mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝⁹ : BorelSpace E
inst✝⁸ : SecondCountableTopology E
p : ℝ≥0∞
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : BorelSpace F
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h :
  ∀ (L : StrongDual ℝ (WithLp p (E × F))),
    charFunDual (Measure.map (toLp p) ξ) L =
      charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
        charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)
L : StrongDual ℝ (WithLp p (E × F))
⊢ charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
      charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) =
    charFunDual (Measure.map (toLp p) (μ.prod ν)) L

Tactic:
charFunDual_prod'

Goal state after:
E : Type u_2
F : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝⁹ : BorelSpace E
inst✝⁸ : SecondCountableTopology E
p : ℝ≥0∞
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : BorelSpace F
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h :
  ∀ (L : StrongDual ℝ (WithLp p (E × F))),
    charFunDual (Measure.map (toLp p) ξ) L =
      charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
        charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)
L : StrongDual ℝ (WithLp p (E × F))
⊢ charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
      charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) =
    charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
      charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ μ B ≤ max (μ univ) (ν univ)

Tactic:
simp [measure_mono]

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

Goal state before:
pp: case refine_2
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ ν B ≤ max (μ univ) (ν univ)

Tactic:
simp [measure_mono]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hf
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
x : X
hx : x ∈ E n
⊢ y n - ε' ≤ f x

Tactic:
dsimp [y]

Goal state after:
case hf
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
x : X
hx : x ∈ E n
⊢ a + ε' * (↑↑n + 1) - ε' ≤ f x
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
⊢ ↑((μ.prod ν).map ⋯) = ↑μ

Tactic:
simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_fst_prod,
             measure_univ, one_smul]

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_2
inst✝² : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
α' : Type u_3
inst✝¹ : MeasurableSpace α'
β' : Type u_4
inst✝ : MeasurableSpace β'
f : α → α'
g : β → β'
f_mble : Measurable f
g_mble : Measurable g
⊢ (μ.map ⋯).prod (ν.map ⋯) = (μ.prod ν).map ⋯

Tactic:
by
  apply Subtype.ext
  simp only [val_eq_to_measure, toMeasure_prod, toMeasure_map]
  rw [Measure.map_prod_map _ _ f_mble g_mble]

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

Goal state before:
pp: case refine_1
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
s₁ : (i : ι) → Set (X i)
hs₁ : s₁ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₁ : (i : κ) → Set (Y i)
ht₁ : t₁ ∈ univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t}
s₂ : (i : ι) → Set (X i)
hs₂ : s₂ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₂ : (i : κ) → Set (Y i)
ht₂ : t₂ ∈ univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t}
⊢ (fun i ↦ s₁ i ∩ s₂ i) ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}

Tactic:
simp only [Set.mem_pi, mem_univ, mem_ofPred_eq, forall_const] at hs₁ hs₂ ⊢
      exact fun i ↦ (hs₁ i).inter (hs₂ i)

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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ ∫ (x : E), g x ∂↑μ ≤ ↑C * ‖g.toBoundedContinuousFunction‖

Tactic:
· calc ∫ x, g x ∂μ
      _ ≤ ∫ (x : E), ‖g.toBoundedContinuousFunction‖ ∂μ := by
        gcongr
        · exact g.continuous.integrable_of_hasCompactSupport g.hasCompactSupport
        · simp
        · intro x
          apply le_of_abs_le
          exact g.toBoundedContinuousFunction.norm_coe_le_norm x
      _ ≤ C * ‖g.toBoundedContinuousFunction‖ := by
        simp only [integral_const, smul_eq_mul]
        gcongr
        exact hμ

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

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ μlim' ∈ {μ | μ.mass ≤ C}

Tactic:
simp only [mem_ofPred_eq, FiniteMeasure.mk_apply, μlim', FiniteMeasure.mass]
    rw [show C = (ENNReal.ofReal ↑C).toNNReal by simp]
    exact ENNReal.toNNReal_mono (by simp) μlim_le

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

Goal state before:
pp: case e'_3.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ i, ‖(x + y) i‖ ^ p) ^ (1 / p) ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) + (∑ i, ‖y i‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ (π * Real.Gamma (2 / p + 1)) ^ card ι = ∫ (x : ι → ℂ), ∏ x_1, rexp (-‖x x_1‖ ^ p)

Tactic:
integral_fintype_prod_volume_eq_pow fun x : ℂ => Real.exp (- ‖x‖ ^ p),

Goal state after:
case e'_3.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ i, ‖(x + y) i‖ ^ p) ^ (1 / p) ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) + (∑ i, ‖y i‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ (π * Real.Gamma (2 / p + 1)) ^ card ι = (∫ (x : ℂ), rexp (-‖x‖ ^ p)) ^ card ι
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case inl
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Subsingleton E
⊢ IsTightMeasureSet S

pp: case inr
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
⊢ IsTightMeasureSet S

Tactic:
· simp only [IsTightMeasureSet, cocompact_eq_bot, smallSets_bot]
    convert! tendsto_pure_nhds (a := ∅) _
    simp

Goal state after:
case inr
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
⊢ IsTightMeasureSet S
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
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), ⋯⟩
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)
⊢ ‖g‖ * ↑(ρ ((partialSups K) n)ᶜ) ≤ ‖g‖ * ↑(ρ (K n)ᶜ)

Tactic:
by gcongr; apply le_partialSups

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

Goal state before:

Tactic:
μ ((⋃ i ∈ Finset.range n, t i) ∆ (⋃ i, f i))
    _ ≤ μ ((⋃ i ∈ Finset.range n, (t i) ∆ (f i)) ∪ ⋃ i ∈ Ici n, f i) := by
      gcongr
      intro x hx
      simp only [Finset.mem_range, mem_symmDiff, mem_iUnion, exists_prop, not_exists, not_and,
        mem_Ici, mem_union] at hx ⊢
      grind
    _ ≤ ∑ i ∈ Finset.range n, μ (t i ∆ f i) + μ (⋃ i ∈ Ici n, f i) := by
      apply (measure_union_le _ _).trans
      gcongr
      apply measure_biUnion_finset_le
    _ ≤ ∑ i ∈ Finset.range n, δ i + μ (⋃ i ∈ Ici n, f i) := by
      gcongr with i; exact (ht i).le
    _ ≤ ∑' i, δ i + μ (⋃ i ∈ Ici n, f i) := by
      gcongr; exact ENNReal.sum_le_tsum (Finset.range n)
    _ < ε / 2 + ε / 2 := by gcongr
    _ = ε := ENNReal.add_halves ε

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)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
⊢ 2⁻¹ * |r| * ‖∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)‖ < ε

Tactic:
have hr : 0 < r := lt_of_lt_of_le (by positivity) hrδ

Goal state after:
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)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
⊢ 2⁻¹ * |r| * ‖∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)‖ < ε
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
this✝ : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
this : infDist a (⇑(algebraMap 𝕜 A) '' μ.support) ≤ ‖(algebraMap 𝕜 A) x - a‖
⊢ ‖(algebraMap 𝕜 A) x - a‖⁻¹ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support))⁻¹

Tactic:
inv_le_inv₀ (by linarith) (by positivity),

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
this✝ : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
this : infDist a (⇑(algebraMap 𝕜 A) '' μ.support) ≤ ‖(algebraMap 𝕜 A) x - a‖
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) ≤ ‖(algebraMap 𝕜 A) x - a‖
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
⊢ μ ≤ ξ + ν

Tactic:
obtain ⟨s, hs⟩ := exists_isHahnDecomposition μ ν

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
s : Set α
hs : IsHahnDecomposition μ ν s
⊢ μ ≤ ξ + ν
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ μ.toFiniteAux[|univ] = μ

Tactic:
Measure.toFiniteAux,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ (if IsFiniteMeasure μ then μ else ⋯.choose)[|univ] = μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.refine_1
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
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
⊢ ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε

Tactic:
have mt := h𝒜₀.measurable t ht

Goal state after:
case refine_2.refine_1
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
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
mt : MeasurableSet t
⊢ ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε
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 j : Set X
hij : i ≠ j
hj : ∃ (x : MeasurableSet j), ⟨j, ⋯⟩ ∈ P.parts
h'i : MeasurableSet i
i_mem : ⟨i, ⋯⟩ ∈ P.parts
⊢ Function.onFun Disjoint id i j

Tactic:
rcases hj with ⟨h'j, j_mem⟩

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 j : Set X
hij : i ≠ j
h'i : MeasurableSet i
i_mem : ⟨i, ⋯⟩ ∈ P.parts
h'j : MeasurableSet j
j_mem : ⟨j, ⋯⟩ ∈ P.parts
⊢ 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 j : Set X
hij : i ≠ j
hj : ∃ (x : MeasurableSet j), ⟨j, ⋯⟩ ∈ P.parts
h'i : MeasurableSet i
i_mem : ⟨i, ⋯⟩ ∈ P.parts
⊢ Function.onFun Disjoint id i j

Tactic:
rcases hj with ⟨h'j, j_mem⟩

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 j : Set X
hij : i ≠ j
h'i : MeasurableSet i
i_mem : ⟨i, ⋯⟩ ∈ P.parts
h'j : MeasurableSet j
j_mem : ⟨j, ⋯⟩ ∈ P.parts
⊢ Function.onFun Disjoint id i j