mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr.refine_3
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
W : eVariationOn f (s ∩ Icc y x ∩ Iio x) = eVariationOn f (s ∩ Icc y x ∩ Iic x)
⊢ eVariationOn f (s ∩ Ico y x) = eVariationOn f (s ∩ Icc y x)

Tactic:
convert! W using 2

Goal state after:
case e'_2
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
W : eVariationOn f (s ∩ Icc y x ∩ Iio x) = eVariationOn f (s ∩ Icc y x ∩ Iic x)
⊢ s ∩ Ico y x = s ∩ Icc y x ∩ Iio x

case e'_3
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
W : eVariationOn f (s ∩ Icc y x ∩ Iio x) = eVariationOn f (s ∩ Icc y x ∩ Iic x)
⊢ s ∩ Icc y x = s ∩ Icc y x ∩ Iic x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : ContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousInv₀ R
inst✝ : UniqueHom R A
f : R → R
a : Aˣ
hf : ContinuousOn f ((fun x ↦ x⁻¹) '' spectrum R ↑a)
ha : p ↑a
⊢ cfc (fun x ↦ f x⁻¹) ↑a = cfc f ↑a⁻¹

Tactic:
by
  rw [cfc_comp' .., cfc_inv_id _]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a)

Tactic:
obtain ⟨ha, hf, h0⟩ := h

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case mp
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R → R
⊢ (fun x ↦ cfcₙ x a) f ∈ ↑(NonUnitalStarAlgHom.range (cfcₙHom ha))

pp: case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
⊢ ↑(NonUnitalAlgHomClass.toNonUnitalAlgHom (cfcₙHom ha)) f ∈ Set.range fun x ↦ cfcₙ x a

Tactic:
· exact cfcₙ_cases _ a f (zero_mem _) fun hf hf₀ ha ↦ ⟨_, rfl⟩

Goal state after:
case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
⊢ ↑(NonUnitalAlgHomClass.toNonUnitalAlgHom (cfcₙHom ha)) f ∈ Set.range fun x ↦ cfcₙ x a
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: f : C(↑I, ℝ)
n : ℕ
i : Fin (n + 1)
⊢ bernstein n ↑i • const (↑I) (f (bernstein.z i)) ∈ polynomialFunctions I

Tactic:
rw [← SetLike.mem_coe, polynomialFunctions_coe]

Goal state after:
f : C(↑I, ℝ)
n : ℕ
i : Fin (n + 1)
⊢ bernstein n ↑i • const (↑I) (f (bernstein.z i)) ∈ Set.range ⇑(Polynomial.toContinuousMapOnAlgHom I)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
obtain ⟨ha, hf, h0⟩ := h

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ ψ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
h : φ.realContinuousMapOfNNReal = ψ.realContinuousMapOfNNReal
f : C(X, ℝ≥0)
⊢ φ f = ψ f

Tactic:
simpa using congr($(h) ((ContinuousMap.mk toReal continuous_coe).comp f))

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
⊢ (-(↑r • f)).toNNReal = r • (-f).toNNReal

Tactic:
← smul_neg,

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
⊢ (↑r • -f).toNNReal = r • (-f).toNNReal
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖b⋆‖ * ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖ ≤
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖

Tactic:
nth_rewrite 2 [← one_mul ‖_ + _‖]
      gcongr
      exact (norm_star b).symm ▸ mem_closedBall_zero_iff.1 hb

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))
SW : A₀.topologicalClosure = ⊤
⊢ (↑I).range ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure

Tactic:
← Submodule.map_top,

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))
SW : A₀.topologicalClosure = ⊤
⊢ Submodule.map ↑I ⊤ ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case star
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f✝ f : C(↑s, 𝕜)
hx✝ : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf : f ∈ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))
⊢ star f ∈ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
    rw [map_star, hf, star_zero]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
⊢ φ = ψ

Tactic:
ext x
  have : x - algebraMap 𝕜 A (ψ x) ∈ RingHom.ker φ := by
    simpa only [h, RingHom.mem_ker, map_sub, AlgHomClass.commutes] using! sub_self (ψ x)
  rwa [RingHom.mem_ker, map_sub, AlgHomClass.commutes, sub_eq_zero] at this

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x : α
t : Set α
⊢ (fun x t ↦ IsOpen[inst✝] t ∧ ((fun x ↦ {x}) x ∩ t).Nonempty) x t →
    ∀ᶠ (x' : α) in 𝓝 x, (fun x t ↦ IsOpen[inst✝] t ∧ ((fun x ↦ {x}) x ∩ t).Nonempty) x' t

Tactic:
⟨ht, hne⟩

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
x : α
t : Set α
ht : IsOpen[inst✝] t
hne : ((fun x ↦ {x}) x ∩ t).Nonempty
⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x t ↦ IsOpen[inst✝] t ∧ ((fun x ↦ {x}) x ∩ t).Nonempty) x' t
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ (((fun x ↦ U ∩ x) '' v).Finite ∧ IsOpen[inst✝] U ∧ ∀ x ∈ (fun x ↦ U ∩ x) '' v, IsOpen[inst✝] x) ∧
    (s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ x ∈ (fun x ↦ U ∩ x) '' v, (s ∩ x).Nonempty) ∧
      {s | s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ x ∈ (fun x ↦ U ∩ x) '' v, (s ∩ x).Nonempty} ⊆
        𝒫 U ∩ ⋂ a ∈ v, {s | (s ∩ a).Nonempty}

Tactic:
forall_mem_image,

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ (((fun x ↦ U ∩ x) '' v).Finite ∧ IsOpen[inst✝] U ∧ ∀ ⦃x : Set α⦄, x ∈ v → IsOpen[inst✝] (U ∩ x)) ∧
    (s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ (U ∩ x)).Nonempty) ∧
      {s | s ⊆ U ∧ (s ∩ U).Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ (U ∩ x)).Nonempty} ⊆
        𝒫 U ∩ ⋂ a ∈ v, {s | (s ∩ a).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
⊢ Continuous[inst✝¹, TopologicalSpace.vietoris β] f ↔
    (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
      ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)

Tactic:
refine ⟨fun hf =>
      ⟨fun U hU => hU.powerset_vietoris.preimage hf, fun F hF => hF.powerset_vietoris.preimage hf⟩,
    fun ⟨h₁, h₂⟩ => ?_⟩
  rw [continuous_generateFrom_iff]
  rintro _ (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)
  · exact h₁ U hU
  · simp_rw [preimage_ofPred_eq, ← not_disjoint_iff_nonempty_inter, ← compl_ofPred,
      isOpen_compl_iff, ← subset_compl_iff_disjoint_right]
    exact h₂ Uᶜ hU.isClosed_compl

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

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
hs : ∅ ∈ (range fun x ↦ {x})ᶜ
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 ∅

Tactic:
(isOpen_singleton_iff_nhds_eq_pure _).mp h₁,

Goal state after:
case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
hs : ∅ ∈ (range fun x ↦ {x})ᶜ
⊢ (range fun x ↦ {x})ᶜ ∈ pure ∅
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
U V : Set α
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
hxU : x ∈ U
hyV : y ∈ V
hUV : Disjoint U V
⊢ ∀ a ∈ {s | (s ∩ U).Nonempty} ∩ {s | (s ∩ V).Nonempty}, a ∈ (range fun x ↦ {x})ᶜ

Tactic:
rintro _ ⟨hzU, hzV⟩ ⟨z, rfl⟩

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
U V : Set α
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
hxU : x ∈ U
hyV : y ∈ V
hUV : Disjoint U V
z : α
hzU : (fun x ↦ {x}) z ∈ {s | (s ∩ U).Nonempty}
hzV : (fun x ↦ {x}) z ∈ {s | (s ∩ V).Nonempty}
⊢ False
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
μ : 𝕜
hμ : μ ∈ frontier (σ 𝕜 x)
⊢ μ ∈ σ 𝕜 ↑x

Tactic:
by_contra h

Goal state after:
𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
μ : 𝕜
hμ : μ ∈ frontier (σ 𝕜 x)
h : μ ∉ σ 𝕜 ↑x
⊢ False
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)
⊢ σ 𝕜 x = σ 𝕜 ↑x ∪ σ 𝕜 x \ σ 𝕜 ↑x

Tactic:
union_sdiff_cancel (spectrum.subset_subalgebra x)

Goal state after:
𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)
⊢ σ 𝕜 x = σ 𝕜 x
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
𝕜 : Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedField 𝕜
inst✝² : NormedDivisionRing R
inst✝¹ : Algebra 𝕜 R
inst✝ : CompleteSpace R
f : C(X, R)
⊢ spectrum 𝕜 f = ⇑(algebraMap 𝕜 R) ⁻¹' Set.range ⇑f

Tactic:
ext x
  simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,
    Classical.not_not, Set.mem_range,
    sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,
    smul_apply, one_apply]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
⊢ idealOfSet 𝕜 (setOfIdeal I) ≤ I.closure

Tactic:
refine (fun f hf => Metric.mem_closure_iff.mpr fun ε hε => ?_)

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ
hε : ε > 0
⊢ ∃ b ∈ ↑I, dist f b < ε
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_3.refine_2.inr
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₂
⊢ 0 < (g₁ + g₂) x

Tactic:
simpa using add_lt_add_of_le_of_lt zero_le (hgt₂ x hx)

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

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ x⁺ * x⁻ = 0 x

Tactic:
simp only [_root_.posPart_def, _root_.negPart_def]

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ max x 0 * max (-x) 0 = 0 x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ x⁻ * x⁺ = 0 x

Tactic:
simp only [_root_.posPart_def, _root_.negPart_def]

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ max (-x) 0 * max x 0 = 0 x
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 : T2Space A
inst✝ : StarModule ℝ A
r : ℝ
hr : r ≤ 0
a : A
⊢ (-(-r • a))⁺ = -r • a⁻

Tactic:
← smul_neg,

Goal state after:
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
hr : r ≤ 0
a : A
⊢ (-r • -a)⁺ = -r • a⁻
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : IsEmpty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

Tactic:
· simp_all [edist_def]

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ 1⁻ = 0

Tactic:
by
  rw [CFC.negPart_def, cfcₙ_eq_cfc]
  simp

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ ((algebraMap ℝ A) r)⁻ = (algebraMap ℝ A) r⁻

Tactic:
by
  rw [CFC.negPart_def, cfcₙ_eq_cfc]
  simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
x : A
hx : x ∈ {a | 0 ≤ a}
hx_pos : 0 < ‖x‖
⊢ (↑r * ↑‖x‖⁻¹)⁻¹ • (↑r * ↑‖x‖⁻¹) • x = x

Tactic:
apply inv_smul_smul₀

Goal state after:
case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
x : A
hx : x ∈ {a | 0 ≤ a}
hx_pos : 0 < ‖x‖
⊢ ↑r * ↑‖x‖⁻¹ ≠ 0