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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ ∃ fseq,
    (∀ (n : ℕ) (x : X), (fseq n) x ≤ 1) ∧
      (∀ (n : ℕ), ∀ x ∈ F, 1 ≤ (fseq n) x) ∧ Tendsto (fun n x ↦ (fseq n) x) atTop (𝓝 (F.indicator fun x ↦ 1))

Tactic:
use fun n ↦ thickenedIndicator (δ := (1 : ℝ) / (n + 1)) Nat.one_div_pos_of_nat F

Goal state after:
case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := ⋯
F : Set X
hF : IsClosed[inst✝¹] F
⊢ (∀ (n : ℕ) (x : X), ((fun n ↦ thickenedIndicator ⋯ F) n) x ≤ 1) ∧
    (∀ (n : ℕ), ∀ x ∈ F, 1 ≤ ((fun n ↦ thickenedIndicator ⋯ F) n) x) ∧
      Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))
Module: Mathlib.Probability.Independence.Basic

Goal state before:

Tactic:
_ = (⋂ i, Y i ⁻¹' t i) ∩ ⋂ i, if i ∈ s then f i else .univ := by
          simp only [Set.iInter_ite, Set.iInter_univ, Set.inter_univ]
        _ = ⋂ i, Y i ⁻¹' t i ∩ (if i ∈ s then f i else .univ) := by rw [Set.iInter_inter_distrib]
        _ = _ := Set.iInter_congr fun i ↦ by by_cases hi : i ∈ s <;> simp [hi, g]

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

Goal state before:
pp: case inr
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : SecondCountableTopology Ω
inst✝¹ : OpensMeasurableSpace Ω
S : Set (Set Ω)
hS : IsPiSystem S
μ : ι → ProbabilityMeasure Ω
ν : ProbabilityMeasure Ω
l : Filter ι
inst✝ : l.IsCountablyGenerated
hmeas : ∀ s ∈ S, MeasurableSet s
h : ∀ (u : Set Ω), IsOpen[inst✝³] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
h' : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))
hl : l.NeBot
G : Set Ω
hG : IsOpen[inst✝³] G
r : ℝ≥0
hr : r < ν G
T : Finset (Set Ω)
TS : ∀ t ∈ T, t ∈ S
T_meas : r < ν (⋃ t ∈ T, t)
TG : ⋃ t ∈ T, t ⊆ G
this : Tendsto (fun i ↦ (μ i) (⋃ t ∈ T, t)) l (𝓝 (ν (⋃ t ∈ T, t)))
⊢ ∀ᶠ (a : ι) in l, r < (μ a) G

Tactic:
filter_upwards [(tendsto_order.1 this).1 r T_meas] with i hi

Goal state after:
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : SecondCountableTopology Ω
inst✝¹ : OpensMeasurableSpace Ω
S : Set (Set Ω)
hS : IsPiSystem S
μ : ι → ProbabilityMeasure Ω
ν : ProbabilityMeasure Ω
l : Filter ι
inst✝ : l.IsCountablyGenerated
hmeas : ∀ s ∈ S, MeasurableSet s
h : ∀ (u : Set Ω), IsOpen[inst✝³] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
h' : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))
hl : l.NeBot
G : Set Ω
hG : IsOpen[inst✝³] G
r : ℝ≥0
hr : r < ν G
T : Finset (Set Ω)
TS : ∀ t ∈ T, t ∈ S
T_meas : r < ν (⋃ t ∈ T, t)
TG : ⋃ t ∈ T, t ⊆ G
this : Tendsto (fun i ↦ (μ i) (⋃ t ∈ T, t)) l (𝓝 (ν (⋃ t ∈ T, t)))
i : ι
hi : r < (μ i) (⋃ t ∈ T, t)
⊢ r < (μ i) G
Module: Mathlib.Probability.Independence.Integration

Goal state before:

Tactic:
_ ≤ C * ∫⁻ ω, ‖X ω‖ₑ * ‖Y ω‖ₑ ∂μ := by
    rw [← lintegral_const_mul'' _ (by fun_prop)]
    gcongr with ω
    simp [← mul_assoc, hB]
  _ = C * ((∫⁻ ω, ‖X ω‖ₑ ∂μ) * (∫⁻ ω, ‖Y ω‖ₑ ∂μ)) := by
    rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.1.enorm hY.1.enorm
        (hXY.comp measurable_enorm measurable_enorm)]
  _ < ∞ := mul_lt_top (by finiteness) (mul_lt_top hX.2 hY.2)

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
𝕜 : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖(B x) y‖ₑ ≤ ↑‖B‖₊ * ‖x‖ₑ * ‖y‖ₑ

Tactic:
rw [← toReal_le_toReal (by finiteness) (by finiteness)]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
𝕜 : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖(B x) y‖ₑ.toReal ≤ (↑‖B‖₊ * ‖x‖ₑ * ‖y‖ₑ).toReal
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
⊢ Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, ∑ j ∈ s, cov[X i, X j; μ]

Tactic:
by
  rw [← covariance_self, covariance_sum_left' (by simpa)]
  · refine Finset.sum_congr rfl fun i hi ↦ ?_
    rw [covariance_sum_right' (by simpa) (hX i hi)]
  · exact memLp_finsetSum' _ (by simpa)
  · exact (memLp_finsetSum' _ (by simpa)).aemeasurable

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

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : AEStronglyMeasurable X μ → ¬eLpNorm X 2 μ < ∞
⊢ ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ = ∞

Tactic:
specialize hℒ hX

Goal state after:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : ¬eLpNorm X 2 μ < ∞
⊢ ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ = ∞
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
⊢ μ {ω | ↑c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} ≤ eVar[X; μ] / ↑c ^ 2

Tactic:
have B : AEStronglyMeasurable (fun _ : Ω => μ[X]) μ := aestronglyMeasurable_const

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ μ {ω | ↑c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} ≤ eVar[X; μ] / ↑c ^ 2
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case e_a
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_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⟩
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = univ
sets_t' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ T then sets_t ⟨i, hi⟩ else univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = univ
h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i)
h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i ↦ f (↑i) ω) ⁻¹' univ.pi sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i ↦ f (↑i) ω) ⁻¹' univ.pi sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (η a) (f i ⁻¹' sets i)
h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)
h_meas_inter : ∀ i ∈ S ∪ T, MeasurableSet (sets_s' i ∩ sets_t' i)
a : α
h_indepS : (η a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i ∈ S, (η a) (f i ⁻¹' sets_s' i)
h_indepT : (η a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i ∈ T, (η a) (f i ⁻¹' sets_t' i)
h_indepST : (η a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i ∈ S ∪ T, (η a) (f i ⁻¹' (sets_s' i ∩ sets_t' i))
⊢ ∏ x ∈ T, (η a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i ∈ T, (η a) (f i ⁻¹' sets_t' i)

Tactic:
refine Finset.prod_congr rfl fun i hi => ?_
    rw [h_sets_s'_univ hi, Set.univ_inter]

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

Goal state before:
pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ ∫ (ω : Ω), id (A.indicator 1 ω) • f (X ω) ∂P = P.real A • ∫ (ω : Ω), f (X ω) ∂P

Tactic:
rw [IndepFun.integral_fun_comp_smul_comp _ _ hX (by fun_prop) hf]

Goal state after:
Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ (∫ (ω : Ω), id (A.indicator 1 ω) ∂P) • ∫ (ω : Ω), f (X ω) ∂P = P.real A • ∫ (ω : Ω), f (X ω) ∂P

Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ A.indicator 1 ⟂ᵢ[P] X

Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P
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
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h : IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) (AEMeasurable.mk (f k) ⋯) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂κ a, f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂κ a, f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
⊢ f j ω = AEMeasurable.mk (f j) ⋯ ω → (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)

Tactic:
hωj

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
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h : IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) (AEMeasurable.mk (f k) ⋯) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂κ a, f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂κ a, f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
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
hf_indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun a ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂κ a, f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂κ a, f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
⊢ f j ω = AEMeasurable.mk (f j) ⋯ ω → (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)

Tactic:
hωj

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
hf_indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
h :
  IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a))
    (fun a ↦ (AEMeasurable.mk (f k) ⋯ a, AEMeasurable.mk (f l) ⋯ a)) κ μ
a : α
hi : ∀ᵐ (ω : Ω) ∂κ a, f i ω = AEMeasurable.mk (f i) ⋯ ω
hj : ∀ᵐ (ω : Ω) ∂κ a, f j ω = AEMeasurable.mk (f j) ⋯ ω
ω : Ω
hωi : f i ω = AEMeasurable.mk (f i) ⋯ ω
hωj : f j ω = AEMeasurable.mk (f j) ⋯ ω
⊢ (AEMeasurable.mk (f i) ⋯ ω, AEMeasurable.mk (f j) ⋯ ω) = (f i ω, f j ω)
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: case neg
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : Preorder ι
x✝ : Set (Filtration ι m)
hn : ¬x✝.Nonempty
⊢ IsGLB x✝ { seq := fun i ↦ m, mono' := ⋯, le' := ⋯ }

Tactic:
rw [Set.not_nonempty_iff_eq_empty] at hn

Goal state after:
case neg
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : Preorder ι
x✝ : Set (Filtration ι m)
hn✝ : ¬x✝.Nonempty
hn : x✝ = ∅
⊢ IsGLB x✝ { seq := fun i ↦ m, mono' := ⋯, le' := ⋯ }
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
n : ℕ
⊢ r ≤ ∫⁻ (a : α), (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a ∂μ

Tactic:
rw [lintegral_const_mul _ <| Finset.measurable_fun_sum _
        fun _ _ ↦ measurable_one.indicator <| hs _,
      lintegral_finsetSum _ fun _ _ ↦ measurable_one.indicator (hs _)]

Goal state after:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
n : ℕ
⊢ r ≤ (↑(n + 1))⁻¹ * ∑ b ∈ Finset.range (n + 1), ∫⁻ (a : α), (s b).indicator 1 a ∂μ
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case refine_1
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInDistribution (fun n ω ↦ (X n ω, c)) l (fun ω ↦ (Z ω, c)) (fun x ↦ μ'') μ'

pp: case refine_2
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInMeasure μ'' ((fun n ω ↦ (X n ω, Y n ω)) - fun n ω ↦ (X n ω, c)) l 0

Tactic:
· exact hXZ.continuous_comp (g := fun x ↦ (x, c)) (by fun_prop)

Goal state after:
case refine_2
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInMeasure μ'' ((fun n ω ↦ (X n ω, Y n ω)) - fun n ω ↦ (X n ω, c)) l 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Count

Goal state before:

Tactic:
‖f i‖ₑ = eLpNorm f p (dirac i) := by rw [eLpNorm_dirac f i hp]
      _ = eLpNorm f p (count.restrict {i}) := by simp
      _ ≤ eLpNorm f p count := eLpNorm_restrict_le ..

Goal state after:
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
R : ℝ≥0
p : ℝ≥0∞
F : Type u_5
inst✝ : NormedAddCommGroup F
ℱ : Filtration ℕ m
f : ℕ → Ω → F
hfm : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
h : ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ f n ω) atTop (𝓝 (g ω))
⊢ MemLp (Classical.choose h) p μ

pp: case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
R : ℝ≥0
p : ℝ≥0∞
F : Type u_5
inst✝ : NormedAddCommGroup F
ℱ : Filtration ℕ m
f : ℕ → Ω → F
hfm : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
h : ¬∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ f n ω) atTop (𝓝 (g ω))
⊢ MemLp 0 p μ

Tactic:
· refine ⟨StronglyMeasurable.aestronglyMeasurable
      ((Classical.choose_spec h).1.mono (sSup_le fun m ⟨n, hn⟩ => hn ▸ ℱ.le _)),
      lt_of_le_of_lt (Lp.eLpNorm_lim_le_liminf_eLpNorm hfm _ (Classical.choose_spec h).2)
        (lt_of_le_of_lt ?_ (ENNReal.coe_lt_top : ↑R < ∞))⟩
    simp_rw [liminf_eq, eventually_atTop]
    exact sSup_le fun b ⟨a, ha⟩ => (ha a le_rfl).trans (hbdd _)

Goal state after:
case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
R : ℝ≥0
p : ℝ≥0∞
F : Type u_5
inst✝ : NormedAddCommGroup F
ℱ : Filtration ℕ m
f : ℕ → Ω → F
hfm : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
h : ¬∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ f n ω) atTop (𝓝 (g ω))
⊢ MemLp 0 p μ
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hg : UnifTight g p μ
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifTight (f - g) p μ

Tactic:
rw [sub_eq_add_neg]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hg : UnifTight g p μ
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifTight (f + -g) p μ
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
f g : α →C_c β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by
    obtain ⟨⟨_, _⟩, _⟩ := f
    obtain ⟨⟨_, _⟩, _⟩ := g
    congr

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ✝ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
f : α →C_c β
⊢ C(α, γ)

Tactic:
by classical exact if g 0 = 0 then g.comp f else 0

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-b * ‖x‖ ^ p) = 2 * π / p * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p)

Tactic:
calc
    _ = ∫ x in Ioi (0 : ℝ) ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p)) := by
      rw [← Complex.integral_comp_polarCoord_symm, polarCoord_target]
      simp_rw [Complex.norm_polarCoord_symm, smul_eq_mul]
    _ = (∫ x in Ioi (0 : ℝ), x * |x| ^ q * rexp (-b * |x| ^ p)) * ∫ _ in Ioo (-π) π, 1 := by
      rw [← setIntegral_prod_mul, volume_eq_prod]
      simp_rw [mul_one]
      congr! 2; ring
    _ = 2 * π * ∫ x in Ioi (0 : ℝ), x * |x| ^ q * rexp (-b * |x| ^ p) := by
      simp_rw [integral_const, measureReal_restrict_apply MeasurableSet.univ, Set.univ_inter,
        volume_real_Ioo_of_le (a := -π) (b := π) (by linarith [pi_nonneg]),
        sub_neg_eq_add, ← two_mul, smul_eq_mul, mul_one, mul_comm]
    _ = 2 * π * ∫ x in Ioi (0 : ℝ), x ^ (q + 1) * rexp (-b * x ^ p) := by
      congr 1
      refine setIntegral_congr_fun measurableSet_Ioi (fun x hx => ?_)
      rw [mem_Ioi] at hx
      rw [abs_eq_self.mpr hx.le, rpow_add hx, rpow_one]
      ring
    _ = (2 * π / p) * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p) := by
      rw [_root_.integral_rpow_mul_exp_neg_mul_rpow (by linarith) (by linarith) hb, add_assoc,
        one_add_one_eq_two]
      ring

Goal state after: