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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
refine ⟨(ε / (2 * C)) ^ ENNReal.toReal p,
    Real.rpow_pos_of_pos (div_pos hε (mul_pos two_pos (NNReal.coe_pos.2 hCpos))) _,
    fun i s hs hμs => ?_⟩

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
⊢ eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos.refine_2.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
x : α
hx : x ∈ {x | ↑R < |μ[f | m] x|}
⊢ ↑‖(fun x ↦ ↑R) x‖₊ ≤ ↑‖μ[f | m] x‖₊

Tactic:
ENNReal.coe_le_coe,

Goal state after:
case pos.refine_2.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |μ[f | m] x|}
x : α
hx : x ∈ {x | ↑R < |μ[f | m] x|}
⊢ ‖(fun x ↦ ↑R) x‖₊ ≤ ‖μ[f | m] x‖₊
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
Heval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v
⊢ Measurable fun a ↦ ((↑a.1).prod ↑a.2) univ

Tactic:
← Set.univ_prod_univ,

Goal state after:
α : Type u_3
β : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
Heval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v
⊢ Measurable fun a ↦ ((↑a.1).prod ↑a.2) (univ ×ˢ univ)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm (s.indicator ((↑n)⁻¹ • f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
this,

Goal state after:
case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ x ∈ Finset.range n, eLpNorm ((↑n)⁻¹ • s.indicator (f x)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ x ∈ Finset.range n, eLpNorm ((↑n)⁻¹ • s.indicator (f x)) p μ ≤ ENNReal.ofReal ε

Tactic:
eLpNorm_const_smul,

Goal state after:
case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ x ∈ Finset.range n, ‖(↑n)⁻¹‖ₑ * eLpNorm (s.indicator (f x)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_3.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
n : ℕ
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm ((↑n)⁻¹ • f i) p μ ≤ ↑C

Tactic:
eLpNorm_const_smul,

Goal state after:
case refine_3.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
n : ℕ
hn : n ≠ 0
⊢ ∑ x ∈ Finset.range n, ‖(↑n)⁻¹‖ₑ * eLpNorm (f x) p μ ≤ ↑C
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = ∅
⊢ (κ a) ∅ = (κ a) s * (κ a) ∅

Tactic:
measure_empty,

Goal state after:
case inr.inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = ∅
⊢ 0 = (κ a) s * 0
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
x✝ : MeasurableSpace Ω
β : ι → Type u_7
m : (x : ι) → MeasurableSpace (β x)
f : (x : ι) → Ω → β x
μ : Measure Ω
⊢ iIndepFun f μ ↔
    ∀ (s : Finset ι) {f' : ι → Set Ω}, (∀ i ∈ s, MeasurableSet (f' i)) → μ (⋂ i ∈ s, f' i) = ∏ i ∈ s, μ (f' i)

Tactic:
simp only [iIndepFun_iff_iIndep, iIndep_iff]

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : MeasurableSet t
iht : ∀ᵐ (a : α) ∂μ, (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
a : α
ha : (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
this : tᶜ ∩ t2 = t2 \ (t ∩ t2)
⊢ (κ a) (tᶜ ∩ t2) = (κ a) tᶜ * (κ a) t2

Tactic:
this,

Goal state after:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : MeasurableSet t
iht : ∀ᵐ (a : α) ∂μ, (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
a : α
ha : (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
this : tᶜ ∩ t2 = t2 \ (t ∩ t2)
⊢ (κ a) (t2 \ (t ∩ t2)) = (κ a) tᶜ * (κ a) t2
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hf : AEMeasurable f (⇑κ ∘ₘ μ)
hg : AEMeasurable g (⇑κ ∘ₘ μ)
hφ : AEMeasurable φ (Measure.map f (⇑κ ∘ₘ μ))
hψ : AEMeasurable ψ (Measure.map g (⇑κ ∘ₘ μ))
h : IndepFun (AEMeasurable.mk φ hφ ∘ f) (AEMeasurable.mk ψ hψ ∘ g) κ μ
hφ_ae : ∀ᵐ (x : Ω) ∂⇑κ ∘ₘ μ, φ (f x) = AEMeasurable.mk φ hφ (f x)
hψ_ae : ∀ᵐ (x : Ω) ∂⇑κ ∘ₘ μ, ψ (g x) = AEMeasurable.mk ψ hψ (g x)
a : α
haψ : ∀ᵐ (ω : Ω) ∂κ a, ψ (g ω) = AEMeasurable.mk ψ hψ (g ω)
ω : Ω
hωψ : ψ (g ω) = AEMeasurable.mk ψ hψ (g ω)
⊢ (AEMeasurable.mk ψ hψ ∘ g) ω = (ψ ∘ g) ω

Tactic:
simp [hωψ]

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (μs i) F) L ≤ μ F
u : ℕ → ι
hu : Tendsto u atTop L
F : Set Ω
hF : IsClosed[inst✝²] F
⊢ limsup (fun i ↦ ((μs ∘ u) i) F) atTop ≤ limsup (fun i ↦ (μs i) F) L

Tactic:
exact (limsup_comp (fun i ↦ μs i F) u _).trans_le
    (limsup_le_limsup_of_le hu (by isBoundedDefault) ⟨1, by simp⟩)

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

Goal state before:
pp: case inr
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
⊢ Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) κ μ

Tactic:
obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
    exists_ae_eq_isMarkovKernel hs.ae_isProbabilityMeasure hμ

Goal state after:
case inr
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
⊢ Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) κ μ
Module: Mathlib.Probability.Density

Goal state before:
pp: Ω : Type u_1
E : Type u_2
inst✝⁵ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝⁴ : MeasurableSpace F
ν : Measure F
X : Ω → E
Y : Ω → F
inst✝³ : IsFiniteMeasure ℙ
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : HasPDF (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)
this✝ : HasPDF X ℙ μ
this : HasPDF Y ℙ ν
s : Set E
t : Set F
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ((μ.prod ν).withDensity fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2) (s ×ˢ t) = (map X ℙ) s * (map Y ℙ) t

Tactic:
by rw [withDensity_apply _ (hs.prod ht), ← prod_restrict,
      lintegral_prod_mul (measurable_pdf X ℙ μ).aemeasurable (measurable_pdf Y ℙ ν).aemeasurable,
      map_eq_setLIntegral_pdf X ℙ μ hs, map_eq_setLIntegral_pdf Y ℙ ν ht]

Goal state after:
Module: Mathlib.Probability.Independence.Integrable

Goal state before:
pp: Ω : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : MeasurableSpace F
f : Ω → E
g : Ω → F
p : ℝ≥0∞
hp : p ≠ 0
hp' : p ≠ ∞
hℒp : MemLp f p μ
hindep : f ⟂ᵢ[μ] g
h'f : ∀ (c : ℝ≥0), 0 < c → μ {ω | c ≤ ‖f ω‖₊} ≤ 0
A : ∀ (c : ℝ≥0), 0 < c → ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖₊ < c
⊢ ∀ᵐ (ω : Ω) ∂μ, f ω = 0

Tactic:
obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), 0 < u n)
      ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto (0 : ℝ≥0)

Goal state after:
Ω : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : MeasurableSpace F
f : Ω → E
g : Ω → F
p : ℝ≥0∞
hp : p ≠ 0
hp' : p ≠ ∞
hℒp : MemLp f p μ
hindep : f ⟂ᵢ[μ] g
h'f : ∀ (c : ℝ≥0), 0 < c → μ {ω | c ≤ ‖f ω‖₊} ≤ 0
A : ∀ (c : ℝ≥0), 0 < c → ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖₊ < c
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ ∀ᵐ (ω : Ω) ∂μ, f ω = 0
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
h : (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) ^ 2) =ᵐ[μ] 0
⊢ ∀ᵐ (ω : Ω) ∂μ, X ω = ∫ (x : Ω), X x ∂μ

Tactic:
filter_upwards [h] with ω hω

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
h : (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) ^ 2) =ᵐ[μ] 0
ω : Ω
hω : (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 = 0 ω
⊢ X ω = ∫ (x : Ω), X x ∂μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝¹⁴ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : MeasurableSpace 𝓧
inst✝¹² : MeasurableSpace 𝓨
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → E
g : 𝓨 → F
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : Integrable f (Measure.map X μ)
hg : Integrable g (Measure.map Y μ)
B : E →L[𝕜] F →L[𝕜] G
h : ∀ᵐ (ω : Ω) ∂μ, f (X ω) = 0
⊢ ∀ᵐ (ω : Ω) ∂μ, (B (f (X ω))) (g (Y ω)) = 0

Tactic:
filter_upwards [h] with ω hω

Goal state after:
Ω : Type u_1
𝕜 : Type u_2
inst✝¹⁴ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : MeasurableSpace 𝓧
inst✝¹² : MeasurableSpace 𝓨
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → E
g : 𝓨 → F
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : Integrable f (Measure.map X μ)
hg : Integrable g (Measure.map Y μ)
B : E →L[𝕜] F →L[𝕜] G
h : ∀ᵐ (ω : Ω) ∂μ, f (X ω) = 0
ω : Ω
hω : f (X ω) = 0
⊢ (B (f (X ω))) (g (Y ω)) = 0
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : 𝓧 → 𝓨
hY : HasLaw Y ν μ
hX : HasLaw X μ P
⊢ AEMeasurable Y (map X P)

Tactic:
rw [hX.map_eq]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : 𝓧 → 𝓨
hY : HasLaw Y ν μ
hX : HasLaw X μ P
⊢ AEMeasurable Y μ
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
P : Measure Ω
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
X : Ω → 𝓧
hX : HasLaw X μ P
f : 𝓧 → E
hf : AEStronglyMeasurable f μ
⊢ ∫ (x : Ω), (f ∘ X) x ∂P = ∫ (x : 𝓧), f x ∂μ

Tactic:
by
  rw [← hX.map_eq, integral_map hX.aemeasurable, Function.comp_def]
  rwa [hX.map_eq]

Goal state after:
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: case hX
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable f (map X P)

Tactic:
rw [hX.map_eq]

Goal state after:
case hX
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable f μ
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: case hY
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable g (map X P)

Tactic:
rw [hX.map_eq]

Goal state after:
case hY
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable g μ
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
P : Measure Ω
c : ℝ≥0∞
inst✝ : MeasurableSingletonClass 𝓧
x : 𝓧
hX : HasLaw X (c • dirac x) P
⊢ ∀ᵐ (y : 𝓧) ∂map X P, y = x

Tactic:
rw [hX.map_eq]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
P : Measure Ω
c : ℝ≥0∞
inst✝ : MeasurableSingletonClass 𝓧
x : 𝓧
hX : HasLaw X (c • dirac x) P
⊢ ∀ᵐ (y : 𝓧) ∂c • dirac x, y = x
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : IsProbabilityMeasure μ
ν : Measure F
inst✝ : IsProbabilityMeasure ν
L : StrongDual ℝ (E × F)
hLμ : MemLp (⇑(L ∘SL ContinuousLinearMap.inl ℝ E F)) 2 μ
hLν : MemLp (⇑(L ∘SL ContinuousLinearMap.inr ℝ E F)) 2 ν
this : ⇑L = fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1 + (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2
⊢ Var[⇑L; μ.prod ν] = Var[⇑(L ∘SL ContinuousLinearMap.inl ℝ E F); μ] + Var[⇑(L ∘SL ContinuousLinearMap.inr ℝ E F); ν]

Tactic:
this,

Goal state after:
E : Type u_3
F : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : IsProbabilityMeasure μ
ν : Measure F
inst✝ : IsProbabilityMeasure ν
L : StrongDual ℝ (E × F)
hLμ : MemLp (⇑(L ∘SL ContinuousLinearMap.inl ℝ E F)) 2 μ
hLν : MemLp (⇑(L ∘SL ContinuousLinearMap.inr ℝ E F)) 2 ν
this : ⇑L = fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1 + (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2
⊢ Var[fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1 + (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2; μ.prod ν] =
    Var[⇑(L ∘SL ContinuousLinearMap.inl ℝ E F); μ] + Var[⇑(L ∘SL ContinuousLinearMap.inr ℝ E F); ν]
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
refine ⟨⟨C, hC₁.le⟩, fun i => le_trans (le_of_eq ?_) hC₂⟩

Goal state after:
case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ eLpNorm ({x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊}.indicator (f i)) p μ = eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
this : {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} = {x | C ≤ ‖f i x‖}
⊢ eLpNorm ({x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊}.indicator (f i)) p μ = eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ

Tactic:
this,

Goal state after:
case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
this : {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} = {x | C ≤ ‖f i x‖}
⊢ eLpNorm ({x | C ≤ ‖f i x‖}.indicator (f i)) p μ = eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ