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
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n
hgg' : g =ᵐ[μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f' n x) atTop (𝓝 (g' x))

Tactic:
filter_upwards [hff'', hgg', hfg] with x hff'x hgg'x hfgx

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n
hgg' : g =ᵐ[μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n ↦ f' n x) atTop (𝓝 (g' x))
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
⊢ ∫ᶜ (x : E) in Path.refl a, ω x = 0

Tactic:
simp [curveIntegral]

Goal state after:
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
a b c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
⊢ ∫ᶜ (x : E) in γ.cast hc hd, ω x = ∫ᶜ (x : E) in γ, ω x

Tactic:
simp [curveIntegral]

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

Goal state before:

Tactic:
_ = ∫ x in s, (ρ x).toReal * f x ∂μ := by
        apply MeasureTheory.integral_congr_ae
        apply h_toReal_ae.mono
        intro x hx
        simp [hx, mul_comm]
      _ = _ := by
        have h := setIntegral_withDensity_eq_setIntegral_toReal_smul₀ hρ_ae hρ_top f hs_meas
        simp [ν, h]

Goal state after:
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
a b : E
γ : Path a b
⊢ curveIntegral 0 γ = 0

Tactic:
simp [curveIntegral]

Goal state after:
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
a b : E
ω : E → E →L[𝕜] F
γ : Path a b
⊢ curveIntegral (-ω) γ = -∫ᶜ (x : E) in γ, ω x

Tactic:
simp [curveIntegral]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c d : E
γ₁ : Path a b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, b_1))) v) u
hcontdiff : ContDiffOn ℝ 2 (fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2) (Icc 0 1)
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
⊢ closure U = Icc 0 1

Tactic:
simp [hU, closure_prod_eq, Prod.mk_zero_zero, Prod.mk_one_one]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
f : ℕ → Set α
hf : ∀ (n : ℕ), ∃ j < i, f n ∈ generateMeasurableRec s j
⊢ ⋃ n, f n ∈ generateMeasurableRec s i

Tactic:
by
  unfold generateMeasurableRec
  exact mem_union_right _ ⟨fun n => ⟨f n, let ⟨j, hj, hf⟩ := hf n; mem_iUnion₂.2 ⟨j, hj, hf⟩⟩, rfl⟩

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
j : Ordinal.{v}
⊢ j < i → #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
hj

Goal state after:
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
j : Ordinal.{v}
hj : j < i
⊢ #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
t : E
⊢ ‖charFun μ t‖ ≤ μ.real Set.univ

Tactic:
rw [charFun_eq_fourierIntegral]
  exact (VectorFourier.norm_fourierIntegral_le_integral_norm _ _ _ _ _).trans_eq (by simp)

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u_1
t₁ t₂ : TopologicalSpace X
ht₁ : CompletelyRegularSpace X
ht₂ : CompletelyRegularSpace X
⊢ CompletelyRegularSpace X

Tactic:
rw [inf_eq_iInf]; apply completelyRegularSpace_iInf; simp [*]

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ κ (thickening s.toReal (thickening r.toReal B)) + (s + r) ≤ κ (thickening (s.toReal + r.toReal) B) + (s + r)

Tactic:
thickening_thickening_subset

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ κ (thickening (s.toReal + r.toReal) B) + (s + r) ≤ κ (thickening (s.toReal + r.toReal) B) + (s + r)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ μ (thickening r.toReal (thickening s.toReal B)) + (s + r) ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r)

Tactic:
thickening_thickening_subset

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ μ (thickening (r.toReal + s.toReal) B) + (s + r) ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hfint
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
⊢ Integrable (fun x ↦ f x) (rieszContent (toNNRealLinear Λ)).measure

Tactic:
exact Continuous.integrable_of_hasCompactSupport f.1.2 f.2

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : BorelSpace Ω
μ ν : LevyProkhorov (ProbabilityMeasure Ω)
h : dist μ ν = 0
⊢ μ.toMeasure = ν.toMeasure

Tactic:
apply ProbabilityMeasure.toMeasure_injective

Goal state after:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : BorelSpace Ω
μ ν : LevyProkhorov (ProbabilityMeasure Ω)
h : dist μ ν = 0
⊢ ↑μ.toMeasure = ↑ν.toMeasure
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : BorelSpace Ω
μ ν : LevyProkhorov (ProbabilityMeasure Ω)
h : dist μ ν = 0
⊢ inst✝³ = MeasurableSpace.generateFrom {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}

pp: case refine_2
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : BorelSpace Ω
μ ν : LevyProkhorov (ProbabilityMeasure Ω)
h : dist μ ν = 0
A : Set Ω
hA : A ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
⊢ ↑μ.toMeasure A = ↑ν.toMeasure A

Tactic:
· rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed]

Goal state after:
case refine_2
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : BorelSpace Ω
μ ν : LevyProkhorov (ProbabilityMeasure Ω)
h : dist μ ν = 0
A : Set Ω
hA : A ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
⊢ ↑μ.toMeasure A = ↑ν.toMeasure A
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
hfV : ∀ (x : X), f x ≤ V.indicator 1 x
⊢ μ.real K ≤ ν.real K + ↑ε

Tactic:
have hfK (x : X) : K.indicator 1 x ≤ f x := by
    by_cases hx : x ∈ K
    · simp [hx, pf1 hx]
    · simp [hx, (pf3 x).1]

Goal state after:
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
hfV : ∀ (x : X), f x ≤ V.indicator 1 x
hfK : ∀ (x : X), K.indicator 1 x ≤ f x
⊢ μ.real K ≤ ν.real K + ↑ε
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
g : G
s : Set A
⊢ ↑((distribHaarChar A) g) * μ s = μ (g • s)

Tactic:
by
  have : (DomMulAct.mk g • μ) s = μ (g • s) := by simp [domSMul_apply]
  rw [eq_comm, ← nnreal_smul_coe_apply, ← addHaarScalarFactor_smul_eq_distribHaarChar μ,
    ← this, ← Measure.smul_apply, ← isAddLeftInvariant_eq_smul_of_regular]

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)
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
n : ℕ
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ ((partialSups K) n)ᶜ ∩ disjointed K n = ∅

Tactic:
apply disjoint_iff.1

Goal state after:
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
n : ℕ
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ Disjoint ((partialSups K) n)ᶜ (disjointed K n)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: 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
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))

Tactic:
← ((stdOrthonormalBasis ℝ E).measurePreserving_repr_symm).measure_preimage
      measurableSet_ball.nullMeasurableSet

Goal state after:
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
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
⊢ x ∈ resolventSet 𝕜 w

Tactic:
contrapose! hw

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
x : 𝕜
hx : x ∈ μ.support
w : A
hw : ¬x ∈ resolventSet 𝕜 w
⊢ w ∉ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ IsFiniteMeasure μ.toFinite

Tactic:
rw [Measure.toFinite]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ IsFiniteMeasure μ.toFiniteAux[|univ]
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
⊢ _root_.SecondCountableTopology ↥(Lp E p μ)

Tactic:
refine @UniformSpace.secondCountable_of_separable _ _ _ ?_

Goal state after:
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
⊢ SeparableSpace ↥(Lp E p μ)
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ volume s = 0

Tactic:
by
  refine measure_mono_null ?_ hs.ordConnected.null_frontier
  rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]
  exact subset_closure

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

Goal state before:
pp: case refine_2
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
i : Set X
hi : i ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
j : Set X
⊢ j ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts) → i ≠ j → Function.onFun Disjoint id i j

Tactic:
hj

Goal state after:
case refine_2
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
i : Set X
hi : i ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
j : Set X
hj : j ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
⊢ i ≠ j → Function.onFun Disjoint id i j
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case refine_2
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
i : Set X
hi : i ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
j : Set X
⊢ j ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts) → i ≠ j → Function.onFun Disjoint id i j

Tactic:
hj

Goal state after:
case refine_2
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
i : Set X
hi : i ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
j : Set X
hj : j ∈ ↑(Finset.map (Function.Embedding.subtype MeasurableSet) P.parts)
⊢ i ≠ j → Function.onFun Disjoint id i j
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case a
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
this : ⋃ i, f i = (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i
i : ℕ
hi : n ≤ i
j : ℕ
⊢ j < n → Disjoint (f j) (f i)

Tactic:
hj

Goal state after:
case a
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
this : ⋃ i, f i = (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i
i : ℕ
hi : n ≤ i
j : ℕ
hj : j < n
⊢ Disjoint (f j) (f i)