mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case compl
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
this✝ : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)
h : (κ a).real (A ×ˢ s) + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = (κ a).real (univ ×ˢ s)
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a + ((κ a) (A ×ˢ s)).toReal = ((κ a) (univ ×ˢ s)).toReal

pp: case compl
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
this✝ : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)
h : (κ a).real (A ×ˢ s) + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = (κ a).real (univ ×ˢ s)
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal ≤ ((κ a) (univ ×ˢ s)).toReal

Tactic:
add_comm

Goal state after:
case compl
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
this✝ : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)
h : (κ a).real (A ×ˢ s) + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = (κ a).real (univ ×ˢ s)
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal

case compl
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
this✝ : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)
h : (κ a).real (A ×ˢ s) + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = (κ a).real (univ ×ˢ s)
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal ≤ ((κ a) (univ ×ˢ s)).toReal
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
⊢ ⇑(condDistrib (fun ω ↦ Y ω.1) (fun ω ↦ X ω.1) (μ.prod ν)) =ᵐ[Measure.map X μ] ⇑(condDistrib Y X μ)

Tactic:
have h_map := condDistrib_map (X := X) (Y := Y) (f := Prod.fst (α := α) (β := γ))
      (ν := μ.prod ν) (mα := inferInstance) (mβ := inferInstance)
      (by simpa) (by simpa) (by fun_prop)

Goal state after:
case pos
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
h_map :
  ⇑(condDistrib Y X (Measure.map Prod.fst (μ.prod ν))) =ᵐ[Measure.map (X ∘ Prod.fst) (μ.prod ν)]
    ⇑(condDistrib (Y ∘ Prod.fst) (X ∘ Prod.fst) (μ.prod ν))
⊢ ⇑(condDistrib (fun ω ↦ Y ω.1) (fun ω ↦ X ω.1) (μ.prod ν)) =ᵐ[Measure.map X μ] ⇑(condDistrib Y X μ)
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
h_map :
  ⇑(condDistrib Y X (Measure.map Prod.fst (μ.prod ν))) =ᵐ[Measure.map (X ∘ Prod.fst) (μ.prod ν)]
    ⇑(condDistrib (Y ∘ Prod.fst) (X ∘ Prod.fst) (μ.prod ν))
⊢ AEMeasurable X (Measure.map Prod.fst (μ.prod ν))

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
⊢ AEMeasurable X (Measure.map Prod.snd (ν.prod μ))

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
⊢ AEMeasurable Y (Measure.map Prod.snd (ν.prod μ))

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
X : α → β
hY : AEMeasurable Y μ
ν : Measure γ
inst✝ : IsProbabilityMeasure ν
hX : AEMeasurable X μ
h_map :
  ⇑(condDistrib Y X (Measure.map Prod.snd (ν.prod μ))) =ᵐ[Measure.map (X ∘ Prod.snd) (ν.prod μ)]
    ⇑(condDistrib (Y ∘ Prod.snd) (X ∘ Prod.snd) (ν.prod μ))
⊢ AEMeasurable X (Measure.map Prod.snd (ν.prod μ))

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
⊢ IicProdIoc a b ⁻¹' Set.univ.pi s = Set.univ.pi (frestrictLe₂ hab s) ×ˢ Set.univ.pi (restrict₂ ⋯ s)

Tactic:
by
  ext x
  simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, IicProdIoc_def, forall_const,
    Subtype.forall, mem_Iic, Set.mem_prod, frestrictLe₂_apply, restrict₂, mem_Ioc]
  refine ⟨fun h ↦ ⟨fun i hi ↦ ?_, fun i ⟨hi1, hi2⟩ ↦ ?_⟩, fun ⟨h1, h2⟩ i hi ↦ ?_⟩
  · convert! h i (hi.trans hab)
    rw [dif_pos hi]
  · convert! h i hi2
    rw [dif_neg (not_le.2 hi1)]
  · split_ifs with hi3
    · exact h1 i hi3
    · exact h2 i ⟨not_le.1 hi3, hi⟩

Goal state after:
Module: Mathlib.Probability.Distributions.SetBernoulli

Goal state before:
pp: ι : Type u_1
u : Set ι
inst✝ : Countable ι
s : Set ι
p : ↑I
hs : ¬s ⊆ u
⊢ {s} ⊆ {x | (fun s ↦ s ⊆ u) x}ᶜ

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.CondVar

Goal state before:
pp: case inl
Ω : Type u_1
m₀ : MeasurableSpace Ω
X : Ω → ℝ
⊢ Var[X; 0 | ⊥] =ᶠ[⊥] fun x ↦ ⨍ (ω : Ω), (X ω - ⨍ (ω' : Ω), X ω' ∂0) ^ 2 ∂0

Tactic:
exact eventually_bot

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case inl.e_a
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
ms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)
x : ℕ
hx : x ∈ Iic a
⊢ (μ ↑⟨x, ⋯⟩) (s ⟨x, ⋯⟩) = (μ ↑⟨x, hx⟩) (frestrictLe₂ hab s ⟨x, hx⟩)

Tactic:
frestrictLe₂,

Goal state after:
case inl.e_a
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
ms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)
x : ℕ
hx : x ∈ Iic a
⊢ (μ ↑⟨x, ⋯⟩) (s ⟨x, ⋯⟩) = (μ ↑⟨x, hx⟩) (restrict₂ ⋯ s ⟨x, hx⟩)
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
a✝¹ : α
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ ((κ.withDensity 1) a✝¹) s✝ = (κ a✝¹) s✝

Tactic:
Kernel.withDensity_apply _ measurable_const

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
a✝¹ : α
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ ((κ a✝¹).withDensity (1 a✝¹)) s✝ = (κ a✝¹) s✝
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
a✝¹ : α
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ ((κ.withDensity 0) a✝¹) s✝ = (0 a✝¹) s✝

Tactic:
Kernel.withDensity_apply _ measurable_const

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
a✝¹ : α
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ ((κ a✝¹).withDensity (0 a✝¹)) s✝ = (0 a✝¹) s✝
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n
a : α
b : β
n : ℕ
hn : ⌈(f a b).toReal⌉₊ ≤ n
this : f a b ≤ ENNReal.ofReal ↑n
⊢ f a b ≤ ↑n

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ (κ a) (s \ κ.mutuallySingularSetSlice η a) + (κ a) (s ∩ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \ κ.mutuallySingularSetSlice η a)

Tactic:
add_comm

Goal state after:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
hm : MeasurableSet (κ.mutuallySingularSetSlice η a)
⊢ (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \ κ.mutuallySingularSetSlice η a) =
    (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \ κ.mutuallySingularSetSlice η a)
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (η.withDensity (κ.rnDeriv η)) a + (κ.singularPart η) a ≪ η a
⊢ (κ.singularPart η) a = 0

Tactic:
rw [Measure.AbsolutelyContinuous.add_left_iff] at h

Goal state after:
case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (η.withDensity (κ.rnDeriv η)) a ≪ η a ∧ (κ.singularPart η) a ≪ η a
⊢ (κ.singularPart η) a = 0
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η ξ : Kernel α γ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel η
h : κ = η.withDensity f + ξ
hf : Measurable (Function.uncurry f)
a : α
hξ : ξ a ⟂ₘ η a
⊢ (η a).withDensity (f a) + ξ a = ξ a + (η a).withDensity (f a)

Tactic:
add_comm

Goal state after:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η ξ : Kernel α γ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel η
h : κ = η.withDensity f + ξ
hf : Measurable (Function.uncurry f)
a : α
hξ : ξ a ⟂ₘ η a
⊢ ξ a + (η a).withDensity (f a) = ξ a + (η a).withDensity (f a)
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
rw [compProd_of_not_sfinite _ _ hν]; simp

Goal state after:
Module: Mathlib.Probability.Kernel.Deterministic

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = (κ a) (s ∩ s)
⊢ (κ a) s = 0 ∨ (κ a) s = 1

pp: case mp.hs
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = ((copy β ∘ₖ κ) a) (s ×ˢ s)
⊢ MeasurableSet s

pp: case mp.ht
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = ((copy β ∘ₖ κ) a) (s ×ˢ s)
⊢ MeasurableSet s

Tactic:
inter_self

Goal state after:
case mp
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = (κ a) s
⊢ (κ a) s = 0 ∨ (κ a) s = 1

case mp.hs
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = ((copy β ∘ₖ κ) a) (s ×ˢ s)
⊢ MeasurableSet s

case mp.ht
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
h : IsDeterministic κ
a : α
s : Set β
hs : MeasurableSet s
this : (κ a) s * (κ a) s = ((copy β ∘ₖ κ) a) (s ×ˢ s)
⊢ MeasurableSet s
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteKernel κ
inst✝¹ : StandardBorelSpace Ω
inst✝ : Nonempty Ω
⊢ ⇑(Kernel.swap 𝓧 Ω ∘ₖ (Kernel.id ×ₖ κ†μ)) ∘ₘ ⇑κ ∘ₘ μ = μ ⊗ₘ κ

Tactic:
← Measure.comp_assoc,

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteKernel κ
inst✝¹ : StandardBorelSpace Ω
inst✝ : Nonempty Ω
⊢ ⇑(Kernel.swap 𝓧 Ω) ∘ₘ ⇑(Kernel.id ×ₖ κ†μ) ∘ₘ ⇑κ ∘ₘ μ = μ ⊗ₘ κ
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.copy 𝓧 ∘ₖ Kernel.deterministic f hf) ∘ₘ μ = ⇑(Kernel.copy 𝓧) ∘ₘ Measure.map f μ

Tactic:
← Measure.comp_assoc,

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.copy 𝓧) ∘ₘ ⇑(Kernel.deterministic f hf) ∘ₘ μ = ⇑(Kernel.copy 𝓧) ∘ₘ Measure.map f μ
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
⊢ ∀ᵐ (a : Ω) ∂μ, κ a ≪ (Kernel.const Ω ν) a

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧
h_ac : ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) b ≪ μ
⊢ ∀ᵐ (a : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) a ≪ (Kernel.const 𝓧 μ) a

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case mt
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Fintype ι
s : (i : ι) → Set (X i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ∀ i ∈ univ, MeasurableSet (s i)

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case mt
ι : Type u_3
κ : ι → Type u_4
X : (i : ι) → κ i → Type u_5
mX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)
μ : (i : ι) → (j : κ i) → Measure (X i j)
hμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)
s : Finset ((i : ι) × κ i)
t : (i : (i : ι) × κ i) → Set (X i.fst i.snd)
ht : ∀ (i : (i : ι) × κ i), MeasurableSet (t i)
⊢ ∀ i ∈ image Sigma.fst s, MeasurableSet ((↑(s.preimage (Sigma.mk i) ⋯)).pi fun j ↦ t ⟨i, j⟩)

Tactic:
· simp only [mem_image, Sigma.exists, exists_and_right, exists_eq_right, forall_exists_index]
    exact fun i j hij ↦ MeasurableSet.pi (countable_toSet _) fun k hk ↦ by simp_all

Goal state after:
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: case hu
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 < (↑α).re

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: case hv
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 < (↑β).re

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 < (↑α).re

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 < (↑β).re

Tactic:
simpa

Goal state after:
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: case inr
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
this :
  ∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {m₀ : MeasurableSpace Ω} (hm : m ≤ m₀) {μ : Measure Ω}
    [inst : IsFiniteMeasure μ] (hXmeas : AEMeasurable X μ) (hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1),
    Measurable X → Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m]
hXmeas : ¬Measurable X
Y : Ω → ℝ
hYmeas : Measurable Y
hXY : X =ᵐ[μ] Y
⊢ Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m]

Tactic:
calc
      Var[X; μ | m]
      _ =ᵐ[μ] Var[Y; μ | m] := condVar_congr_ae hXY
      _ =ᵐ[μ] μ[Y | m] * μ[1 - Y | m] := by
        refine this hm hYmeas.aemeasurable ?_ hYmeas
        filter_upwards [hX, hXY] with ω hXω hXYω
        simp [hXω, ← hXYω]
      _ =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by
        refine .mul ?_ ?_ <;>
          exact condExp_congr_ae <| by filter_upwards [hXY] with ω hω; simp [hω]

Goal state after:
Module: Mathlib.Probability.Kernel.Condexp

Goal state before:
pp: case neg.refine_2.inr.inl
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
t : Set Ω
ht : MeasurableSet t
h : t = s
⊢ ∫ (x : Ω) in s, f x ∂μ = ∫ (x : Ω) in s ∩ s, f x ∂μ

pp: case neg.refine_2.inr.inl.ha
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
t : Set Ω
ht : MeasurableSet t
h : t = s
⊢ μ.real s ≠ 0

Tactic:
inter_self

Goal state after:
case neg.refine_2.inr.inl
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
t : Set Ω
ht : MeasurableSet t
h : t = s
⊢ ∫ (x : Ω) in s, f x ∂μ = ∫ (x : Ω) in s, f x ∂μ

case neg.refine_2.inr.inl.ha
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
t : Set Ω
ht : MeasurableSet t
h : t = s
⊢ μ.real s ≠ 0
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
this : ⨆ a, ⨆ n ∈ ns a, s n = ⨆ n, s n
⊢ Indep (⨆ n, s n) (limsup s f) κ μα

Tactic:
← this

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
this : ⨆ a, ⨆ n ∈ ns a, s n = ⨆ n, s n
⊢ Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ Indep (limsup s atTop) (limsup s atTop) κ μα

Tactic:
refine indep_limsup_self h_le h_indep ?_ ?_ hnsp ?_

Goal state after:
case refine_1
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ ∀ (t : Set ι), BddAbove t → tᶜ ∈ atTop

case refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) ns

case refine_3
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ ∀ (n : ι), ∃ a, n ∈ ns a