mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).im) μ

Tactic:
exact SignedMeasure.integrable_rnDeriv _ _

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
s : Set α
hs : MeasurableSet s
f : α → E
hm : m ≤ m0
hfint : Integrable f μ
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, |μ[f | m] x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

Tactic:
calc
  _ = ∫ x in s, |(μ.restrict s)[f | m] x| ∂μ :=
    (integral_congr_ae ((condExp_restrict_ae_eq_restrict hm hs hfint).fun_comp abs)).symm
  _  ≤ _ := integral_abs_condExp_le f

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

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
ν : ProbabilityMeasure Ω
⊢ ν ∅ = 0

Tactic:
by simp [coeFn_def]

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : BorelSpace Ω
μ ν : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂↑μ = ∫⁻ (x : Ω), ↑(f x) ∂↑ν
⊢ μ = ν

Tactic:
apply Subtype.ext
  change (μ : Measure Ω) = (ν : Measure Ω)
  exact ext_of_forall_lintegral_eq_of_IsFiniteMeasure h

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ (fun a ↦ ↑(f a)) ω ≤ (fun a ↦ ↑(g a) + edist f g) ω

Tactic:
have le : (f ω : ℝ≥0∞) ≤ (g ω : ℝ≥0∞) + nndist f g := by
    simpa only [← ENNReal.coe_add] using (by exact_mod_cast le')

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
le : ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)
⊢ (fun a ↦ ↑(f a)) ω ≤ (fun a ↦ ↑(g a) + edist f g) ω
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ∞
C : ℝ≥0∞ := (Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ).max' ⋯
i : ι
⊢ eLpNorm (f i) p μ ≤ ↑C.toNNReal

Tactic:
rw [ENNReal.coe_toNNReal]

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ∞
C : ℝ≥0∞ := (Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ).max' ⋯
i : ι
⊢ eLpNorm (f i) p μ ≤ C

case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ∞
C : ℝ≥0∞ := (Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ).max' ⋯
i : ι
⊢ C ≠ ∞
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: case mp
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass) → Tendsto μs F (𝓝 μ)

pp: case mpr
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto μs F (𝓝 μ) → Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)

Tactic:
· rintro ⟨normalized_lim, mass_lim⟩
    exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim

Goal state after:
case mpr
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto μs F (𝓝 μ) → Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_3
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))

Tactic:
by
  rw [tendsto_iff_forall_toWeakDualBCNN_tendsto]
  simp_rw [toWeakDualBCNN_apply _ _, ← testAgainstNN_coe_eq, ENNReal.tendsto_coe,
    ENNReal.toNNReal_coe]

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
⊢ μ Eᶜ = 1 - μ E

Tactic:
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i ↦ (μs i) E) L
hne : L.NeBot
⊢ μ Eᶜ = 1 - μ E

Tactic:
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
have mbles : ∀ r : ℝ, MeasurableSet (frontier (Metric.thickening r s)) :=
    fun r ↦ isClosed_frontier.measurableSet
  have disjs := Metric.frontier_thickening_disjoint s
  have key := Measure.countable_meas_pos_of_disjoint_iUnion (μ := μ) mbles disjs
  have aux := measure_sdiff_null (s := Ioo a b) (Set.Countable.measure_zero key volume)
  have len_pos : 0 < ENNReal.ofReal (b - a) := by simp only [hab, ENNReal.ofReal_pos, sub_pos]
  rw [← Real.volume_Ioo, ← aux] at len_pos
  simpa [Set.Nonempty] using nonempty_of_measure_ne_zero len_pos.ne'

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
μ : Measure α
inst✝¹ : Subsingleton ι
m : ι → Set (Set Ω)
κ : Kernel α Ω
inst✝ : IsMarkovKernel κ
f : ι → Set Ω
hf : ∀ i ∈ ∅, f i ∈ m i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ ∅, f i) = ∏ i ∈ ∅, (κ a) (f i)

pp: case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
μ : Measure α
inst✝¹ : Subsingleton ι
m : ι → Set (Set Ω)
κ : Kernel α Ω
inst✝ : IsMarkovKernel κ
f : ι → Set Ω
i : ι
hf : ∀ i_1 ∈ {i}, f i_1 ∈ m i_1
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i_1 ∈ {i}, f i_1) = ∏ i ∈ {i}, (κ a) (f i)

Tactic:
all_goals simp

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
ι' : Type u_5
g : ι' → ι
hg : Function.Injective g
h : iIndepSets π κ μ
s : Finset ι'
f : ι' → Set Ω
hf : ∀ i ∈ s, f i ∈ (π ∘ g) i
f' : ι → Set Ω := Function.extend g f fun x ↦ ∅
f'_apply : ∀ (x : ι'), f' (g x) = f x
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)

Tactic:
have hf' : ∀ i ∈ s.image g, f' i ∈ π i := by
    simp_rw [Finset.forall_mem_image, f'_apply]
    exact hf

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
ι' : Type u_5
g : ι' → ι
hg : Function.Injective g
h : iIndepSets π κ μ
s : Finset ι'
f : ι' → Set Ω
hf : ∀ i ∈ s, f i ∈ (π ∘ g) i
f' : ι → Set Ω := Function.extend g f fun x ↦ ∅
f'_apply : ∀ (x : ι'), f' (g x) = f x
hf' : ∀ i ∈ Finset.image g s, f' i ∈ π i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ↑μ (range f)ᶜ = 0

Tactic:
by exact μ.2

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
a : α
ha : ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
⊢ (κ a) (⋃ i, t1 ∩ f i) = (κ a) t1 * (κ a) (⋃ i, f i)

Tactic:
measure_iUnion,

Goal state after:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
a : α
ha : ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
⊢ ∑' (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (⋃ i, f i)

case hn
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
a : α
ha : ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
⊢ Pairwise (Function.onFun Disjoint fun i ↦ t1 ∩ f i)

case h
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
a : α
ha : ∀ (i : ℕ), (κ a) (t1 ∩ f i) = (κ a) t1 * (κ a) (f i)
⊢ ∀ (i : ℕ), MeasurableSet (t1 ∩ f i)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
ι : Type u_2
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) L
⊢ Tendsto (fun i ↦ μs i) L (𝓝 μ)

Tactic:
apply Filter.tendsto_of_seq_tendsto fun u hu ↦ ?_
  apply tendsto_of_forall_isOpen_le_liminf_nat fun G hG ↦ (h_opens G hG).trans ?_
  change _ ≤ atTop.liminf ((fun i ↦ μs i G) ∘ u)
  rw [liminf_comp]
  refine liminf_le_liminf_of_le hu (by isBoundedDefault) ?_
  exact isBoundedUnder_of ⟨1, by simp⟩ |>.isCoboundedUnder_ge

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
t : Set Ω
ht : t ∈ piiUnionInter (fun k ↦ {s k}) T
⊢ ∀ (n : ι), generateFrom {s n} ≤ _mΩ

pp: case inr.refine_3
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IsPiSystem (piiUnionInter (fun k ↦ {s k}) S)

pp: case inr.refine_4
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IsPiSystem (piiUnionInter (fun k ↦ {s k}) T)

pp: case inr.refine_5
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IndepSets (piiUnionInter (fun k ↦ {s k}) S) (piiUnionInter (fun k ↦ {s k}) T) η μ

Tactic:
· exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k

Goal state after:
case inr.refine_3
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IsPiSystem (piiUnionInter (fun k ↦ {s k}) S)

case inr.refine_4
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IsPiSystem (piiUnionInter (fun k ↦ {s k}) T)

case inr.refine_5
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ IndepSets (piiUnionInter (fun k ↦ {s k}) S) (piiUnionInter (fun k ↦ {s k}) T) η μ
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : ↥T) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : ↥S) → β ↑i)
sets_s : (i : ↥S) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : ↥T) → β ↑i)
sets_t : (i : ↥T) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
i : ι
hi : i ∈ S
⊢ sets_s' i = sets_s ⟨i, hi⟩

Tactic:
sets_s',

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : ↥T) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : ↥S) → β ↑i)
sets_s : (i : ↥S) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : ↥T) → β ↑i)
sets_t : (i : ↥T) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
i : ι
hi : i ∈ S
⊢ (if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ) = sets_s ⟨i, hi⟩
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : ↥T) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : ↥S) → β ↑i)
sets_s : (i : ↥S) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : ↥T) → β ↑i)
sets_t : (i : ↥T) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩
i : ι
hi : i ∈ T
⊢ sets_s' i = univ

Tactic:
sets_s',

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : ↥T) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : ↥S) → β ↑i)
sets_s : (i : ↥S) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : ↥T) → β ↑i)
sets_t : (i : ↥T) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩
i : ι
hi : i ∈ T
⊢ (if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ) = univ
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: γ : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
F : Filter γ
inst✝ : F.IsCountablyGenerated
μs : γ → ProbabilityMeasure Ω
μ : ProbabilityMeasure Ω
hne : F.NeBot
h :
  ∀ (f : Ω → ℝ),
    (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →
      (∃ L, LipschitzWith L f) → Tendsto (fun i ↦ ∫ (ω : Ω), f ω ∂↑(μs i)) F (𝓝 (∫ (ω : Ω), f ω ∂↑μ))
s : Set Ω
hs : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
fs : ℕ → Ω → ℝ := fun n ω ↦ ↑((thickenedIndicator ⋯ s) ω)
key₁ : Tendsto (fun n ↦ ∫ (ω : Ω), fs n ω ∂↑μ) atTop (𝓝 ((↑μ).real s))
room₁ : (↑μ).real s < (↑μ).real s + ε / 2
M : ℕ
hM : ∀ (b : ℕ), M ≤ b → ∫ (ω : Ω), fs b ω ∂↑μ < (↑μ).real s + ε / 2
key₂ : Tendsto (fun i ↦ ∫ (ω : Ω), fs M ω ∂↑(μs i)) F (𝓝 (∫ (ω : Ω), fs M ω ∂↑μ))
room₂ : ∫ (a : Ω), fs M a ∂↑μ < ∫ (a : Ω), fs M a ∂↑μ + ε / 2
ev_near : ∀ᶠ (x : γ) in F, (↑(μs x)).real s ≤ ∫ (a : Ω), fs M a ∂↑μ + ε / 2
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F fun n ↦ (↑(μs n)).real s

Tactic:
· exact isCoboundedUnder_le_of_le F (x := 0) (by simp)

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
inst✝ : MeasurableSingletonClass Ω
x : Ω
⊢ AEMeasurable X (Measure.dirac x)

Tactic:
· exact aemeasurable_dirac

Goal state after:
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
ι' : Type u_3
Y : ι' → Ω → ℝ
t : Finset ι'
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : ∀ i ∈ t, MemLp (Y i) 2 μ
i : ι
hi : i ∈ s
⊢ cov[X i, ∑ j ∈ t, Y j; μ] = ∑ j ∈ t, cov[X i, Y j; μ]

Tactic:
covariance_sum_right' hY (hX i hi)

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
ι' : Type u_3
Y : ι' → Ω → ℝ
t : Finset ι'
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : ∀ i ∈ t, MemLp (Y i) 2 μ
i : ι
hi : i ∈ s
⊢ ∑ i_1 ∈ t, cov[X i, Y i_1; μ] = ∑ j ∈ t, cov[X i, Y j; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: case e'_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
ι' : Type u_3
Y : ι' → Ω → ℝ
t : Finset ι'
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : ∀ i ∈ t, MemLp (Y i) 2 μ
x✝ : Ω
⊢ ∑ i ∈ s, X i x✝ = (∑ i ∈ s, X i) x✝

pp: case e'_4
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
ι' : Type u_3
Y : ι' → Ω → ℝ
t : Finset ι'
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : ∀ i ∈ t, MemLp (Y i) 2 μ
x✝ : Ω
⊢ ∑ j ∈ t, Y j x✝ = (∑ j ∈ t, Y j) x✝

Tactic:
all_goals simp

Goal state after: