mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: 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 : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (spectrum R a)
x : { x // x ∈ spectrum S a }
⊢ Function.extend Subtype.val
      (⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          ({ toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ }.comp
            { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })))
      0 ↑x =
    (algebraMap R S) (g (f ↑x))

Tactic:
simp [Function.comp]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: 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
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Icc y x)) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))

Tactic:
by
  rcases eq_or_neBot (𝓝[s ∩ Iio x] x) with h | h
  · simp [h]
  suffices H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l)
      (𝓝[s ∩ Iio x] x) (𝓝 (0 + edist (f x) l)) by
    simp only [zero_add] at H
    apply Tendsto.congr' _ H
    filter_upwards [self_mem_nhdsWithin] with y hy
    have N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x := by
      rw [show s ∩ Ici y ∩ Iio x = s ∩ Iio x ∩ Ici y by grind, nhdsWithin_inter_of_mem']
      exact mem_nhdsWithin_of_mem_nhds (Ici_mem_nhds hy.2)
    rw [show s ∩ Icc y x = (s ∩ Ici y) ∩ Iic x by grind,
      eVariationOn_on_inter_Iic_eq_Iio_add_edist (l := l)]
    · congr 2; grind
    · convert h using 1
    · exact ⟨hx, hy.2.le⟩
    · convert h'f
  apply Tendsto.add ?_ tendsto_const_nhds
  exact (hf.tendsto_eVariationOn_Ico_zero x).mono_left (nhdsWithin_mono _ inter_subset_left)

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

Goal state before:
pp: case e'_3
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousInv₀ R
a : Aˣ
ha : p ↑a
⊢ ↑a = cfc id ↑a

pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousInv₀ R
a : Aˣ
ha : p ↑a
⊢ ∀ x ∈ spectrum R ↑a, id x ≠ 0

Tactic:
· exact (cfc_id R (a : A)).symm

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousInv₀ R
a : Aˣ
ha : p ↑a
⊢ ∀ x ∈ spectrum R ↑a, id x ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: 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
⊢ cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a)

Tactic:
by_cases h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
  · obtain ⟨ha, hf, h0⟩ := h
    rw [cfcₙ_apply f a, ← map_star, cfcₙ_apply _ a]
    congr
  · simp only [not_and_or] at h
    obtain (ha | hf | h0) := h
    · simp [cfcₙ_apply_of_not_predicate a ha]
    · rw [cfcₙ_apply_of_not_continuousOn a hf, cfcₙ_apply_of_not_continuousOn, star_zero]
      exact fun hf_star ↦ hf <| by simpa using hf_star.star
    · rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, star_zero]
      exact fun hf0 ↦ h0 <| by simpa using congr(star $(hf0))

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

Goal state before:
pp: 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
⊢ cfcₙ (fun x ↦ star x) a = star a

Tactic:
by
  rw [cfcₙ_star _ a, cfcₙ_id' R a]

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

Goal state before:
pp: a b : ℝ
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ (polynomialFunctions (Set.Icc a b)).topologicalClosure

Tactic:
polynomialFunctions_closure_eq_top _ _

Goal state after:
a b : ℝ
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ ⊤
Module: Mathlib.Analysis.Normed.Algebra.Unitization

Goal state before:
pp: case inl_add_inr
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
h : Function.Injective ⇑(mul 𝕜 A)
a✝ : A
hx : (mul 𝕜 A) a✝ = (mul 𝕜 A) 0
⊢ ↑a✝ = 0

Tactic:
rw [h hx, inr_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
x : Unitization 𝕜 A
a✝ : Nontrivial A
⊢ ‖(mul 𝕜 A) x.toProd.2‖ ≤ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (mul 𝕜 A) x.toProd.2‖ + ‖x.toProd.1‖

Tactic:
simpa only [add_comm _ (mul 𝕜 A x.snd), norm_algebraMap'] using
          norm_le_add_norm_add (mul 𝕜 A x.snd) (algebraMap 𝕜 _ x.fst)

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
k : 𝕜
x : Unitization 𝕜 A
⊢ ‖k‖ * ‖(splitMul 𝕜 A) x‖ ≤ ‖k‖ * ‖x‖

Tactic:
← norm_def

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
k : 𝕜
x : Unitization 𝕜 A
⊢ ‖k‖ * ‖x‖ ≤ ‖k‖ * ‖x‖
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
a b : A
⊢ { toFun := fun φ ↦ φ (a + b), continuous_toFun := ⋯ } =
    { toFun := fun φ ↦ φ a, continuous_toFun := ⋯ } + { toFun := fun φ ↦ φ b, continuous_toFun := ⋯ }

Tactic:
ext; simp only [map_add, coe_mk, coe_add, Pi.add_apply]

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

Goal state before:
pp: case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
this : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 ↑s) (𝓝 x)
⊢ x ∈ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))

Tactic:
refine mem_closure_of_tendsto this <| eventually_inf_principal.mpr ?_

Goal state after:
case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
this : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 ↑s) (𝓝 x)
⊢ ∀ᶠ (x : A) in 𝓝 x, x ∈ ↑s → x - φ x • 1 ∈ ↑s ∩ ↑(RingHom.ker φ)
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
h : ∀ t ∈ 𝓝 (f x✝), ∀ᶠ (x' : α) in 𝓝[s] x✝, t ∈ 𝓝 (f x')
t : Set β
ht : t ∈ 𝓝 (f x✝)
x : α
⊢ t ∈ 𝓝 (f x) → f x ∈ t

Tactic:
exact mem_of_mem_nhds

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
F : Set α
h : IsClosed[inst✝] F
⊢ IsOpen[TopologicalSpace.vietoris α] {a | ¬a ⊆ F}

Tactic:
← inter_compl_nonempty_iff

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
F : Set α
h : IsClosed[inst✝] F
⊢ IsOpen[TopologicalSpace.vietoris α] {a | (a ∩ Fᶜ).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ 𝒫 U
x : α
hx : x ∈ s
⊢ x ∈ U

Tactic:
obtain ⟨y, hyt, hxy⟩ := hst x hx

Goal state after:
case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ 𝒫 U
x : α
hx : x ∈ s
y : α
hyt : y ∈ t
hxy : x ⤳ y
⊢ x ∈ U
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
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x) \ σ 𝕜 ↑x)

Tactic:
closure_eq_interior_union_frontier,

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
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((interior (σ 𝕜 x) ∪ frontier (σ 𝕜 x)) \ σ 𝕜 ↑x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case map_spec
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σₙ 𝕜 (ψ f) = Set.range ⇑f

pp: case isStarNormal
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ p (ψ f)

Tactic:
case map_spec =>
      exact quasispectrum_eq_spectrum_inr' 𝕜 𝕜 (ψ f) ▸ coe_ψ _ ▸ spec_cfcₙAux hp₁ a ha f

Goal state after:
case isStarNormal
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ p (ψ f)
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:
rw [cfcₙ_nnreal_eq_real ..]

Goal state after:
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).toNNReal) a
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
⊢ ‖a‖ * ‖a‖ = ‖a‖ ^ 2

Tactic:
← pow_two

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
⊢ ‖a‖ ^ 2 = ‖a‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
⊢ IsGreatest (f '' σₙ ℝ≥0 a) ‖cfcₙ f a‖₊

Tactic:
rw [cfcₙ_nnreal_eq_real ..]

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
⊢ IsGreatest (f '' σₙ ℝ≥0 a) ‖cfcₙ (fun x ↦ ↑(f x.toNNReal)) a‖₊
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 : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a⋆ * a)

Tactic:
· exact IsSelfAdjoint.star_mul_self a

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

Goal state before:
pp: case inl
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
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
hK : K ∈ (fun a ↦ {a_1 | ↑a_1 ⊆ a}) U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hKV : ↑K ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | ¬↑a ⊆ U} ⊆ {K | (↑K ∩ W).Nonempty}

Tactic:
← inter_compl_nonempty_iff

Goal state after:
case inl
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
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
hK : K ∈ (fun a ↦ {a_1 | ↑a_1 ⊆ a}) U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hKV : ↑K ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | (↑a ∩ Uᶜ).Nonempty} ⊆ {K | (↑K ∩ W).Nonempty}
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
ε : ℝ
hε : ε > 0
g : α →C₀ β
hg : g.toBCF ∈ ball f (ε / 2)
⊢ ∀ᶠ (x : α) in cocompact α, dist (f x) 0 < ε

Tactic:
refine (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
      (Eventually.of_forall fun x hx => ?_)

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
ε : ℝ
hε : ε > 0
g : α →C₀ β
hg : g.toBCF ∈ ball f (ε / 2)
x : α
hx : dist (g x) 0 < ε / 2
⊢ dist (f x) 0 < ε
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x✝¹ : E
x✝ : x✝¹ ∈ -s
⊢ x✝¹ ∈ balancedHull ℝ s

Tactic:
by rw [mem_balancedHull_iff]; use -1; simp_all

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

Goal state before:
pp: case f
α : Type u_1
inst✝ : TopologicalSpace α
h : ∀ (s : Set (NonemptyCompacts α)), IsClopen s → s = ∅ ∨ s = univ
s : Set α
hs : IsClopen s
⊢ {K | ↑K ⊆ s} = ∅ → s = ∅

Tactic:
simp only [Set.eq_empty_iff_forall_notMem]
    exact fun h x hx => h {x} (Set.singleton_subset_iff.mpr hx)

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
⊢ φ ∈ characterSpace 𝕜 A → φ ∈ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖

Tactic:
hφ

Goal state after:
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
⊢ φ ∈ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖
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 := ⋯ } ∈ characterSpace 𝕜 C(X, 𝕜)

Tactic:
CharacterSpace.eq_set_map_one_map_mul

Goal state after:
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}
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
ι : Type u_2
f : ι → A
s : Finset ι
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊

Tactic:
induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih =>
    suffices f j * ∑ i ∈ s, f i = 0 by simp_all [nnnorm_add_eq_max this]
    simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind)

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ a⁻ = -a ↔ a ≤ 0

Tactic:
rw [← neg_inj, neg_neg, eq_comm]
  refine ⟨fun ha ↦ by rw [ha, neg_nonpos]; exact negPart_nonneg a, fun ha ↦ ?_⟩
  rw [← neg_nonneg] at ha
  rw [negPart_def, ← cfcₙ_neg]
  have _ : IsSelfAdjoint a := neg_neg a ▸ (IsSelfAdjoint.neg <| .of_nonneg ha)
  conv_lhs => rw [← cfcₙ_id ℝ a]
  refine cfcₙ_congr fun x hx ↦ ?_
  rw [Unitization.quasispectrum_eq_spectrum_inr ℝ, ← neg_neg x, ← Set.mem_neg,
    spectrum.neg_eq, ← Unitization.inr_neg, ← Unitization.quasispectrum_eq_spectrum_inr ℝ] at hx
  rw [← neg_eq_iff_eq_neg, eq_comm]
  simpa using quasispectrum_nonneg_of_nonneg _ ha _ hx

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

Goal state before:
pp: case mul
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f g : C(↑s, 𝕜)
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcHomSuperset ⋯ ⋯) f
hg : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcHomSuperset ⋯ ⋯) g
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcHomSuperset ⋯ ⋯) (f * g)

Tactic:
simpa using! hf.mul hg

Goal state after: