mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Process.Basic

Goal state before:
pp: S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X : (i : S) → (j : T i) → Ω → 𝓧 i j
hX : ∀ (i : S) (j : T i), Measurable (X i j)
h : ∀ (I : Finset S) (J : (i : ↥I) → Finset (T ↑i)), iIndepFun (fun i ω j ↦ X (↑i) (↑j) ω) κ P
hμ : P ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[P] ⇑η
hη✝ : IsMarkovKernel η
π : S → Set (Set Ω) := fun i ↦ {s | ∃ t ∈ squareCylinders fun j ↦ {s | MeasurableSet s}, (fun ω j ↦ X i j ω) ⁻¹' t = s}
π_pi : ∀ (i : S), IsPiSystem (π i)
π_gen : ∀ (i : S), MeasurableSpace.comap (fun ω j ↦ X i j ω) MeasurableSpace.pi = generateFrom (π i)
I : Finset S
s : S → Set Ω
J : (i : S) → Finset (T i)
t : (i : S) → (i_1 : T i) → Set (𝓧 i i_1)
ht : ∀ i ∈ I, ∀ (i_1 : T i), MeasurableSet (t i i_1)
hs : ∀ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) = s i
this :
  ⋂ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) =
    ⋂ i ∈ Finset.univ, (fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j
h' : ∀ i ∈ Finset.univ, MeasurableSet ((↑Finset.univ).pi fun j ↦ t ↑i ↑j)
ω : α
hω :
  (κ ω) (⋂ i ∈ Finset.univ, (fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j) =
    ∏ i, (κ ω) ((fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j)
hη : κ ω = η ω
x✝¹ : ↥I
x✝ : Ω
⊢ (x✝ ∈ (fun ω j ↦ X (↑x✝¹) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑x✝¹ ↑j) ↔
    x✝ ∈ (fun ω j ↦ X (↑x✝¹) j ω) ⁻¹' (↑(J ↑x✝¹)).pi (t ↑x✝¹)

Tactic:
;

Goal state after:
S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X : (i : S) → (j : T i) → Ω → 𝓧 i j
hX : ∀ (i : S) (j : T i), Measurable (X i j)
h : ∀ (I : Finset S) (J : (i : ↥I) → Finset (T ↑i)), iIndepFun (fun i ω j ↦ X (↑i) (↑j) ω) κ P
hμ : P ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[P] ⇑η
hη✝ : IsMarkovKernel η
π : S → Set (Set Ω) := fun i ↦ {s | ∃ t ∈ squareCylinders fun j ↦ {s | MeasurableSet s}, (fun ω j ↦ X i j ω) ⁻¹' t = s}
π_pi : ∀ (i : S), IsPiSystem (π i)
π_gen : ∀ (i : S), MeasurableSpace.comap (fun ω j ↦ X i j ω) MeasurableSpace.pi = generateFrom (π i)
I : Finset S
s : S → Set Ω
J : (i : S) → Finset (T i)
t : (i : S) → (i_1 : T i) → Set (𝓧 i i_1)
ht : ∀ i ∈ I, ∀ (i_1 : T i), MeasurableSet (t i i_1)
hs : ∀ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) = s i
this :
  ⋂ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) =
    ⋂ i ∈ Finset.univ, (fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j
h' : ∀ i ∈ Finset.univ, MeasurableSet ((↑Finset.univ).pi fun j ↦ t ↑i ↑j)
ω : α
hω :
  (κ ω) (⋂ i ∈ Finset.univ, (fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j) =
    ∏ i, (κ ω) ((fun ω j ↦ X (↑i) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑i ↑j)
hη : κ ω = η ω
x✝¹ : ↥I
x✝ : Ω
⊢ (x✝ ∈ (fun ω j ↦ X (↑x✝¹) (↑j) ω) ⁻¹' (↑Finset.univ).pi fun j ↦ t ↑x✝¹ ↑j) ↔
    x✝ ∈ (fun ω j ↦ X (↑x✝¹) j ω) ⁻¹' (↑(J ↑x✝¹)).pi (t ↑x✝¹)
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case pos.refine_1.hc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ ↑⟨q, htq⟩ < 0

Tactic:
rwa [Subtype.coe_mk]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
a : α
t : ℝ
⊢ stieltjesFunctionAux f a t = ⨅ r, f a ↑r

Tactic:
;

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
a : α
t : ℝ
⊢ stieltjesFunctionAux f a t = ⨅ r, f a ↑r
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
r : { q' // y < ↑q' }
z : ℝ
⊢ (z ∈ range fun q ↦ f a ↑q) → 0 ≤ z

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
r : { q' // y < ↑q' }
z : ℝ
⊢ (z ∈ range fun q ↦ f a ↑q) → 0 ≤ z
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
r : { q' // y < ↑q' }
u : { q' // x < ↑q' }
⊢ 0 ≤ (fun q ↦ f a ↑q) u

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
r : { q' // y < ↑q' }
u : { q' // x < ↑q' }
⊢ 0 ≤ (fun q ↦ f a ↑q) u
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
    Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (sInf (stieltjesFunctionAux f a '' Ioi x)))

Tactic:
sInf_image'

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
    Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (⨅ a_1, stieltjesFunctionAux f a ↑a_1))
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
z : ℝ
⊢ (z ∈ range fun r ↦ f a ↑r) → 0 ≤ z

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
z : ℝ
⊢ (z ∈ range fun r ↦ f a ↑r) → 0 ≤ z
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
u : { r' // x < ↑r' }
⊢ 0 ≤ (fun r ↦ f a ↑r) u

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
u : { r' // x < ↑r' }
⊢ 0 ≤ (fun r ↦ f a ↑r) u
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x - 1 < ↑q ∧ ↑q < x
qs : ℝ → ℚ := fun x ↦ ⋯.choose
⊢ Tendsto qs atTop atTop

Tactic:
rw [tendsto_atTop_atTop]

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
h_exists : ∀ (x : ℝ), ∃ q, x - 1 < ↑q ∧ ↑q < x
qs : ℝ → ℚ := fun x ↦ ⋯.choose
⊢ ∀ (b : ℚ), ∃ i, ∀ (a : ℝ), i ≤ a → b ≤ qs a
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
⊢ (ρ.restrict (univ ×ˢ Iic r)).fst ≤ (ρ.restrict (univ ×ˢ Iic r')).fst

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
⊢ (ρ.restrict (univ ×ˢ Iic r)).fst ≤ (ρ.restrict (univ ×ˢ Iic r')).fst
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
h : ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑(-i)
i : ℚ
h' : x.1 ∈ s ∧ x.2 ≤ ↑i
⊢ x.2 ≤ ↑i

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
h : ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑(-i)
i : ℚ
h' : x.1 ∈ s ∧ x.2 ≤ ↑i
⊢ x.2 ≤ ↑i
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (⋂ i, s ×ˢ Iic ↑↑i) = (κ a) (s ×ˢ Iic x)

Tactic:
rw [← prod_iInter]
    congr with y
    simp only [mem_iInter, mem_Iic, Subtype.forall]
    exact ⟨le_of_forall_lt_rat_imp_le, fun hyx q hq ↦ hyx.trans hq.le⟩

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝¹ : Finite ι
κ : ι → Type u_4
inst✝ : ∀ (i : ι), Finite (κ i)
X : (i : ι) → κ i → Ω → ℝ
hX : HasGaussianLaw (fun ω i j ↦ X i j ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0
this : IsProbabilityMeasure P
x✝² : ι
x✝¹ : Ω
x✝ : κ x✝²
⊢ X x✝² x✝ x✝¹ = (ofLp ∘ toLp 2 ∘ fun ω j ↦ X x✝² j ω) x✝¹ x✝

Tactic:
;

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝¹ : Finite ι
κ : ι → Type u_4
inst✝ : ∀ (i : ι), Finite (κ i)
X : (i : ι) → κ i → Ω → ℝ
hX : HasGaussianLaw (fun ω i j ↦ X i j ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0
this : IsProbabilityMeasure P
x✝² : ι
x✝¹ : Ω
x✝ : κ x✝²
⊢ X x✝² x✝ x✝¹ = (ofLp ∘ toLp 2 ∘ fun ω j ↦ X x✝² j ω) x✝¹ x✝
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (f (a, t)) atTop (𝓝 1)

Tactic:
suffices ∀ᵐ t ∂(ν a), Tendsto (fun (n : ℕ) ↦ f (a, t) n) atTop (𝓝 1) by
    filter_upwards [this, hf.mono a] with t ht h_mono
    rw [tendsto_iff_tendsto_subseq_of_monotone h_mono tendsto_natCast_atTop_atTop]
    exact ht

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (fun n ↦ f (a, t) ↑n) atTop (𝓝 1)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
⊢ ∀ (r : ↑(Ioi q)), ∫ (t : β) in A, ⨅ r', f (a, t) ↑r' ∂ν a ≤ (κ a).real (A ×ˢ Iic ↑↑r)

Tactic:
intro r

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
r : ↑(Ioi q)
⊢ ∫ (t : β) in A, ⨅ r', f (a, t) ↑r' ∂ν a ≤ (κ a).real (A ×ˢ Iic ↑↑r)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
t : ℝ≥0
x y : ↥{t₀, t₀ + t} → ℝ
⊢ x ⟨t₀ + t, ⋯⟩ + y ⟨t₀ + t, ⋯⟩ - (x ⟨t₀, ⋯⟩ + y ⟨t₀, ⋯⟩) = x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩ + (y ⟨t₀ + t, ⋯⟩ - y ⟨t₀, ⋯⟩)

Tactic:
;

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
t : ℝ≥0
x y : ↥{t₀, t₀ + t} → ℝ
⊢ x ⟨t₀ + t, ⋯⟩ + y ⟨t₀ + t, ⋯⟩ - (x ⟨t₀, ⋯⟩ + y ⟨t₀, ⋯⟩) = x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩ + (y ⟨t₀ + t, ⋯⟩ - y ⟨t₀, ⋯⟩)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
t : ℝ≥0
c : ℝ
x : ↥{t₀, t₀ + t} → ℝ
⊢ c * x ⟨t₀ + t, ⋯⟩ - c * x ⟨t₀, ⋯⟩ = c * (x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩)

Tactic:
;

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
t : ℝ≥0
c : ℝ
x : ↥{t₀, t₀ + t} → ℝ
⊢ c * x ⟨t₀ + t, ⋯⟩ - c * x ⟨t₀, ⋯⟩ = c * (x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
t✝ t : Set ℝ
ht : MeasurableSet t
iht : ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
x : β
⊢ ((IsCondKernelCDF.toKernel f hf) (a, x)) tᶜ =
    ((IsCondKernelCDF.toKernel f hf) (a, x)) univ - ((IsCondKernelCDF.toKernel f hf) (a, x)) t

Tactic:
;

Goal state after:
case e_f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
t✝ t : Set ℝ
ht : MeasurableSet t
iht : ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
x : β
⊢ ((IsCondKernelCDF.toKernel f hf) (a, x)) tᶜ =
    ((IsCondKernelCDF.toKernel f hf) (a, x)) univ - ((IsCondKernelCDF.toKernel f hf) (a, x)) t
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝ : SFinite μ
hκ : ¬IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy α) ∘ₘ μ

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝ : SFinite μ
hκ : IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy α) ∘ₘ μ

Tactic:
;

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝ : SFinite μ
hκ : ¬IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy α) ∘ₘ μ

case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝ : SFinite μ
hκ : IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy α) ∘ₘ μ
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ ((Kernel.id ∥ₖ κ) a) s) (μ.prod ν)

pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ∫⁻ (a : β), (κ a) (Prod.mk x ⁻¹' s) ∂ν ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((Kernel.id ∥ₖ κ) (x, y)) s ∂ν ∂μ

Tactic:
;

Goal state after:
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ ((Kernel.id ∥ₖ κ) a) s) (μ.prod ν)

α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ∫⁻ (a : β), (κ a) (Prod.mk x ⁻¹' s) ∂ν ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((Kernel.id ∥ₖ κ) (x, y)) s ∂ν ∂μ
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
x✝ : Ω
⊢ (⇑L ∘ fun ω ↦ (X ω, Y ω)) x✝ = (⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y) x✝

Tactic:
;

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
x✝ : Ω
⊢ (⇑L ∘ fun ω ↦ (X ω, Y ω)) x✝ = (⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y) x✝
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
hμ : ¬SFinite μ
⊢ ⇑(Kernel.id ∥ₖ η) ∘ₘ μ ⊗ₘ κ = μ ⊗ₘ (η ∘ₖ κ)

pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
hμ : SFinite μ
⊢ ⇑(Kernel.id ∥ₖ η) ∘ₘ μ ⊗ₘ κ = μ ⊗ₘ (η ∘ₖ κ)

Tactic:
;

Goal state after:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
hμ : ¬SFinite μ
⊢ ⇑(Kernel.id ∥ₖ η) ∘ₘ μ ⊗ₘ κ = μ ⊗ₘ (η ∘ₖ κ)

case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
hμ : SFinite μ
⊢ ⇑(Kernel.id ∥ₖ η) ∘ₘ μ ⊗ₘ κ = μ ⊗ₘ (η ∘ₖ κ)
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: case e'_6.e'_2
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 Var[X 0; P].toNNReal) P'
hX : MemLp (X 0) 2 P
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
h : Var[X 0; P] ≠ 0
⊢ ↑1 = ↑(Var[X 0; P].toNNReal / NNReal.mk (√Var[X 0; P] ^ 2) ⋯)

Tactic:
;

Goal state after:
case e'_6.e'_2
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 Var[X 0; P].toNNReal) P'
hX : MemLp (X 0) 2 P
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
h : Var[X 0; P] ≠ 0
⊢ ↑1 = ↑(Var[X 0; P].toNNReal / NNReal.mk (√Var[X 0; P] ^ 2) ⋯)
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝ : ρ.IsCondKernel ρCond
hρ : ¬IsSFiniteKernel ρCond
⊢ ρ = 0

Tactic:
;

Goal state after:
α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝ : ρ.IsCondKernel ρCond
hρ : ¬IsSFiniteKernel ρCond
⊢ ρ = 0
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρCond : Kernel α Ω
inst✝² : MeasurableSingletonClass α
x : α
s : Set Ω
hs : MeasurableSet s
inst✝¹ : IsCondKernel 0 ρCond
inst✝ : IsFiniteMeasure 0
hx : (fst 0) {x} ≠ 0
⊢ False

Tactic:
;

Goal state after:
α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρCond : Kernel α Ω
inst✝² : MeasurableSingletonClass α
x : α
s : Set Ω
hs : MeasurableSet s
inst✝¹ : IsCondKernel 0 ρCond
inst✝ : IsFiniteMeasure 0
hx : (fst 0) {x} ≠ 0
⊢ False
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝² : ρ.IsCondKernel ρCond
inst✝¹ : IsFiniteMeasure ρ
inst✝ : MeasurableSingletonClass α
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this : IsSFiniteKernel ρCond
a : α
hax : a ≠ x
y : Ω
⊢ y ∈ Prod.mk a ⁻¹' Prod.mk x '' s ↔ y ∈ ∅

Tactic:
;

Goal state after:
α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝² : ρ.IsCondKernel ρCond
inst✝¹ : IsFiniteMeasure ρ
inst✝ : MeasurableSingletonClass α
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this : IsSFiniteKernel ρCond
a : α
hax : a ≠ x
y : Ω
⊢ y ∈ Prod.mk a ⁻¹' Prod.mk x '' s ↔ y ∈ ∅
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))

pp: case pos
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))

Tactic:
;

Goal state after:
case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))

case pos
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : 0 = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))

Tactic:
;

Goal state after:
case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : 0 = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))