mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ μ s ≤ μ (s ∩ o ∪ oᶜ)

Tactic:
by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ c ∈ w,
      ∀ d ∈ w,
        c < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)

Tactic:
filter_upwards [B]

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ c ∈ w,
      ∀ d ∈ w,
        c < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∀ (a : α),
    (∀ c ∈ w,
        ∀ d ∈ w,
          c < d → ¬((∃ᶠ (a : Set α) in v.filterAt a, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt a, d < ρ a / μ a)) →
      ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a) (𝓝 c)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: case hgf
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (↑hf.stieltjesFunction b - f x) / (b - x) ≤ (f b - f x) / (b - x)

Tactic:
filter_upwards [self_mem_nhdsWithin]
      rintro y hy
      rw [mem_Iio, ← sub_neg] at hy
      apply div_le_div_of_nonpos_of_le hy.le
      exact (sub_le_sub_iff_right _).2 (hf.le_rightLim (le_refl _))

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

Goal state before:
pp: case a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
⊢ ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i)) ≤ eVariationOn f s

Tactic:
· apply iSup_le
    rintro ⟨n, u, u_mono, u_mem⟩
    apply sum_le_of_monotoneOn_Iic (by grind [MonotoneOn, StrictMonoOn]) (by grind)

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : ρ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = ∞} ∩ o
ρs : ρ s ≠ ∞
A : ∀ (q : ℝ≥0), 1 ≤ q → μ s ≤ (↑q)⁻¹ * ρ s
⊢ μ s ≤ 0

Tactic:
have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by
    apply ENNReal.Tendsto.mul_const _ (Or.inr ρs)
    exact tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id)

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : ρ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = ∞} ∩ o
ρs : ρ s ≠ ∞
A : ∀ (q : ℝ≥0), 1 ≤ q → μ s ≤ (↑q)⁻¹ * ρ s
B : Tendsto (fun q ↦ (↑q)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s))
⊢ μ s ≤ 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
⊢ μ + (ν - ξ) = μ + ν - ξ

Tactic:
have h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s := by
    rw [h.restrict_nullSet, restrict_sub_eq_restrict_sub_restrict hs]
    simp

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
⊢ μ + (ν - ξ) = μ + ν - ξ
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case mp.add_left
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
p✝ p : C(α, R)
p_mem : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp' : ∀ (f g : C(α, R)₀), ↑g = ↑f + p → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
f g : C(α, R)₀
s : C(α, R)
hp : ↑g = ↑f + (star s * s + p)
⊢ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p

Tactic:
have h₀ : (star s * s + p) 0 = 0 := by simpa using congr($(hp) 0).symm

Goal state after:
case mp.add_left
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
p✝ p : C(α, R)
p_mem : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp' : ∀ (f g : C(α, R)₀), ↑g = ↑f + p → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
f g : C(α, R)₀
s : C(α, R)
hp : ↑g = ↑f + (star s * s + p)
h₀ : (star s * s + p) 0 = 0
⊢ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ eVariationOn f (s ∪ t) = eVariationOn f s + edist (f x) (f y) + eVariationOn f t

Tactic:
(by grind [hs.1, ht.1] : s ∪ t = (s ∪ {x, y}) ∪ t),

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ eVariationOn f (s ∪ {x, y} ∪ t) = eVariationOn f s + edist (f x) (f y) + eVariationOn f t
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
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
f' : R → R
h : ⇑f = (σₙ R a).domRestrict f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ (cfcₙHom ha) f = cfcₙ f' a

Tactic:
rw [cfcₙ_apply ..]

Goal state after:
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
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
f' : R → R
h : ⇑f = (σₙ R a).domRestrict f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ (cfcₙHom ha) f = (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f', continuous_toFun := ⋯, map_zero' := ⋯ }
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing 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 : ∀ x ∈ σₙ R a, 0 ≤ f x
⊢ 0 ≤ cfcₙ f a

Tactic:
by
  by_cases hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
  · obtain ⟨h₁, h₂⟩ := hf
    simpa using cfcₙ_mono h
  · simp only [not_and_or] at hf
    obtain (hf | hf) := hf
    · simp only [cfcₙ_apply_of_not_continuousOn _ hf, le_rfl]
    · simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]

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✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing 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 : ∀ x ∈ σₙ R a, f x ≤ 0
⊢ cfcₙ f a ≤ 0

Tactic:
by
  by_cases hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
  · obtain ⟨h₁, h₂⟩ := hf
    simpa using cfcₙ_mono h
  · simp only [not_and_or] at hf
    obtain (hf | hf) := hf
    · simp only [cfcₙ_apply_of_not_continuousOn _ hf, le_rfl]
    · simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]

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
⊢ LipschitzWith 2 ⇑(addEquiv 𝕜 A)

Tactic:
rw [← Real.toNNReal_ofNat]

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
⊢ LipschitzWith 2.toNNReal ⇑(addEquiv 𝕜 A)
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y}

Tactic:
lift φ to characterSpace 𝕜 A using hφ

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : { x // x ∈ characterSpace 𝕜 A }
⊢ ↑φ ∈ {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y}
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) =
    closure (↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)))

Tactic:
AlgHom.closure_ker_inter (φ := evalStarAlgHom 𝕜 𝕜 (X := s) ⟨0, h0⟩) (continuous_eval_const _) _

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) =
    closure ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ V.preimage s ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)

Tactic:
hst,

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ : (s, t) ∈ hausdorffEntourage V
hst : s ⊆ V.preimage t
hts : t ⊆ V.image s
⊢ V.preimage (V.preimage t) ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ closure[inst✝.toTopologicalSpace] s ⊆ SetRel.preimage U t

Tactic:
hst,

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
x✝¹ : Set α × Set α
s t : Set α
x✝ :
  (s, t) ∈
    (fun x ↦ (closure[inst✝.toTopologicalSpace] x.1, closure[inst✝.toTopologicalSpace] x.2)) ⁻¹' hausdorffEntourage V
hst : closure[inst✝.toTopologicalSpace] s ⊆ V.preimage (closure[inst✝.toTopologicalSpace] t)
hts : closure[inst✝.toTopologicalSpace] t ⊆ V.image (closure[inst✝.toTopologicalSpace] s)
⊢ V.preimage (closure[inst✝.toTopologicalSpace] t) ⊆ SetRel.preimage U t
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a ^ n‖₊ * r ^ n ≤ max 1 ‖1‖₊

Tactic:
rcases n with - | n

Goal state after:
case zero
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
⊢ ‖a ^ 0‖₊ * r ^ 0 ≤ max 1 ‖1‖₊

case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a ^ (n + 1)‖₊ * r ^ (n + 1) ≤ max 1 ‖1‖₊
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : ∀ i ∈ 𝓤 α, ∃ x ∈ Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i, x ∈ {s | TotallyBounded s}
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
t : Set α
ht : t ∈ {s | TotallyBounded s}
hst : s ⊆ V.preimage t
u : Set α
hu : u.Finite
htu : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ V}
⊢ s ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}

Tactic:
hst,

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : ∀ i ∈ 𝓤 α, ∃ x ∈ Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i, x ∈ {s | TotallyBounded s}
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
t : Set α
ht : t ∈ {s | TotallyBounded s}
hst : s ⊆ V.preimage t
u : Set α
hu : u.Finite
htu : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ V}
⊢ V.preimage t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}
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 → β
hf : IsInducing f
this : {U | IsOpen[inst✝¹] U} = (fun x ↦ f ⁻¹' x) '' {V | IsOpen[inst✝] V}
⊢ TopologicalSpace.vietoris α = induced (fun x ↦ f '' x) (TopologicalSpace.vietoris β)

Tactic:
simp_rw [TopologicalSpace.vietoris, this, induced_generateFrom_eq, image_union, image_image,
    powerset, preimage_ofPred_eq, image_subset_iff, image_inter_nonempty_iff]

Goal state after:
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
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
t : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ V.preimage t
⊢ ∃ t, t.Finite ∧ U.preimage t ∈ l

Tactic:
refine ⟨t, ht₁, Filter.mem_of_superset (Filter.mem_lift' hK) ?_⟩

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 : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ V.preimage t
⊢ ⋃ K_1 ∈ {x | (fun K' ↦ (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)) x}, ↑K_1 ⊆ U.preimage t
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
⊢ frontier (σ 𝕜 x) ⊆ frontier (σ 𝕜 ↑x)

Tactic:
frontier_eq_closure_inter_closure (s := σ 𝕜 (x : A)),

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
⊢ frontier (σ 𝕜 x) ⊆
    closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 ↑x) ∩
      closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 ↑x)ᶜ
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
⊢ IsOpen {p | Disjoint p.1 p.2}

Tactic:
simpa only [disjoint_coe_iff] using isOpen_setOfPred_disjoint_coe

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
⊢ √(spectralRadius ℂ (a * a⋆)).toReal = ‖a‖

Tactic:
simp [toReal_spectralRadius_self_mul_star_eq_norm_sq]

Goal state after:
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: 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
⊢ ∀
    t ∈
      (fun a ↦ {a_1 | ↑a_1 ⊆ a}) '' {U | IsOpen[inst✝³] U} ∪
        (fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) '' {U | IsOpen[inst✝³] U},
    ∀ a ∈ t, Disjoint (𝓝ˢ tᶜ) (𝓝 a)

Tactic:
Filter.disjoint_iff

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✝ : RegularSpace α
⊢ ∀
    t ∈
      (fun a ↦ {a_1 | ↑a_1 ⊆ a}) '' {U | IsOpen[inst✝³] U} ∪
        (fun a ↦ {a_1 | (↑a_1 ∩ a).Nonempty}) '' {U | IsOpen[inst✝³] U},
    ∀ a ∈ t, ∃ s ∈ 𝓝ˢ tᶜ, ∃ t ∈ 𝓝 a, Disjoint s t
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
U : Set α
hU : U ∈ u
⊢ ∃ M, IsCompact M ∧ M ⊆ L ∧ M ⊆ U ∧ (↑K ∩ interior M).Nonempty

Tactic:
obtain ⟨x, hxK, hxU⟩ := hKu₂ U hU
      obtain ⟨M, hM, _⟩ := exists_compact_subset (U := U ∩ interior L)
        ((hu₂ U hU).inter isOpen_interior) ⟨hxU, hLK hxK⟩
      exact ⟨M, hM, by grind [interior_subset], by grind, x, by grind⟩

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
⊢ SemiconjBy x (star a) (star b)

Tactic:
have (a : A) : HasDerivAt (fun z : ℂ ↦ exp (z • a)) a 0 := by
      simpa using hasDerivAt_exp_smul_const a (0 : ℂ)

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
this : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
⊢ SemiconjBy x (star a) (star b)
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
this : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
⊢ SemiconjBy x (star a) (star b)

Tactic:
apply (this (star a)).const_mul x |>.unique

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
this : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
⊢ HasDerivAt (fun y ↦ x * NormedSpace.exp (y • star a)) (star b * x) 0