mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ U ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
t : Set α
htK : t ⊆ K
x : α
hxt : x ∈ t
hxU : x ∉ U
⊢ t ∈ ⋃ a ∈ T, {s | (s ∩ a).Nonempty}

Tactic:
obtain ⟨V, hVT, hxV⟩ := mem_iUnion₂.mp <| hKT ⟨htK hxt, hxU⟩

Goal state after:
case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ U ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
t : Set α
htK : t ⊆ K
x : α
hxt : x ∈ t
hxU : x ∉ U
V : Set α
hVT : V ∈ T
hxV : x ∈ V
⊢ t ∈ ⋃ a ∈ T, {s | (s ∩ a).Nonempty}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σ 𝕜
      ((cfcHom ⋯)
        ((Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯)) (ContinuousMapZero.toContinuousMapHom f))) =
    Set.range ⇑f

Tactic:
cfcHom_map_spectrum (hp₁.mpr ha) (R := 𝕜) _

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ Set.range ⇑((Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯)) (ContinuousMapZero.toContinuousMapHom f)) =
    Set.range ⇑f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖cfc f a‖ = (fun x ↦ ‖f x‖) x

Tactic:
cfc_apply f a,

Goal state after:
case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖(cfcHom ha) { toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ }‖ = (fun x ↦ ‖f x‖) x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖cfc f a‖ ≤ c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A

Goal state after:
case inl
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
h✝ : Subsingleton A
⊢ ‖cfc f a‖ ≤ c

case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
h✝ : Nontrivial A
⊢ ‖cfc f a‖ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
⊢ ‖cfc f a‖ < c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A

Goal state after:
case inl
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
h✝ : Subsingleton A
⊢ ‖cfc f a‖ < c

case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
h✝ : Nontrivial A
⊢ ‖cfc f a‖ < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
hx : 0 ≤ x
⊢ Set.RightInvOn (⇑ContinuousMap.realToNNReal) (⇑(algebraMap NNReal ℝ)) (σₙ ℝ (x * (a * x))) ∧
    Function.LeftInverse ⇑ContinuousMap.realToNNReal ⇑(algebraMap NNReal ℝ)

Tactic:
← quasispectrumRestricts_iff,

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
hx : 0 ≤ x
⊢ QuasispectrumRestricts (x * (a * x)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
⊢ ‖cfc f a‖₊ ≤ c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A

Goal state after:
case inl
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h✝ : Subsingleton A
⊢ ‖cfc f a‖₊ ≤ c

case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h✝ : Nontrivial A
⊢ ‖cfc f a‖₊ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
hc : 0 < c
h : ∀ x ∈ σ ℝ≥0 a, f x < c
⊢ ‖cfc f a‖₊ < c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A

Goal state after:
case inl
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
hc : 0 < c
h : ∀ x ∈ σ ℝ≥0 a, f x < c
h✝ : Subsingleton A
⊢ ‖cfc f a‖₊ < c

case inr
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
hc : 0 < c
h : ∀ x ∈ σ ℝ≥0 a, f x < c
h✝ : Nontrivial A
⊢ ‖cfc f a‖₊ < c
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: case lower
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ {z | z.im < 0} ⊆ (σ ℂ a)ᶜ ∩ {z | z.im ≤ 0}

Tactic:
refine Set.subset_inter (fun z hz hz' ↦ ?_) (fun _ ↦ by simpa using le_of_lt)

Goal state after:
case lower
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ {z | z.im < 0}
hz' : z ∈ σ ℂ a
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: case upper
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ {z | 0 < z.im} ⊆ (σ ℂ a)ᶜ ∩ {z | 0 ≤ z.im}

Tactic:
refine Set.subset_inter (fun z hz hz' ↦ ?_) (fun _ ↦ by simpa using le_of_lt)

Goal state after:
case upper
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ {z | 0 < z.im}
hz' : z ∈ σ ℂ a
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
f : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜
ι : C(↑(σ 𝕜 a), ↑(σₙ 𝕜 a)) := { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ }
⊢ ∀ (x : ↑(σₙ 𝕜 a)), ‖↑f x‖ ≤ ‖(↑f).comp ι‖

Tactic:
· rintro ⟨x, hx⟩
      obtain (rfl | hx') : x = 0 ∨ x ∈ σ 𝕜 a := by
        simpa [quasispectrum_eq_spectrum_union_zero] using hx
      · change ‖f 0‖ ≤ _
        simp
      · exact (f : C(σₙ 𝕜 a, 𝕜)).comp ι |>.norm_coe_le_norm ⟨x, hx'⟩

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : RegularSpace α
⊢ RegularSpace (Compacts α)

Tactic:
simp_rw [regularSpace_generateFrom induced_generateFrom_eq, image_union, image_image, powerset,
    preimage_ofPred_eq, Filter.disjoint_iff]
  rintro _ (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩) K hK
  · obtain ⟨V, W, hV, hW, hKV, hUW, hVW⟩ :=
      SeparatedNhds.of_isCompact_isClosed K.isCompact hU.isClosed_compl
        (disjoint_compl_right_iff_subset.mpr hK)
    refine ⟨{K | (↑K ∩ W).Nonempty}, ?_, {K | ↑K ⊆ V},
      (isOpen_subsets_of_isOpen hV).mem_nhds_iff.mpr hKV, by grind [Set.Nonempty]⟩
    simp_rw [(isOpen_inter_nonempty_of_isOpen hW).mem_nhdsSet, compl_ofPred,
      ← inter_compl_nonempty_iff]
    grw [hUW]
  · obtain ⟨x, hx₁, hx₂⟩ := hK
    obtain ⟨V, W, hV, hW, hxV, hUW, hVW⟩ :=
      SeparatedNhds.of_isCompact_isClosed (isCompact_singleton (x := x)) hU.isClosed_compl
        (by simpa)
    refine ⟨{K | ↑K ⊆ W}, ?_, {K | (↑K ∩ V).Nonempty}, ?_, by grind [Set.Nonempty]⟩
    · simp_rw [(isOpen_subsets_of_isOpen hW).mem_nhdsSet, compl_ofPred, not_nonempty_iff_eq_empty,
        ← disjoint_iff_inter_eq_empty, ← subset_compl_iff_disjoint_right]
      gcongr
    · rw [(isOpen_inter_nonempty_of_isOpen hV).mem_nhds_iff]
      exact ⟨x, hx₁, hxV <| Set.mem_singleton x⟩

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
hU :
  ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝³] U},
    K ∈ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ∧ {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty} ⊆ U
⊢ ∃ s ∈ 𝓝 K, s ⊆ U ∧ IsCompact s

Tactic:
obtain ⟨u, ⟨hu₁, hu₂⟩, ⟨hKu₁, hKu₂⟩, huU⟩ := hU

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
U : Set (Compacts α)
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
huU : {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty} ⊆ U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
⊢ ∃ s ∈ 𝓝 K, s ⊆ U ∧ IsCompact s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ IsPreconnected U

Tactic:
by
    refine this.subset_closure inter_subset_left ?_
    grw [← (basis.isOpen hU).inter_closure, dense_setOfPred_finite.closure_eq, inter_univ]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
⊢ expMulMulExp a b x z = ↑(expUnitary (2 • ℑ (z • star b))) * x * ↑(expUnitary (2 • ℑ (star z • a)))

Tactic:
let _ : NormedAlgebra ℚ A := .restrictScalars ℚ ℂ A
  nth_rw 1 [expMulMulExp, ← (h.smul_right (star z)).exp_neg_mul_mul_exp_eq_self]
  simp_rw [← mul_assoc, mul_assoc (_ * _ * x)]
  congr!
  all_goals
    simp [imaginaryPart_apply_coe, smul_comm (2 : ℝ) I, smul_smul I I, sub_eq_add_neg]
    grind [exp_add_of_commute, Commute.smul_right, Commute.neg_right]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : LocallyConvexSpace 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0 ∧ Convex 𝕜 s
⊢ Convex 𝕜 ((convexHull 𝕜) (balancedCore 𝕜 s))

Tactic:
exact convex_convexHull 𝕜 (balancedCore 𝕜 s)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
| deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2

Tactic:
enter [

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
| deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
K L : NonemptyCompacts α
hL : IsPreconnected ↑L
⊢ IsPreconnected (Ioc K L)

Tactic:
rw [← isEmbedding_toCompacts.isPreconnected_image, ← coe_toCompactsOrderEmbedding,
    OrderEmbedding.image_Ioc _ (by simpa [← Set.Ioi_bot] using ordConnected_Ioi)]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
K L : NonemptyCompacts α
hL : IsPreconnected ↑L
⊢ IsPreconnected (Ioc (toCompactsOrderEmbedding K) (toCompactsOrderEmbedding L))
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
this : Continuous[t₁, t𝕜] ⇑φ ↔ ∃ s, Continuous[t₂ s, t𝕜] ⇑φ
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) ↔ Continuous[⨅ i, induced (⇑(f i)) inferInstance, inferInstance] ⇑φ

Tactic:
simp_rw [this, ← mem_span_iff_continuous_of_finite, Submodule.span_range_eq_iSup,
      iSup_subtype]
    rw [Submodule.mem_iSup_iff_exists_finset]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁷ : NontriviallyNormedField 𝕜₁
inst✝¹⁶ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : Module 𝕜₂ F
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformSpace F
inst✝⁸ : IsUniformAddGroup E
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : ContinuousSMul 𝕜₁ E
inst✝⁵ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝⁴ : PolynormableSpace 𝕜₂ F
inst✝³ : ContinuousSMul 𝕜₂ F
inst✝² : T2Space F
l : Filter α
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
g : α → E →SL[σ₁₂] F
f : E → F
h : Tendsto (fun n x ↦ (g n) x) l (𝓝 f)
u : ℕ → α
hu : Tendsto u atTop l
⊢ Continuous[inst✝¹⁰.toTopologicalSpace, inst✝⁹.toTopologicalSpace] (linearMapOfTendsto f (fun a ↦ ↑(g a)) h).toFun

Tactic:
refine (h.comp hu).continuous_of_equicontinuous
      (PolynormableSpace.banach_steinhaus ?_).equicontinuous

Goal state after:
α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁷ : NontriviallyNormedField 𝕜₁
inst✝¹⁶ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : Module 𝕜₂ F
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformSpace F
inst✝⁸ : IsUniformAddGroup E
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : ContinuousSMul 𝕜₁ E
inst✝⁵ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝⁴ : PolynormableSpace 𝕜₂ F
inst✝³ : ContinuousSMul 𝕜₂ F
inst✝² : T2Space F
l : Filter α
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
g : α → E →SL[σ₁₂] F
f : E → F
h : Tendsto (fun n x ↦ (g n) x) l (𝓝 f)
u : ℕ → α
hu : Tendsto u atTop l
⊢ ∀ (x : E), IsVonNBounded 𝕜₂ (range fun i ↦ (g (u i)) x)
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
inst✝¹ : SeminormedAddCommGroup V
inst✝ : NormedSpace ℝ V
p₁ p₂ p₃ p₄ : V
⊢ dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2

Tactic:
simpa using dist_midpoint_midpoint_le' (𝕜 := ℝ) p₁ p₂ p₃ p₄

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed[inst✝¹] s
ht : IsClosed[inst✝¹] t
hd : Disjoint s t
P : Set X → Set X → Prop := fun x x_1 ↦ True
⊢ ∀ {c u : Set X},
    IsClosed[inst✝¹] c →
      P c u →
        IsOpen[inst✝¹] u →
          c ⊆ u → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u ∧ P c v ∧ P (closure[inst✝¹] v) u

Tactic:
rintro c u c_closed - u_open cu

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed[inst✝¹] s
ht : IsClosed[inst✝¹] t
hd : Disjoint s t
P : Set X → Set X → Prop := fun x x_1 ↦ True
c u : Set X
c_closed : IsClosed[inst✝¹] c
u_open : IsOpen[inst✝¹] u
cu : c ⊆ u
⊢ ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u ∧ P c v ∧ P (closure[inst✝¹] v) u
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : CommRing 𝕜
inst✝³ : TopologicalSpace 𝕜
inst✝² : IsTopologicalRing 𝕜
inst✝¹ : Nontrivial 𝕜
inst✝ : NoZeroDivisors 𝕜
x : X
⊢ { toFun := fun f ↦ f x, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } ∈
    {φ | φ 1 = 1 ∧ ∀ (x y : C(X, 𝕜)), φ (x * y) = φ x * φ y}

Tactic:
exact ⟨rfl, fun f g => rfl⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺

Tactic:
have := (not_iff_not.mpr <| (IsSelfAdjoint.all r).smul_iff hr.isUnit (x := a)) |>.mpr ha
      simp [CFC.posPart_def, cfcₙ_apply_of_not_predicate a ha,
        cfcₙ_apply_of_not_predicate _ this]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a c : A
hac : a⁺ - a⁻ = a⁺ - c
hc : 0 ≤ c
ha : IsSelfAdjoint a
⊢ a⁻ = c

Tactic:
simpa using hac

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Fintype ↑𝔖
f g : α →ᵤ[𝔖] β
⊢ ⨆ s ∈ 𝔖, ⨆ x ∈ s, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x) =
    edist (fun s ↦ UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) f))) fun s ↦
      UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) g))

Tactic:
iSup_subtype',

Goal state after:
α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Fintype ↑𝔖
f g : α →ᵤ[𝔖] β
⊢ ⨆ x, ⨆ x_1, edist ((toFun 𝔖) f ↑x_1) ((toFun 𝔖) g ↑x_1) =
    edist (fun s ↦ UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) f))) fun s ↦
      UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) g))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : ‖a‖₊ ≤ t
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ‖(algebraMap ℝ A) ↑t - a‖₊ ≤ t

Tactic:
have : IsSelfAdjoint (algebraMap ℝ A t - a) := IsSelfAdjoint.algebraMap A (.all (t : ℝ)) |>.sub ha
  rw [← ENNReal.coe_le_coe, ← IsSelfAdjoint.spectralRadius_eq_nnnorm,
    ← SpectrumRestricts.spectralRadius_eq (f := Complex.reCLM)] at ht ⊢
  · exact SpectrumRestricts.nnreal_iff_spectralRadius_le ht
  all_goals
    try apply IsSelfAdjoint.spectrumRestricts
    assumption

Goal state after: