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

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
h_int_X : Integrable (fun x ↦ F (X n x)) μ''
h_int_sub : Integrable (fun a ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ ∫ (a : Ω'') in {x | ‖Y n x - X n x‖ < ε / 2}, ‖F (Y n a) - F (X n a)‖ ∂μ'' +
      ∫ (a : Ω'') in {x | ε / 2 ≤ ‖Y n x - X n x‖}, ‖F (Y n a) - F (X n a)‖ ∂μ'' ≤
    ∫ (a : Ω'') in {x | ‖Y n x - X n x‖ < ε / 2}, ↑L * (ε / 2) ∂μ'' +
      ∫ (a : Ω'') in {x | ε / 2 ≤ ‖Y n x - X n x‖}, M ∂μ''

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
h_int_X : Integrable (fun x ↦ F (X n x)) μ''
h_int_sub : Integrable (fun a ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ ∫ (a : Ω'') in {x | ‖Y n x - X n x‖ < ε / 2}, ↑L * (ε / 2) ∂μ'' +
      ∫ (a : Ω'') in {x | ε / 2 ≤ ‖Y n x - X n x‖}, M ∂μ'' =
    ↑L * (ε / 2) * μ''.real {x | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖}

Tactic:
by

Goal state after:
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
u : Finset ℕ
⊢ ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
i : ι
hi : i ∈ s
A : Set α
hA : MeasurableSet A
hA' : X i ⁻¹' A = f i
⊢ (fun ω ↦ (X i ω, Y i ω)) ⁻¹' A ×ˢ univ = f i

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this : TopologicalSpace ι := Preorder.topology ι
i j : ι
hij : i ≤ j
⊢ (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i) i ≤
    (fun i ↦ if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i) j

Tactic:
by

Goal state after:
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
⊢ ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this : TopologicalSpace ι := Preorder.topology ι
i : ι
⊢ (if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i) ≤ m

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
h_int_X : Integrable (fun x ↦ F (X n x)) μ''
h_int_sub : Integrable (fun a ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ ↑L * (ε / 2) * μ''.real {x | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} ≤
    ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝² : PartialOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
𝓕 : Filtration ι m
i : ι
⊢ ↑𝓕₊ i = if (𝓝[>] i).NeBot then ⨅ j, ⨅ (_ : j > i), ↑𝓕 j else ↑𝓕 i

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
⊢ |∫ (ω : E), F ω ∂Measure.map (Y n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'| ≤
    ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} +
      |∫ (ω : E), F ω ∂Measure.map (X n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'|

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
i : ι
⊢ MeasurableSet (g i)

Tactic:
by

Goal state after:
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
n : ℕ
⊢ Measurable (f n)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝² : PartialOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
𝓕 : Filtration ι m
i : ι
hi : 𝓝[>] i = ⊥
⊢ ↑𝓕₊ i = ↑𝓕 i

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |∫ (ω : E), F ω ∂Measure.map (Y n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'| ≤
      ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |∫ (ω : E), F ω ∂Measure.map (X n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'|
this :
  Tendsto
    (fun n ↦
      ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |∫ (ω : E), F ω ∂Measure.map (X n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'|)
    l (𝓝 (↑L * ε / 2 + M * 0 + 0))
⊢ Tendsto
    (fun n ↦
      ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |∫ (ω : E), F ω ∂Measure.map (X n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'|)
    l (𝓝 (↑L * ε / 2))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : SuccOrder ι
𝓕 : Filtration ι m
⊢ 𝓕₊ = 𝓕

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
i : ι
hi : IsMax i
⊢ ↑𝓕₊ i = ↑𝓕 i

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
𝓕 : Filtration ι m
i : ι
hi : ∃ j > i, Set.Ioo i j = ∅
⊢ ↑𝓕₊ i = ↑𝓕 i

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝³ : PartialOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
𝓕 : Filtration ι m
i : ι
inst✝ : (𝓝[>] i).NeBot
⊢ ↑𝓕₊ i = ⨅ j, ⨅ (_ : j > i), ↑𝓕 j

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : DenselyOrdered ι
𝓕 : Filtration ι m
i : ι
hi : ¬IsMax i
⊢ ↑𝓕₊ i = ⨅ j, ⨅ (_ : j > i), ↑𝓕 j

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
⊢ 𝓕 ≤ 𝓕₊

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
u : ι
hu : u > i
⊢ Set.Ioo i u ∈ 𝓝[>] i

Tactic:
by

Goal state after:
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)
n : ℕ
⊢ f n ≤ 1

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
u : ι
hu : u > i
hiou : Set.Ioo i u ∈ 𝓝[>] i
v : ι
hv : v ∈ Set.Ioo i u
hle₁ : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ↑𝓕₊ v
⊢ ↑𝓕₊ v ≤ ↑𝓕 u

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
⊢ ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
⊢ 𝓕₊₊ = 𝓕₊

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝³ : (i : ι) → TopologicalSpace (β i)
inst✝² : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝¹ : ∀ (i : ι), BorelSpace (β i)
inst✝ : Preorder ι
u : (i : ι) → Ω → β i
hum : ∀ (i : ι), StronglyMeasurable (u i)
i : ι
⊢ ⨆ j, ⨆ (_ : j ≤ i), MeasurableSpace.comap (u j) (mβ j) ≤ m

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝³ : (i : ι) → TopologicalSpace (β i)
inst✝² : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝¹ : ∀ (i : ι), BorelSpace (β i)
inst✝ : Preorder ι
u : (i : ι) → Ω → β i
hum : ∀ (i : ι), StronglyMeasurable (u i)
i : ι
⊢ ↑(natural u hum) i = MeasurableSpace.comap (fun ω j ↦ u (↑j) ω) inferInstance

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
⊢ (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ((κ a) (⋂ i, Y i ⁻¹' t i))⁻¹ * (κ a) ((⋂ i, Y i ⁻¹' t i) ∩ ⋂ i ∈ s, f i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
⊢ (⋂ i, Y i ⁻¹' t i) ∩ ⋂ i ∈ s, f i = (⋂ i, Y i ⁻¹' t i) ∩ ⋂ i, if i ∈ s then f i else univ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |∫ (ω : E), F ω ∂Measure.map (Y n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'| ≤
      ↑L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |∫ (ω : E), F ω ∂Measure.map (X n) μ'' - ∫ (ω : E), F ω ∂Measure.map Z μ'|
hXY : ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ''.real {x | ε ≤ ‖(Y - X) i x‖}) l (𝓝 0)
⊢ 0 < ε / 2

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t : Set Ω
n : ι
ht : MeasurableSet t
this : MeasurableSpace.generateFrom {t | n ≤ i ∧ MeasurableSet t} ≤ MeasurableSpace.generateFrom {t | ∃ j ≤ i, s j = t}
⊢ MeasurableSet t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
⊢ ((⋂ i, Y i ⁻¹' t i) ∩ ⋂ i, if i ∈ s then f i else univ) = ⋂ i, Y i ⁻¹' t i ∩ if i ∈ s then f i else univ

Tactic:
by

Goal state after:
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
⊢ ↑(n + 1) ≠ 0 ∨ ∑ x ∈ Finset.range (n + 1), μ (s x) ≠ 0

Tactic:
by

Goal state after:
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
⊢ ↑(n + 1) ≠ ∞ ∨ ∑ x ∈ Finset.range (n + 1), μ (s x) ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
i : ι
⊢ (Y i ⁻¹' t i ∩ if i ∈ s then f i else univ) = g i

Tactic:
by

Goal state after: