A
inst_7 : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : Membership.mem (γ := Filter 𝕜) (𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))) s
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ (i : X) (i_1 : i ∈ t), s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t
have this : ∀ (x : (↑t : Type u_1)), ∃ (S : Set 𝕜), IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a (↑x : X)), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
Tactic: have (x : t) : ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a x), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s := by
    obtain ⟨S, ⟨hS₁, hS₂⟩, hS₃⟩ :=
      spectrum.isCompact (𝕜 := 𝕜) (a x) |>.nhdsSet_basis_isCompact.mem_iff.mp (hs' x x.2)
    refine ⟨S, hS₂, ?_, hS₃⟩
    exact upperHemicontinuous_spectrum 𝕜 A |>.upperHemicontinuousAt (a x) _ hS₁ |>.mono
      fun _ ↦ subset_of_mem_nhdsSet