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
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
⊢ ∃ c ∈ t, ‖f x - c‖ₑ < ε / 2

Tactic:
simp_rw [← edist_eq_enorm_sub]

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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
⊢ ∃ c ∈ t, edist (f x) c < ε / 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
a : A
⊢ cfc 0 a = 0

Tactic:
by
  by_cases ha : p a
  · exact cfc_apply (0 : R → R) a ▸ map_zero (cfcHom ha)
  · rw [cfc_apply_of_not_predicate a ha]

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

Goal state before:
pp: case succ
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
n : ℕ
ih : v n ∈ s
⊢ v (n + 1) ∈ s

Tactic:
| succ n ih =>

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : CompleteSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
hf : BoundedVariationOn f univ
x : α
⊢ Tendsto f (𝓝[<] x) (𝓝 (Function.leftLim f x))

Tactic:
by
  apply tendsto_leftLim_of_tendsto
  convert! hf.exists_tendsto_left x
  simp

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 : LocallyBoundedVariationOn f s
x : α
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
y : α
h'f : Tendsto f (𝓝[s ∩ Icc y x ∩ Iio x] x) (𝓝 l)
hy : y ∈ s ∩ Iio x
this✝ : 𝓝[s ∩ Iio x] x = 𝓝[s ∩ Icc y x ∩ Iio x] x
this : BoundedVariationOn f (s ∩ Icc y x)
⊢ (fun y_1 ↦ eVariationOn f (s ∩ Icc y x ∩ Icc y_1 x)) =ᶠ[𝓝[s ∩ Icc y x ∩ Iio x] x] fun y ↦ eVariationOn f (s ∩ Icc y x)

Tactic:
filter_upwards [self_mem_nhdsWithin] with z hz

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
l : E
hf : LocallyBoundedVariationOn f s
x : α
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
y : α
h'f : Tendsto f (𝓝[s ∩ Icc y x ∩ Iio x] x) (𝓝 l)
hy : y ∈ s ∩ Iio x
this✝ : 𝓝[s ∩ Iio x] x = 𝓝[s ∩ Icc y x ∩ Iio x] x
this : BoundedVariationOn f (s ∩ Icc y x)
z : α
hz : z ∈ s ∩ Icc y x ∩ Iio x
⊢ eVariationOn f (s ∩ Icc y x ∩ Icc z x) = eVariationOn f (s ∩ Icc z x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case neg.inl
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 : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
this : ¬ContinuousOn (fun x ↦ (algebraMap R S) (g (f x))) (σₙ S a)
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn a this]

Goal state after:
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 > 0, ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2

Tactic:
by
    have := CompactSpace.uniformContinuous_of_continuous (map_continuous f)
    rw [Metric.uniformity_basis_dist.uniformContinuous_iff
      (basis_sets _).uniformity_of_nhds_zero_swapped] at this
    exact this {z | gauge U z < 1 / 2} <| tendsto_gauge_nhds_zero hU₀
      |>.eventually_lt_const <| by positivity

Goal state after:
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
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (x : ↑I), gauge U ((bernsteinApproximation n f) x - f x) < 1

Tactic:
obtain ⟨δ, hδ₀, hδ⟩ : ∃ δ > 0, ∀ x y : I, dist x y < δ → gauge U (f x - f y) < 1 / 2 := by
    have := CompactSpace.uniformContinuous_of_continuous (map_continuous f)
    rw [Metric.uniformity_basis_dist.uniformContinuous_iff
      (basis_sets _).uniformity_of_nhds_zero_swapped] at this
    exact this {z | gauge U z < 1 / 2} <| tendsto_gauge_nhds_zero hU₀
      |>.eventually_lt_const <| by positivity

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
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (x : ↑I), gauge U ((bernsteinApproximation n f) x - f x) < 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case mpr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : Semifield R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
x : ↑(σₙ R a)
hx : ↑x ∈ {0}
⊢ f x ∈ {0}

Tactic:
push _ ∈ _ at hx ⊢

Goal state after:
case mpr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : Semifield R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
x : ↑(σₙ R a)
hx : ↑x = 0
⊢ f x = 0
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : CompactSpace ↑s
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
frequently : ∀ (f : C(↑s, 𝕜)), (∃ᶠ (g : C(↑s, 𝕜)) in 𝓝 f, p g) → p f
f✝ : C(↑s, 𝕜)
h : ∀ f ∈ (polynomialFunctions s).starClosure, p f
f : C(↑s, 𝕜)
this : ∃ᶠ (x : C(↑s, 𝕜)) in 𝓝 f, x ∈ ↑(polynomialFunctions s).starClosure
⊢ ∃ᶠ (g : C(↑s, 𝕜)) in 𝓝 f, p g

Tactic:
exact this.mp <| .of_forall h

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

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

Tactic:
by
  intro f hf
  induction hf using adjoin_induction with
  | mem f hf =>
    obtain rfl := Set.mem_singleton_iff.mp hf
    rfl
  | add f g _ _ hf hg => exact add_mem hf hg
  | zero => exact zero_mem _
  | mul f g _ _ _ hg => exact Ideal.mul_mem_left _ f hg
  | smul r f _ hf =>
    rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
    rw [map_smul, hf, smul_zero]
  | star f _ hf =>
    rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
    rw [map_star, hf, star_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
⊢ closure ↑(adjoin 𝕜 {ContinuousMapZero.id s}) = Set.univ

Tactic:
← isClosedEmbedding_toContinuousMap.injective.preimage_image (closure _),

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
⊢ _root_.toContinuousMap ⁻¹' _root_.toContinuousMap '' closure ↑(adjoin 𝕜 {ContinuousMapZero.id s}) = Set.univ
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case id
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
⊢ φ (ContinuousMapZero.id s) * a = 0

Tactic:
| id =>

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

Goal state before:
pp: case id
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : a * φ (ContinuousMapZero.id s) = 0
hmul_star_id : a * φ (star (ContinuousMapZero.id s)) = 0
hφ : Continuous ⇑φ
⊢ a * φ (ContinuousMapZero.id s) = 0

Tactic:
| id =>

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
⊢ IsOpen[inst✝.toTopologicalSpace] u → f x₀ ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x₀, f x' ⊆ u

Tactic:
hu

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
⊢ f x₀ ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x₀, f x' ⊆ u
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ inst✝ =
    generateFrom
      ((fun a ↦ {a_1 | {a_1} ⊆ a}) '' {U | IsOpen[inst✝] U} ∪
        (fun a ↦ {a_1 | ({a_1} ∩ a).Nonempty}) '' {U | IsOpen[inst✝] U})

Tactic:
singleton_subset_iff,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
⊢ inst✝ =
    generateFrom
      ((fun a ↦ {a_1 | a_1 ∈ a}) '' {U | IsOpen[inst✝] U} ∪
        (fun a ↦ {a_1 | ({a_1} ∩ a).Nonempty}) '' {U | IsOpen[inst✝] U})
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 ⇑(cfcₙAux ⋯ a ha)

Tactic:
simp only [cfcₙAux, NonUnitalStarAlgHom.coe_comp]

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
    ((⇑↑(cfcHom ⋯) ∘ ⇑↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))) ∘
      ⇑ContinuousMapZero.toContinuousMapHom)
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
⊢ CompleteSpace (Compacts α)

Tactic:
refine ⟨fun {f} ⟨_, hf⟩ => ?_⟩
  grw [← Filter.curry_le_prod, (𝓤 α).basis_sets.uniformity_compacts.ge_iff] at hf
  change ∀ {U} (hU : U ∈ 𝓤 α), ∀ᶠ K in f, ∀ᶠ K' in f, (↑K, ↑K') ∈ hausdorffEntourage U at hf
  let l : Filter α := f.lift' fun s => ⋃ K ∈ s, K
  have hl : l.TotallyBounded := by
    intro U hU
    obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
    obtain ⟨K, hK⟩ := hf (symm_le_uniformity hV) |>.exists
    obtain ⟨t, ht₁, ht₂⟩ := K.isCompact.totallyBounded V hV
    rw [← SetRel.preimage_eq_biUnion] at ht₂
    refine ⟨t, ht₁, Filter.mem_of_superset (Filter.mem_lift' hK) ?_⟩
    rw [Set.iUnion₂_subset_iff]
    intro K' ⟨_, (hK' : ↑K' ⊆ V.preimage K)⟩
    grw [← hVU, SetRel.preimage_comp, ← ht₂, hK']
  let L : Compacts α := ⟨{x | ClusterPt x l}, hl.isCompact_setOfPred_clusterPt⟩
  exists L
  simp_rw [nhds_eq_comap_uniformity']
  rw [uniformity_hasBasis_closed.uniformity_compacts.comap _ |>.ge_iff]
  intro U ⟨hU₁, hU₂⟩
  filter_upwards [hf hU₁] with K hK
  simp_rw [Set.mem_preimage, Prod.map, id, mem_hausdorffEntourage]
  constructor
  · intro x hx
    set lx := l ⊓ 𝓟 (UniformSpace.ball x U) with le_def
    have hlx : lx.TotallyBounded := hl.mono inf_le_left
    have : lx.NeBot := by
      rw [le_def, Filter.lift'_inf_principal_eq, Filter.lift'_neBot_iff fun _ _ h =>
        Set.inter_subset_inter_left _ <| Set.biUnion_subset_biUnion_left h]
      intro s hs
      obtain ⟨K', ⟨h₁, -⟩, h₂⟩ := Filter.nonempty_of_mem <| Filter.inter_mem hK hs
      obtain ⟨y, hy, hxy⟩ := h₁ hx
      exact ⟨y, Set.mem_iUnion₂_of_mem h₂ hy, hxy⟩
    obtain ⟨y, hy⟩ := hlx.exists_clusterPt
    have hy₁ : ClusterPt y l := .of_inf_left hy
    have hy₂ : ClusterPt y (𝓟 (UniformSpace.ball x U)) := .of_inf_right hy
    rw [← mem_closure_iff_clusterPt, (UniformSpace.isClosed_ball x hU₂).closure_eq] at hy₂
    exact ⟨y, hy₁, hy₂⟩
  · intro x (hx : ClusterPt x l)
    rw [← (hU₂.relImage_of_isCompact K.isCompact).closure_eq, mem_closure_iff_clusterPt]
    refine hx.mono ?_
    rw [Filter.le_principal_iff]
    refine Filter.mem_of_superset (Filter.mem_lift' hK) ?_
    rw [Set.iUnion₂_subset_iff]
    exact fun _ ⟨_, h⟩ => h

Goal state after:
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
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
⊢ f ≤ 𝓝 x

Tactic:
rw [(nhds_basis_opens x).ge_iff]

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 : α
hx : x ∈ ↑K
⊢ ∀ (i' : Set α), x ∈ i' ∧ IsOpen[inst✝.toTopologicalSpace] i' → i' ∈ f
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t

Tactic:
rw [nnreal_iff]

Goal state after:
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
⊢ (∀ x ∈ spectrum ℝ a, 0 ≤ x) ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
x : A
hx : IsSelfAdjoint x
ha₁ : IsSelfAdjoint (x * x)
ha₂ : QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
⊢ 0 ≤ x * x

Tactic:
simpa [sq, hx.star_eq] using star_mul_self_nonneg x

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

Goal state before:
pp: A : Type u_4
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
a : A
ha : (spectrum ℝ a).Nonempty
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx₁ : x ∈ spectrum ℝ a
hx₂ : ↑‖x‖₊ = spectralRadius ℝ a
⊢ (algebraMap ℝ≥0 ℝ) ‖x‖₊ ∈ spectrum ℝ a

Tactic:
NNReal.algebraMap_eq_coe

Goal state after:
A : Type u_4
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
a : A
ha : (spectrum ℝ a).Nonempty
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx₁ : x ∈ spectrum ℝ a
hx₂ : ↑‖x‖₊ = spectralRadius ℝ a
⊢ ↑‖x‖₊ ∈ spectrum ℝ a
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
this : IsConnected ((σ ℂ a)ᶜ ∩ {a | 0 ≤ a.im ∨ a.im ≤ 0})
⊢ Set.univ = {a | 0 ≤ a.im ∨ a.im ≤ 0}

Tactic:
exact Eq.symm <| Set.eq_univ_of_forall (fun z ↦ le_total 0 z.im)

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : ℕ → E' →SL[σ₁₂] F
g : E' →SL[σ₁₂] F
hg : Tendsto (fun n x ↦ (f n) x) atTop (𝓝 ⇑g)
hf : CauchySeq f
b : ℕ → ℝ
hb₀ : ∀ (n : ℕ), 0 ≤ b n
hfb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
hb_lim : Tendsto b atTop (𝓝 0)
⊢ Tendsto f atTop (𝓝 g)

Tactic:
simp_rw [dist_eq_norm] at hfb

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : ℕ → E' →SL[σ₁₂] F
g : E' →SL[σ₁₂] F
hg : Tendsto (fun n x ↦ (f n) x) atTop (𝓝 ⇑g)
hf : CauchySeq f
b : ℕ → ℝ
hb₀ : ∀ (n : ℕ), 0 ≤ b n
hb_lim : Tendsto b atTop (𝓝 0)
hfb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → ‖f n - f m‖ ≤ b N
⊢ Tendsto f atTop (𝓝 g)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed[inst✝¹] s
ht : IsClosed[inst✝¹] t
hd : Disjoint s t
P : Set X → Set X → Prop := fun x x_1 ↦ True
c : Urysohns.CU P := { C := s, U := tᶜ, P_C_U := trivial, closed_C := hs, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ ∃ f, EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Icc 0 1

Tactic:
exact ⟨⟨c.lim, c.continuous_lim⟩, c.lim_of_mem_C, fun x hx => c.lim_of_notMem_U _ fun h => h hx,
    c.lim_mem_Icc⟩

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case refine_2
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
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
⊢ { toFun := fun x ↦ 1 - f x, continuous_toFun := ⋯ } x ∈ Icc 0 1

Tactic:
· have : 0 ≤ f x ∧ f x ≤ 1 := by simpa using h'f x
    simp [this]

Goal state after: