mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_1
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } ∈
    ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra))

Tactic:
rw [SetLike.mem_coe, Subalgebra.mem_comap]
    convert! (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A)
    ext1
    simp [← RCLike.mul_conj]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
x : α
⊢ (∀ t ∈ 𝓝 (f x), ∀ᶠ (x' : α) in 𝓝[s] x, t ∈ 𝓝 (f x')) → ∀ s_1 ∈ 𝓝 (f x), ∀ᶠ (x : α) in 𝓝[s] x, f x ∈ s_1

Tactic:
intro h

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
x : α
h : ∀ t ∈ 𝓝 (f x), ∀ᶠ (x' : α) in 𝓝[s] x, t ∈ 𝓝 (f x')
⊢ ∀ s_1 ∈ 𝓝 (f x), ∀ᶠ (x : α) in 𝓝[s] x, f x ∈ s_1
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
x : α
⊢ (∀ (u : Set β), IsOpen[inst✝] u → f x ∈ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ∈ u) →
    ∀ s_1 ∈ 𝓝 (f x), ∀ᶠ (x : α) in 𝓝[s] x, f x ∈ s_1

Tactic:
intro h

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
x : α
h : ∀ (u : Set β), IsOpen[inst✝] u → f x ∈ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ∈ u
⊢ ∀ s_1 ∈ 𝓝 (f x), ∀ᶠ (x : α) in 𝓝[s] x, f x ∈ s_1
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ IsUnit ((algebraMap 𝕜 A) ↑u⁻¹ - a)

Tactic:
← mem_resolventSet_iff

Goal state after:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ ↑u⁻¹ ∈ resolventSet 𝕜 a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}) ⊆
    ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))

Tactic:
rw [← NonUnitalStarSubalgebra.coe_map, SetLike.coe_subset_coe, NonUnitalStarSubalgebra.map_le]

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)} ≤
    NonUnitalStarSubalgebra.comap (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
U : Set α
hxU : x ∈ U
hU : IsOpen[inst✝.toTopologicalSpace] U
⊢ U ∈ f

Tactic:
filter_upwards [hK <| (isOpen_inter_nonempty_of_isOpen hU).mem_nhds ⟨x, hx, hxU⟩]

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
U : Set α
hxU : x ∈ U
hU : IsOpen[inst✝.toTopologicalSpace] U
⊢ ∀ a ∈ (fun x ↦ {x}) ⁻¹' {K | (↑K ∩ U).Nonempty}, a ∈ U
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (g₁ x) (g₂ x) ≤
    dist
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))

Tactic:
apply le_of_eq_of_le ?_ <| ContinuousMap.dist_apply_le_dist x'

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (g₁ x) (g₂ x) =
    dist
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        x')
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        x')
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ IsGreatest ((fun x ↦ ‖f x‖₊) '' σₙ 𝕜 a) ‖cfcₙ f a‖₊

Tactic:
convert! Real.toNNReal_monotone.map_isGreatest (.norm_cfcₙ f a)

Goal state after:
case e'_3
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ (fun x ↦ ‖f x‖₊) '' σₙ 𝕜 a = Real.toNNReal '' (fun x ↦ ‖f x‖) '' σₙ 𝕜 a

case e'_4
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ ‖cfcₙ f a‖₊ = ‖cfcₙ f a‖.toNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (↑g₁ x) (↑g₂ x) ≤
    dist
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))

Tactic:
apply le_of_eq_of_le ?_ <| ContinuousMap.dist_apply_le_dist x'

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (↑g₁ x) (↑g₂ x) =
    dist
      (↑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
        x')
      (↑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
        x')
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝¹¹ : TopologicalSpace A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : PartialOrder A
inst✝⁸ : StarRing A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : IsSemitopologicalRing A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : SMulCommClass ℝ A A
inst✝² : T2Space A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ → ℝ
a : A
hf_nonneg : ∀ x ∈ σₙ ℝ a, 0 ≤ f x
ha : 0 ≤ a
⊢ cfcₙ f a = cfcₙ (fun x ↦ (f ↑x).toNNReal) a

Tactic:
by
  rw [cfcₙ_nnreal_eq_real ..]
  refine cfcₙ_congr fun x hx ↦ ?_
  rw [x.coe_toNNReal (quasispectrum_nonneg_of_nonneg _ ha _ hx),
    (f x).coe_toNNReal (hf_nonneg x hx)]

Goal state after:
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
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
⊢ f x ≤ ‖cfc f a‖₊

Tactic:
revert hx
  nontriviality A
  exact (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.2 ⟨x, ·, rfl⟩)

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
⊢ (↑z.re).im = 0

Tactic:
ofReal_im

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
⊢ 0 = 0
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

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

Tactic:
have := Filter.NeBot.nonempty_of_mem inferInstance <| Filter.mem_map.mp <|
      Complex.isometry_ofReal.antilipschitz.tendsto_cobounded (spectrum.isBounded a |>.compl)

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
this : (ofReal ⁻¹' (σ ℂ a)ᶜ).Nonempty
⊢ ((σ ℂ a)ᶜ ∩ {z | 0 ≤ z.im} ∩ ((σ ℂ a)ᶜ ∩ {z | z.im ≤ 0})).Nonempty
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit ((↑a)⋆ * ↑a)
⊢ IsUnit a

Tactic:
have ha₂ := ha.mul ha.star

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit ((↑a)⋆ * ↑a)
ha₂ : IsUnit (↑a * (↑a)⋆)
⊢ IsUnit a
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✝ : DiscreteTopology α
K : Compacts α
⊢ {K} = {a | ↑a = ↑K}

Tactic:
SetLike.coe_set_eq,

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✝ : DiscreteTopology α
K : Compacts α
⊢ {K} = {a | a = K}
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
⊢ B.polar {0} = Set.univ

Tactic:
simp only [polar_singleton, map_zero, zero_apply, norm_zero, zero_le_one, Set.ofPred_true]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ (absConvexHull 𝕜) s = ∅ → s = ∅

Tactic:
intro h

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
h : (absConvexHull 𝕜) s = ∅
⊢ s = ∅
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ {y | ∀ x ∈ m, (B x) y = 0} → y ∈ B.polar ↑m

Tactic:
intro h

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
h : y ∈ {y | ∀ x ∈ m, (B x) y = 0}
⊢ y ∈ B.polar ↑m
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
y x : ℝ
| NormedSpace.exp (x • a) * NormedSpace.exp (x • b)

Tactic:
1

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
y x : ℝ
| NormedSpace.exp (x • a)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (NonemptyCompacts α)
hs : IsCompact S
⊢ ⋃ K ∈ S, ↑K = ⋃ K ∈ toCompacts '' S, ↑K

Tactic:
biUnion_image,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
S : Set (NonemptyCompacts α)
hs : IsCompact S
⊢ ⋃ K ∈ S, ↑K = ⋃ y ∈ S, ↑y.toCompacts
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
⊢ IsPreconnected (⋃ L ∈ V, ↑L)

pp: case refine_3
α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
⊢ id (⋃ L ∈ V, ↑L) ⊆ U

Tactic:
· exact vietoris.isPreconnected_biUnion hV₂ (by fun_prop) ⟨{x}, hxV, isPreconnected_singleton⟩

Goal state after:
case refine_3
α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
⊢ id (⋃ L ∈ V, ↑L) ⊆ U
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp : ∀ (x : ↥S), f x ≤ p ↑x
⊢ ?m.34

Tactic:
refine exists_extension_of_le_sublinear ⟨S, f⟩ p (fun _ hc _ => ?_) ?_ hp

Goal state after:
case refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp : ∀ (x : ↥S), f x ≤ p ↑x
x✝¹ : ℝ
hc : 0 < x✝¹
x✝ : E
⊢ p (x✝¹ • x✝) = x✝¹ * p x✝

case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp : ∀ (x : ↥S), f x ≤ p ↑x
⊢ ∀ (x y : E), p (x + y) ≤ p x + p y
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : ℕ → E' →SL[σ₁₂] F
g : E' →SL[σ₁₂] F
hg : Tendsto (fun n x ↦ (f n) x) atTop (𝓝 ⇑g)
hf : CauchySeq f
⊢ Tendsto f atTop (𝓝 g)

Tactic:
by
  /- Since `f` is a Cauchy sequence, there exists `b → 0` such that `‖f n - f m‖ ≤ b N` for any
    `m, n ≥ N`. -/
  rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩
  simp_rw [dist_eq_norm] at hfb
  -- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
  suffices ∀ n x, ‖f n x - g x‖ ≤ b n * ‖x‖ from
    tendsto_iff_norm_sub_tendsto_zero.2
    (squeeze_zero (fun n => norm_nonneg _) (fun n => opNorm_le_bound _ (hb₀ n) (this n)) hb_lim)
  intro n x
  -- Note that `f m x → g x`, hence `‖f n x - f m x‖ → ‖f n x - g x‖` as `m → ∞`
  have : Tendsto (fun m => ‖f n x - f m x‖) atTop (𝓝 ‖f n x - g x‖) :=
    (tendsto_const_nhds.sub <| tendsto_pi_nhds.1 hg _).norm
  -- Thus it suffices to verify `‖f n x - f m x‖ ≤ b n * ‖x‖` for `m ≥ n`.
  refine le_of_tendsto this (eventually_atTop.2 ⟨n, fun m hm => ?_⟩)
  -- This inequality follows from `‖f n - f m‖ ≤ b n`.
  exact (f n - f m).le_of_opNorm_le (hfb _ _ _ le_rfl hm) _

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ ‖toStrongDual φ‖ ≤ ‖1‖

Tactic:
exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ :)

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
c : CU P
⊢ c.Uᶜ.indicator 1 x ≤ c.left.Uᶜ.indicator 1 x

Tactic:
grw [left_U_subset]

Goal state after:
case hf
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
c : CU P
⊢ 0 ≤ 1 x
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
x : E
⊢ (k⁻¹ • fun x ↦ x -ᵥ c) ((fun x ↦ k • x +ᵥ c) x) = x

Tactic:
simp [inv_smul_smul₀ hk]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r' : k
⊢ 0 < r' * (b - a) → (f ((lineMap b a) r') ≤ (lineMap (f b) (f a)) r' ↔ slope f a b ≤ slope f ((lineMap b a) r') b)

Tactic:
intro h

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r' : k
h : 0 < r' * (b - a)
⊢ f ((lineMap b a) r') ≤ (lineMap (f b) (f a)) r' ↔ slope f a b ≤ slope f ((lineMap b a) r') b
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : Semiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : T2Space R
I : Ideal C(X, R)
⊢ IsOpen (setOfIdeal I)

Tactic:
simp only [setOfIdeal, Set.ofPred_forall, isOpen_compl_iff]
  exact
    isClosed_iInter fun f =>
      isClosed_iInter fun _ => isClosed_eq (map_continuous f) continuous_const

Goal state after: