mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
⊢ UnifTight f p μ

Tactic:
intro ε hε
  rw [ENNReal.tendsto_atTop_zero] at hf_tendsto
  obtain ⟨N, hNε⟩ := hf_tendsto ε (by simpa only [gt_iff_lt, ENNReal.coe_pos])
  let F : Fin N → α → β := fun n => f n
  have hF : ∀ n, MemLp (F n) p μ := fun n => hf n
  obtain ⟨s, hμs, hFε⟩ := unifTight_fin hp' hF hε
  refine ⟨s, hμs, fun n => ?_⟩
  by_cases! hn : n < N
  · exact hFε ⟨n, hn⟩
  · exact (eLpNorm_indicator_le _).trans (hNε n hn)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous (circleTransformDeriv R z w f)

Tactic:
rw [circleTransformDeriv_eq]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous fun θ ↦ (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f g : α →C_c ℝ
h : α →C_c ℝ≥0
hh : (f + g).nnrealPart + h = f.nnrealPart + g.nnrealPart ∧ (-f + -g).nnrealPart + h = (-f).nnrealPart + (-g).nnrealPart
⊢ ↑(Λ (f.nnrealPart + g.nnrealPart)) - ↑(Λ ((-g + -f).nnrealPart + h)) =
    ↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart) + (↑(Λ g.nnrealPart) - ↑(Λ (-g).nnrealPart))

Tactic:
add_comm (-g) (-f),

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f g : α →C_c ℝ
h : α →C_c ℝ≥0
hh : (f + g).nnrealPart + h = f.nnrealPart + g.nnrealPart ∧ (-f + -g).nnrealPart + h = (-f).nnrealPart + (-g).nnrealPart
⊢ ↑(Λ (f.nnrealPart + g.nnrealPart)) - ↑(Λ ((-f + -g).nnrealPart + h)) =
    ↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart) + (↑(Λ g.nnrealPart) - ↑(Λ (-g).nnrealPart))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
ω : E → E →L[𝕜] F
a : E
⊢ CurveIntegrable ω (Path.refl a)

Tactic:
by
  simp [CurveIntegrable, Pi.zero_def]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
t : ℝ
ht : t ∈ Ioo 0 1
⊢ ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖

Tactic:
rw [DifferentiableAt.lineDeriv_eq_fderiv]

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
t : ℝ
ht : t ∈ Ioo 0 1
⊢ ‖(fderiv ℝ f ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖

E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
t : ℝ
ht : t ∈ Ioo 0 1
⊢ DifferentiableAt ℝ f ((lineMap a b) t)
Module: Mathlib.MeasureTheory.Integral.TorusIntegral

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin n → ℂ) → E
c : Fin n → ℂ
R : Fin n → ℝ
⊢ (∯ (x : Fin n → ℂ) in T(c, R), -f x) = -∯ (x : Fin n → ℂ) in T(c, R), f x

Tactic:
simp [torusIntegral, integral_neg]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
⊢ ContDiff ℝ (↑n) (charFun μ)

Tactic:
by
  simp_rw [funext charFun_eq_fourierIntegral']
  refine (contDiff_fourierIntegral (L := innerSL ℝ) fun k hk ↦ ?_).comp (by fun_prop)
  simp only [Pi.one_apply, one_mem, CStarRing.norm_of_mem_unitary, mul_one]
  refine MemLp.integrable_norm_pow' ((hint k).mono_exponent (by simp_all))

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ (fun t ↦ charFun (Measure.map X P) t - (1 - ↑t ^ 2 / 2)) =o[𝓝 0] fun t ↦ t ^ 2

Tactic:
simp_rw [← taylorWithinEval_charFun_two_zero' (by fun_prop) h0 h1]
  convert! taylor_isLittleO_univ ?_
  · simp
  refine contDiff_charFun <|
    (memLp_two_iff_integrable_sq (by fun_prop)).2 (.of_integral_ne_zero ?_)
  rw [integral_map]
  any_goals fun_prop
  simp_all

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: case refine_2
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFunDual μ = charFunDual ν
⊢ Continuous fun p ↦ ((ContinuousLinearMap.toLinearMap₁₂ ⋯.toContinuousLinearMap) p.1) p.2

Tactic:
· exact isBoundedBilinearMap_apply.symm.continuous

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ
δ_nn : 0 ≤ δ
h : ∀ (ε : ℝ) (B : Set Ω), δ < ε → MeasurableSet B → μ B ≤ ν (thickening ε B) + ENNReal.ofReal ε
⊢ levyProkhorovEDist μ ν ≤ ENNReal.ofReal δ

Tactic:
apply levyProkhorovEDist_le_of_forall_le

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ
δ_nn : 0 ≤ δ
h : ∀ (ε : ℝ) (B : Set Ω), δ < ε → MeasurableSet B → μ B ≤ ν (thickening ε B) + ENNReal.ofReal ε
⊢ ∀ (ε : ℝ≥0∞) (B : Set Ω), ENNReal.ofReal δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) (Ioc 0 ‖f‖) volume

Tactic:
exact intble₂.add <| integrable_const ε

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h₂
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := ⋯
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := ⋯
T : Set (FiniteMeasure ↑K) := ⋯
⊢ F '' T ⊆ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}

Tactic:
· simp only [null_iff_toMeasure_null, image_subset_iff, preimage_ofPred_eq, toMeasure_map,
        ofPred_subset_ofPred, F, T]
      intro μ hμ
      rw [Measure.map_apply hf.continuous.measurable hK.measurableSet.compl]
      refine ⟨(mass_map_le _ _).trans hμ, by simp [f]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: x : EuclideanSpace ℝ (Fin 3)
r : ℝ
⊢ volume (closedBall x r) = ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3)

Tactic:
by
  rw [addHaar_closedBall_eq_addHaar_ball, volume_ball_fin_three x]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}

pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦
    2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖

Tactic:
· exact IsCoboundedUnder.of_frequently_ge <| .of_forall fun _ ↦ ENNReal.toReal_nonneg

Goal state after:
case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦
    2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⋯
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 : ∀ᶠ (n : ℕ) in atTop, dist (∫ (x : E), g x ∂∑ i ∈ Finset.range (n + 1), ↑(ν i)) (∫ (x : E), g x ∂↑μ) < ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * ↑(u n) < ε / 3
n : ℕ
hn : dist (∫ (x : E), g x ∂∑ i ∈ Finset.range (n + 1), ↑(ν i)) (∫ (x : E), g x ∂↑μ) < ε / 3
h'n : ‖g‖ * ↑(u n) < ε / 3
this✝ :
  Tendsto (fun ρ ↦ ∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i))) (↑f)
    (𝓝 (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ν i)))
ρ : FiniteMeasure E
hρ :
  dist (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)))
      (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ν i)) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
this : ρ = ρ.restrict ((partialSups K) n)ᶜ + ∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)
⊢ K n ⊆ (partialSups K) n

Tactic:
apply le_partialSups

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case refine_2.inl
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : NormalSpace E
this : (↑μ).InnerRegular
⊢ μ (K n)ᶜ ≤ u n

Tactic:
rw [← ENNReal.coe_le_coe, ennreal_coeFn_eq_coeFn_toMeasure,
      (hK n).isClosed.isOpen_compl.measure_eq_biSup_integral_continuous]

Goal state after:
case refine_2.inl
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : NormalSpace E
this : (↑μ).InnerRegular
⊢ ⨆ f,
      ⨆ (_ : Continuous[inst✝², _] f),
        ⨆ (_ : EqOn f 0 (K n)ᶜᶜ), ⨆ (_ : 0 ≤ f), ⨆ (_ : f ≤ 1), ENNReal.ofReal (∫ (x : E), f x ∂↑μ) ≤
    ↑(u n)
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:

Tactic:
∑ i ∈ Finset.range n, ∑ p ∈ (P i).parts, f p
    _ = ∑ p ∈ Q.parts, f p := (Finpartition.sum_combine P hs_disj.supIndep (fun p => f p)).symm
    _ ≤ ∑ p ∈ R.parts, f p := Finset.sum_le_sum_of_subset (Q.parts_subset_extendOfLE hQ_le)
    _ ≤ preVariationFun f (⋃ i, s i) := sum_le f (MeasurableSet.iUnion hs) R

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this : ↑μ (K n)ᶜ ≤ ↑(u n)
⊢ μ (K n)ᶜ ≤ u n

Tactic:
apply ENNReal.coe_le_coe.1
      convert! this
      simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case inr
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
this : TopologicalSpace.UpgradedIsCompletelyMetrizableSpace E := TopologicalSpace.upgradeIsCompletelyMetrizable E
h✝ : 𝓕.NeBot
U : Ultrafilter ι
hU : ↑U ≤ 𝓕
h_compact : IsCompact (closure[instTopologicalSpace] {x | ∃ n, μ n = x})
μ' : ProbabilityMeasure E
hμ' : Tendsto μ (↑U) (𝓝 μ')
⊢ ↑μ' = ↑μ₀

Tactic:
refine ext_of_forall_mem_subalgebra_integral_eq_of_pseudoEMetric_complete_countable hA
    fun g hg ↦ tendsto_nhds_unique ?_ ((hμ g hg).comp hU)

Goal state after:
case inr
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
this : TopologicalSpace.UpgradedIsCompletelyMetrizableSpace E := TopologicalSpace.upgradeIsCompletelyMetrizable E
h✝ : 𝓕.NeBot
U : Ultrafilter ι
hU : ↑U ≤ 𝓕
h_compact : IsCompact (closure[instTopologicalSpace] {x | ∃ n, μ n = x})
μ' : ProbabilityMeasure E
hμ' : Tendsto μ (↑U) (𝓝 μ')
g : E →ᵇ 𝕜
hg : g ∈ A
⊢ Tendsto ((fun n ↦ ∫ (x : E), g x ∂↑(μ n)) ∘ fun x ↦ x) U.1 (𝓝 (∫ (x : E), g x ∂↑μ'))
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case hx
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this : Filter.Tendsto (⇑μ ∘ fun n ↦ T n ∩ s) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)))
⊢ μ (⋃ n, T n ∩ s) ≠ ∞

Tactic:
exact ne_top_of_le_ne_top hμs
        (measure_mono (iUnion_subset (fun i ↦ inter_subset_right ..)))

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < (variation μ) s
P : Finset (Set X)
Ps : ∀ t ∈ P, t ⊆ s
P_disj : (↑P).PairwiseDisjoint id
P_meas : ∀ t ∈ P, MeasurableSet t
hP : a < ∑ p ∈ P, ‖μ p‖ₑ
I : ∑ p ∈ P with 0 ≤ μ p, ‖μ p‖ₑ = ‖μ (⋃ p ∈ {p ∈ P | 0 ≤ μ p}, p)‖ₑ
⊢ ∑ x ∈ P with μ x < 0, |μ x| = ∑ x ∈ P with μ x < 0, -μ x

Tactic:
refine Finset.sum_congr rfl (fun p hp ↦ ?_)
        simp only [Finset.mem_filter] at hp
        simp [hp.2.le]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case inr.inr.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
hbound : ∀ (m : ℕ), ∀ μ ∈ S, 1 - ε * 2 ^ (-↑m) < ↑(μ (⋃ i, ⋃ (_ : i ≤ km m), ball (D i) (u m)))
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
bigcalc : ∀ μ ∈ S, ↑μ bigKᶜ ≤ ε
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] bigK

Tactic:
simp_rw [bigK, ← Set.mem_Iic]

Goal state after:
case inr.inr.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
hbound : ∀ (m : ℕ), ∀ μ ∈ S, 1 - ε * 2 ^ (-↑m) < ↑(μ (⋃ i, ⋃ (_ : i ≤ km m), ball (D i) (u m)))
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
bigcalc : ∀ μ ∈ S, ↑μ bigKᶜ ≤ ε
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (⋂ m, ⋃ i ∈ Iic (km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case intro
ι : Type u_1
X : Type u_2
E : Type u_3
F : Type u_4
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
μ : VectorMeasure X F
f : X → E
inst✝ : Finite ι
t : ι → Set X
ht : ∀ (i : ι), MeasurableSet (t i)
h't : ∀ (i : ι), μ.IntegrableOn f (t i)
val✝ : Fintype ι
⊢ μ.IntegrableOn f (⋃ i, t i)

Tactic:
simpa using IntegrableOn.biUnion_finset (f := f) (μ := μ) (s := Finset.univ) (t := t)
    (fun i hi ↦ ht i) (fun i hi ↦ h't i)

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := AddContent.onIoc (Function.rightLim f)
u v : α
huv : u ≤ v
⊢ ‖Function.rightLim f v - Function.rightLim f u‖ₑ ≤
    ENNReal.ofReal
      (variationOnFromTo (Function.rightLim f) univ ⋯.some v - variationOnFromTo (Function.rightLim f) univ ⋯.some u)

Tactic:
rw [← variationOnFromTo.add hf.rightLim.locallyBoundedVariationOn
      (mem_univ h'α.some) (mem_univ u) (mem_univ v)]

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := AddContent.onIoc (Function.rightLim f)
u v : α
huv : u ≤ v
⊢ ‖Function.rightLim f v - Function.rightLim f u‖ₑ ≤
    ENNReal.ofReal
      (variationOnFromTo (Function.rightLim f) univ h'α.some u + variationOnFromTo (Function.rightLim f) univ u v -
        variationOnFromTo (Function.rightLim f) univ ⋯.some u)