mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ ∑ k ∈ Sᶜ, (bernstein n ↑k) x * gauge U (f k/ₙ - f x) ≤ ∑ k ∈ Sᶜ, C * (bernstein n ↑k) x

Tactic:
simp only [mul_comm (bernstein n _ x)]

Goal state after:
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ ∑ x_1 ∈ Sᶜ, gauge U (f x_1/ₙ - f x) * (bernstein n ↑x_1) x ≤ ∑ k ∈ Sᶜ, C * (bernstein n ↑k) x
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NonUnitalNormedRing A
inst✝ : NormedSpace 𝕜 A
x : A
⊢ ‖toLp 1 ↑x‖₊ = ‖x‖₊

Tactic:
simp [unitization_nnnorm_def]

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ IsTopologicalBasis
    ((fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
      {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1})

Tactic:
refine isTopologicalBasis.isTopologicalBasis_of_exists_subset ?_ ?_ <;>
    rw [forall_mem_image]

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set α × Set (Set α)⦄,
    x ∈ {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} →
      IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty}

case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} →
      ∀ x_1 ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty},
        ∃
          v ∈
            (fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
              {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1},
          x_1 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
⊢ ∃ t, t.Finite ∧ U.preimage t ∈ l

Tactic:
obtain ⟨K, hK⟩ := hf (symm_le_uniformity hV) |>.exists

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
⊢ ∃ t, t.Finite ∧ U.preimage t ∈ l
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
⊢ ∃ x, f ≤ 𝓝 x

Tactic:
obtain ⟨K, hK⟩ := CompleteSpace.complete <| hf.map uniformContinuous_singleton

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, f ≤ 𝓝 x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 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
⊢ ∃ φ,
    Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑φ ∧
      Function.Injective ⇑φ ∧
        φ { toContinuousMap := ContinuousMap.restrict (σₙ 𝕜 a) (ContinuousMap.id 𝕜), map_zero' := ⋯ } = a ∧
          (∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), σₙ 𝕜 (φ f) = Set.range ⇑f) ∧ ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), p (φ f)

Tactic:
by
    let ψ : C(σₙ 𝕜 a, 𝕜)₀ →⋆ₙₐ[𝕜] A := comp (inrRangeEquiv 𝕜 A).symm <|
      codRestrict (cfcₙAux hp₁ a ha) _ (cfcₙAux_mem_range_inr hp₁ a ha)
    have coe_ψ (f : C(σₙ 𝕜 a, 𝕜)₀) : ψ f = cfcₙAux hp₁ a ha f :=
      congr_arg Subtype.val <| (inrRangeEquiv 𝕜 A).apply_symm_apply
        ⟨cfcₙAux hp₁ a ha f, cfcₙAux_mem_range_inr hp₁ a ha f⟩
    refine ⟨ψ, ?continuous, ?injective, ?map_id, fun f ↦ ?map_spec, fun f ↦ ?isStarNormal⟩
    case continuous =>
      rw [isometry_inr (𝕜 := 𝕜) |>.isEmbedding.continuous_iff]
      have := continuous_cfcₙAux hp₁ a ha
      simp only [coe_comp, NonUnitalStarAlgHom.coe_coe, Function.comp_def,
        inrRangeEquiv_symm_apply, coe_codRestrict, ψ]
      fun_prop
    case injective =>
      have h₁ : Function.Injective ⇑(codRestrict (cfcₙAux hp₁ a ha) _
          (cfcₙAux_mem_range_inr hp₁ a ha)) :=
        (Set.injective_codRestrict _).mpr (cfcₙAux_injective hp₁ a ha)
      simpa [ψ] using (inrRangeEquiv 𝕜 A).symm.injective.comp h₁
    case map_id => exact inr_injective (R := 𝕜) <| coe_ψ _ ▸ cfcₙAux_id hp₁ a ha
    case map_spec =>
      exact quasispectrum_eq_spectrum_inr' 𝕜 𝕜 (ψ f) ▸ coe_ψ _ ▸ spec_cfcₙAux hp₁ a ha f
    case isStarNormal => exact hp₁.mp <| coe_ψ _ ▸ cfcHom_predicate (R := 𝕜) (hp₁.mpr ha) _

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

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : QuasispectrumRestricts a ⇑Complex.reCLM
x : ℝ
hx : (algebraMap ℝ ℂ) x ∈ σₙ ℂ a
⊢ star ((algebraMap ℝ ℂ) x) = id ((algebraMap ℝ ℂ) x)

Tactic:
exact Complex.conj_ofReal _

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

Goal state before:
pp: 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
⊢ Isometry ⇑(cfcₙHom ha)

Tactic:
rw [cfcₙHom_eq_restrict f ha ha' haf]

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
⊢ Isometry ⇑(nonUnitalStarAlgHom (cfcₙHom ha') haf)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.refine_1
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
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | ↑a ∩ U = ∅} ⊆ {K | ↑K ⊆ W}

Tactic:
← disjoint_iff_inter_eq_empty,

Goal state after:
case inr.refine_1
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
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | Disjoint (↑a) U} ⊆ {K | ↑K ⊆ W}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (Compacts α)
⊢ IsCompact univ

Tactic:
convert! isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (Compacts α)
⊢ univ = ⋃ K ∈ univ, ↑K
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}})
⊢ IsTopologicalBasis {s | IsOpen s ∧ IsPreconnected s}

Tactic:
refine basis.of_isOpen_of_subset (by grind) (fun U hU => ⟨basis.isOpen 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✝ : 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}}
⊢ IsPreconnected U
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 : ℝ
| fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)

Tactic:
x,

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) * NormedSpace.exp (x • b)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (NonemptyCompacts α)
⊢ IsCompact univ

Tactic:
convert! isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (NonemptyCompacts α)
⊢ univ = ⋃ K ∈ univ, ↑K
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ LocallyCompactSpace (NonemptyCompacts α) ↔ LocallyCompactSpace α

Tactic:
refine ⟨fun _ => ⟨fun x U hU => ?_⟩, fun _ => inferInstance⟩
  rw [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, ← coe_singleton] at hU
  obtain ⟨K, hK, hxK, hKU⟩ := exists_compact_subset (isOpen_subsets_of_isOpen isOpen_interior) hU
  refine ⟨⋃ L ∈ K, L, ?_, iUnion₂_subset <| by grind [interior_subset],
    isCompact_biUnion_coe_of_isCompact hK⟩
  rw [mem_interior_iff_mem_nhds] at hxK
  filter_upwards [continuous_singleton.tendsto x hxK] with y hy using mem_iUnion₂_of_mem hy rfl

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

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
φ_cont : Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ
x : E
hx : ∀ (i : ι), (f i) x = 0
⊢ Inseparable x 0

Tactic:
simp_rw [Inseparable, nhds_iInf, nhds_induced, hx, map_zero]

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)
h : ∀ (x : E), Tendsto (fun i ↦ (g i) x) l (𝓝 (f x))
__LinearMap✝ : E →ₛₗ[σ₁₂] F := linearMapOfTendsto f (fun a ↦ ↑(g a)) h✝
u : ℕ → α
hu : Tendsto u atTop l
x : E
⊢ IsVonNBounded 𝕜₂ (range fun i ↦ (g (u i)) x)

Tactic:
exact ((h x).comp hu).isVonNBounded_range 𝕜₂

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ p₃ p₄ : P
⊢ dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖

Tactic:
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, midpoint_vsub_midpoint]
  rw [midpoint_eq_smul_add, norm_smul, invOf_eq_inv, norm_inv, ← div_eq_inv_mul]
  grw [norm_add_le]

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
⊢ approx 0 c x ∈ Icc (approx 0 c.right x) (approx 0 c.left x)

Tactic:
simp only [approx]
    refine ⟨le_rfl, ?_⟩
    grw [left_U_subset]
    rw [Pi.one_apply]; positivity

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
I : Ideal C(X, R)
x : X
⊢ x ∈ {x | ∀ f ∈ I, f x = 0}ᶜᶜ ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0

Tactic:
compl_compl,

Goal state after:
X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
I : Ideal C(X, R)
x : X
⊢ x ∈ {x | ∀ f ∈ I, f x = 0} ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
⊢ ∃ f, EqOn (⇑f) 0 t ∧ EqOn (⇑f) 1 s ∧ ∀ (x : X), f x ∈ Icc 0 1

Tactic:
obtain ⟨g, hgs, hgt, (hicc : ∀ x, 0 ≤ g x ∧ g x ≤ 1)⟩ := exists_continuous_zero_one_of_isCompact
    hs ht hd

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
⊢ ∃ f, EqOn (⇑f) 0 t ∧ EqOn (⇑f) 1 s ∧ ∀ (x : X), f x ∈ Icc 0 1
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_4
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
x : X
hx : x ∈ (setOfIdeal I)ᶜᶜ
⊢ ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x

Tactic:
compl_compl,

Goal state after:
case refine_4
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
x : X
hx : x ∈ setOfIdeal I
⊢ ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsSelfAdjoint a
hb : IsSelfAdjoint b
hab : a * b = 0
⊢ Commute a b

Tactic:
grind [commute_of_mul_eq_zero]

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
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
⊢ id x = -x⁻

Tactic:
Unitization.quasispectrum_eq_spectrum_inr ℝ,

Goal state after:
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
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : x ∈ spectrum ℝ ↑a
⊢ id x = -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
⊢ cfcₙ (fun x ↦ x⁺) 1 = 1

Tactic:
cfcₙ_eq_cfc

Goal state after:
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
⊢ cfc (fun x ↦ x⁺) 1 = 1
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
⊢ cfcₙ (fun x ↦ x⁻) 1 = 0

Tactic:
cfcₙ_eq_cfc

Goal state after:
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
⊢ cfc (fun x ↦ x⁻) 1 = 0
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
⊢ cfcₙ (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺

Tactic:
cfcₙ_eq_cfc

Goal state after:
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
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺
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
⊢ cfcₙ (fun x ↦ x⁻) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁻

Tactic:
cfcₙ_eq_cfc

Goal state after:
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
⊢ cfc (fun x ↦ x⁻) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁻
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
have h_c_spec₁ : SpectrumRestricts (c * star c) ContinuousMap.realToNNReal := by
    rw [eq_sub_iff_add_eq'.mpr <| star_mul_self_add_self_mul_star c, sub_eq_add_neg, ← sq, ← sq]
    refine SpectrumRestricts.nnreal_add ?_ ?_ ?_ h_c_spec₀
    · exact .smul (star_trivial _) <| ((ℜ c).prop.pow 2).add ((ℑ c).prop.pow 2)
    · exact .neg <| .star_mul_self c
    · rw [← Nat.cast_smul_eq_nsmul ℝ]
      refine (ℜ c).2.sq_spectrumRestricts.nnreal_add ((ℜ c).2.pow 2) ((ℑ c).2.pow 2)
        (ℑ c).2.sq_spectrumRestricts |>.smul_of_nonneg <| by simp

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«2».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖

Tactic:
exact CStarAlgebra.norm_negPart_le _ |>.trans <| realPart.norm_le a

Goal state after: