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
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))

Tactic:
filter_upwards [A, B, C] with _ Ax Bx Cx

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
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
⊢ Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a✝) (𝓝 (ρ.rnDeriv μ a✝))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case refine_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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this✝¹ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this✝ :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
⊢ ∀ᶠ (c : α) in 𝓝[s ∩ Iio a] a,
    ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f c) ≤ eVariationOn f (s ∩ Iic a)

Tactic:
filter_upwards [this, self_mem_nhdsWithin] with y hy h'y

Goal state after:
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
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this✝¹ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this✝ :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
h'y : y ∈ s ∩ Iio a
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y) ≤ eVariationOn f (s ∩ Iic a)
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
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
rcases hf.exists_nat_integrableOn with ⟨u, u_open, u_univ, hu⟩

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Field S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalRing S
inst✝¹² : ContinuousStar S
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : IsClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : IsUniformEmbedding ⇑(algebraMap R S)
⊢ IsClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)

Tactic:
have : h.homeomorph.symm 0 = 0 := Subtype.ext (map_zero <| algebraMap _ _)
  refine hφ.comp <| IsUniformEmbedding.isClosedEmbedding <| .comp
    (ContinuousMapZero.isUniformEmbedding_comp _ halg)
    (UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.isUniformEmbedding)

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
r : R
a : A
ha : p a
⊢ cfc id a ≤ (algebraMap R A) r ↔ ∀ x ∈ spectrum R a, x ≤ r

Tactic:
exact cfc_le_algebraMap_iff id r a

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
a b : α
hf : MonotoneOn f s
as : a ∈ s
bs : b ∈ s
hab : a ≤ b
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s ∩ Set.Icc a b
⊢ ∀ i ∈ Finset.range n, 0 ≤ f (u (i + 1)) - f (u i)

Tactic:
exact fun i _ ↦ sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))

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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
⊢ cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a

pp: case neg
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : ¬p a
⊢ cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a

Tactic:
· have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
    have hf' : ContinuousOn (∑ i : s, f i) (σₙ R a) := by
      rw [sum_coe_sort s, hsum]
      exact continuousOn_finsetSum s fun i hi => hf i hi
    rw [← sum_coe_sort s, ← sum_coe_sort s]
    rw [cfcₙ_apply_pi _ a ha (fun ⟨i, hi⟩ => hf i hi), ← map_sum, cfcₙ_apply _ a hf']
    congr 1
    ext
    simp

Goal state after:
case neg
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : ¬p a
⊢ cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inl
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
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a)

Tactic:
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

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) ≠ 0

Tactic:
by
  simp only [bernsteinPolynomial.iterate_derivative_at_0, Ne]
  simp only [← ascPochhammer_eval_cast]
  norm_cast
  apply ne_of_gt
  obtain rfl | h' := Nat.eq_zero_or_pos ν
  · simp
  · rw [← Nat.succ_pred_eq_of_pos h'] at h
    exact ascPochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h))

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

Goal state before:
pp: case neg
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:
· 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, neg_zero]
      exact fun hf_neg ↦ hf <| by simpa using hf_neg.fun_neg
    · rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, neg_zero]
      exact (h0 <| neg_eq_zero.mp ·)

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f g : ↥A
⊢ |↑g - ↑f| ∈ A.topologicalClosure

Tactic:
exact mod_cast abs_mem_subalgebra_closure A _

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case calc_1.e_a
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
k : ℕ
m : k ∈ Finset.range (n + 1)
⊢ (↑n * X - ↑k) ^ 2 = ↑k * ↑(k - 1) + (1 - 2 * ↑n * X) * ↑k + ↑n ^ 2 * X ^ 2

Tactic:
cases k

Goal state after:
case calc_1.e_a.zero
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
m : 0 ∈ Finset.range (n + 1)
⊢ (↑n * X - ↑0) ^ 2 = ↑0 * ↑(0 - 1) + (1 - 2 * ↑n * X) * ↑0 + ↑n ^ 2 * X ^ 2

case calc_1.e_a.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
n✝ : ℕ
m : n✝ + 1 ∈ Finset.range (n + 1)
⊢ (↑n * X - ↑(n✝ + 1)) ^ 2 = ↑(n✝ + 1) * ↑(n✝ + 1 - 1) + (1 - 2 * ↑n * X) * ↑(n✝ + 1) + ↑n ^ 2 * X ^ 2
Module: Mathlib.Analysis.Normed.Algebra.Unitization

Goal state before:
pp: 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
x : Unitization 𝕜 A
⊢ (x.toProd.1 + 0, (lift (NonUnitalAlgHom.Lmul 𝕜 A)).toRingHom x) =
    (x.toProd.1, (algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (mul 𝕜 A) x.toProd.2)

Tactic:
rw [add_zero]; rfl

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
⊢ ∃ g, ‖↑g - f‖ < ε

Tactic:
have w :=
    mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)
  rw [Metric.nhds_basis_ball.frequently_iff] at w
  obtain ⟨g, H, m⟩ := w ε pos
  rw [Metric.mem_ball, dist_eq_norm] at H
  exact ⟨⟨g, m⟩, H⟩

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Fact (0 ∈ s)
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id s)
star_id : p (star (ContinuousMapZero.id s))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id s}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
⊢ p 0

Tactic:
exact zero

Goal state after:
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 (V.preimage t) ⊆ SetRel.preimage U (closure[inst✝.toTopologicalSpace] t)

Tactic:
← subset_closure,

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 t
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A✝ : Type u_2
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜 A✝
inst✝³ : CompleteSpace A✝
A : Type u_3
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : A
inst✝ : IsCompact (spectrum ℝ a)
⊢ IsCompact (spectrum ℝ a)

Tactic:
by assumption

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NormedField 𝕜
B : Type u_3
inst✝⁵ : NonUnitalNormedRing B
inst✝⁴ : NormedSpace 𝕜 B
inst✝³ : CompleteSpace B
inst✝² : IsScalarTower 𝕜 B B
inst✝¹ : SMulCommClass 𝕜 B B
inst✝ : ProperSpace 𝕜
a : B
⊢ IsCompact (quasispectrum 𝕜 a)

Tactic:
Unitization.quasispectrum_eq_spectrum_inr' 𝕜 𝕜,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NormedField 𝕜
B : Type u_3
inst✝⁵ : NonUnitalNormedRing B
inst✝⁴ : NormedSpace 𝕜 B
inst✝³ : CompleteSpace B
inst✝² : IsScalarTower 𝕜 B B
inst✝¹ : SMulCommClass 𝕜 B B
inst✝ : ProperSpace 𝕜
a : B
⊢ IsCompact (σ ↑a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹³ : NormedField 𝕜
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra 𝕜 A
inst✝¹⁰ : CompleteSpace A
B : Type u_3
inst✝⁹ : NonUnitalNormedRing B
inst✝⁸ : NormedSpace 𝕜 B
inst✝⁷ : CompleteSpace B
inst✝⁶ : IsScalarTower 𝕜 B B
inst✝⁵ : SMulCommClass 𝕜 B B
inst✝⁴ : ProperSpace 𝕜
inst✝³ : NormedSpace ℝ B
inst✝² : IsScalarTower ℝ B B
inst✝¹ : SMulCommClass ℝ B B
a : B
inst✝ : IsCompact (quasispectrum ℝ a)
⊢ IsCompact (quasispectrum ℝ a)

Tactic:
by assumption

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case right
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.image (V.image s) ⊆ SetRel.image U (closure[inst✝.toTopologicalSpace] s)

Tactic:
← subset_closure,

Goal state after:
case right
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.image (V.image s) ⊆ SetRel.image U s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
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
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s

Tactic:
rcases h.exists_eq_singleton_or_nontrivial with ⟨x, rfl⟩ | ⟨x, hx, y, hy, hxy⟩

Goal state after:
case inr.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}
x : α
hs : {x} ∈ (range fun x ↦ {x})ᶜ
h : {x}.Nonempty
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 {x}

case inr.inr
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
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹
⊢ ‖y‖₊ * ‖a‖₊ < 1

Tactic:
← NNReal.lt_inv_iff_mul_lt h

Goal state after:
case neg
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
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹
⊢ ‖y‖₊ < ‖a‖₊⁻¹
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
⊢ IsClosedEmbedding
    ((⇑↑(cfcHom ⋯) ∘ ⇑↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))) ∘
      ⇑ContinuousMapZero.toContinuousMapHom)

Tactic:
refine ((cfcHom_isClosedEmbedding (hp₁.mpr ha)).comp ?_).comp
    ContinuousMapZero.isClosedEmbedding_toContinuousMap

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
⊢ IsClosedEmbedding ⇑↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
f : ℂ → ℂ
a : A
hf_real : ∀ x ∈ σₙ ℂ a, star (f x) = f x
ha : IsSelfAdjoint a
⊢ cfcₙ f a = cfcₙ (fun x ↦ (f ↑x).re) a

Tactic:
rw [cfcₙ_real_eq_complex ..]

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
f : ℂ → ℂ
a : A
hf_real : ∀ x ∈ σₙ ℂ a, star (f x) = f x
ha : IsSelfAdjoint a
⊢ cfcₙ f a = cfcₙ (fun x ↦ ↑(f ↑x.re).re) a
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.hKV
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃ i ∈ w, i
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
w' : Set (Set α)
h✝ : w' ⊆ w ∧ w'.Finite ∧ ↑K ⊆ ⋃ i ∈ w', i
⊢ ↑K ⊆ ⋃₀ w'

Tactic:
grind [sUnion_eq_biUnion]

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

Goal state before:
pp: case inr.hwB
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃ i ∈ w, i
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
w' : Set (Set α)
h✝ : w' ⊆ w ∧ w'.Finite ∧ ↑K ⊆ ⋃ i ∈ w', i
⊢ w' ⊆ B

Tactic:
grind [sUnion_eq_biUnion]

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

Goal state before:
pp: case inr.hwV
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃ i ∈ w, i
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
w' : Set (Set α)
h✝ : w' ⊆ w ∧ w'.Finite ∧ ↑K ⊆ ⋃ i ∈ w', i
⊢ ⋃₀ w' ⊆ V

Tactic:
grind [sUnion_eq_biUnion]

Goal state after: