mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: case e_f.hs
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ MeasurableSet ((fun x ↦ (frestrictLe b x, x)) ⁻¹' s)

Tactic:
fun_prop

Goal state after:
case e_f.hs
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ MeasurableSet ((fun x ↦ (frestrictLe b x, x)) ⁻¹' s)
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: case e_f.hs
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ MeasurableSet (Prod.mk x ⁻¹' s)

Tactic:
fun_prop

Goal state after:
case e_f.hs
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ MeasurableSet (Prod.mk x ⁻¹' s)
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ Measurable fun x ↦ (frestrictLe b x, x)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
x : (i : ↥(Iic b)) → X ↑i
⊢ Measurable (Prod.mk x)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
⊢ Measurable fun x ↦ (frestrictLe b x, x)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
⊢ Measurable fun x ↦ x

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
⊢ Measurable ?m.371

Tactic:
fun_prop

Goal state after:
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
⊢ Measurable ?m.371
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: case hf
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
s : Set (((i : ↥(Iic b)) → X ↑i) × ((n : ℕ) → X n))
ms : MeasurableSet s
⊢ Measurable fun x ↦ (frestrictLe b x, x)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
x₀ : (n : ↥(Iic a)) → X ↑n
hf : (fun x ↦ (frestrictLe b x, x (b + 1))) = (Prod.map id fun x ↦ x (b + 1)) ∘ fun x ↦ (frestrictLe b x, x)
⊢ Measurable (Prod.map id fun x ↦ x (b + 1))

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
x₀ : (n : ↥(Iic a)) → X ↑n
hf : (fun x ↦ (frestrictLe b x, x (b + 1))) = (Prod.map id fun x ↦ x (b + 1)) ∘ fun x ↦ (frestrictLe b x, x)
⊢ Measurable fun x ↦ (frestrictLe b x, x)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
x₀ : (n : ↥(Iic a)) → X ↑n
hf : (fun x ↦ (frestrictLe b x, x (b + 1))) = (Prod.map id fun x ↦ x (b + 1)) ∘ fun x ↦ (frestrictLe b x, x)
⊢ Measurable fun x ↦ x (b + 1)

Tactic:
fun_prop

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

Goal state before:
pp: R : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace R
inst✝ : AddMonoidWithOne R
m : MeasurableSpace Ω
P : Measure Ω
X : Ω → R
n : ℕ
p : ↑I
⊢ AEMeasurable ncard setBer(Iio n, p)

Tactic:
fun_prop

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

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
P : Measure Ω
n : ℕ
p : ↑I
X : Ω → ℕ
hX : HasLaw X Bin(n, p) P
⊢ Measurable fun x ↦ x ≤ n

Tactic:
fun_prop

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

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
P : Measure Ω
n : ℕ
p : ↑I
X : Ω → ℕ
hX : HasLaw X Bin(n, p) P
⊢ AEMeasurable ncard setBer(Iio n, p)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝³ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)
hf' : Integrable (uncurry f) (Measure.map (fun x ↦ (frestrictLe b x, x)) ((traj κ a) x₀))
⊢ AEMeasurable (fun x ↦ (frestrictLe b x, x)) ((traj κ a) x₀)

Tactic:
fun_prop

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

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ Measurable fun x ↦ π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹

Tactic:
fun_prop

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

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ StronglyMeasurable (cauchyPDFReal x₀ γ)

Tactic:
fun_prop

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

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ Measurable fun x ↦ ENNReal.ofReal (cauchyPDFReal x₀ γ x)

Tactic:
fun_prop

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

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ StronglyMeasurable (cauchyPDF x₀ γ)

Tactic:
fun_prop

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

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n k : ℕ
p : ↑I
⊢ Measurable Nat.cast

Tactic:
fun_prop

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

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
⊢ AEStronglyMeasurable (cauchyPDFReal x₀ γ) volume

Tactic:
fun_prop

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

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ ∫⁻ (x : ℝ) in Icc 0 x, ENNReal.ofReal (r * rexp (-(r * x))) ∂volume = ENNReal.ofReal (1 - rexp (-(r * x)))

Tactic:
← ENNReal.toReal_eq_toReal_iff' _ ENNReal.ofReal_ne_top,

Goal state after:
r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ (∫⁻ (x : ℝ) in Icc 0 x, ENNReal.ofReal (r * rexp (-(r * x))) ∂volume).toReal =
    (ENNReal.ofReal (1 - rexp (-(r * x)))).toReal

r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ ∫⁻ (x : ℝ) in Icc 0 x, ENNReal.ofReal (r * rexp (-(r * x))) ∂volume ≠ ∞
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
⊢ p.toOuterMeasure {a} = p a

Tactic:
by
  refine (p.toOuterMeasure_apply {a}).trans ((tsum_eq_single a fun b hb => ?_).trans ?_)
  · classical exact ite_eq_right_iff.2 fun hb' => False.elim <| hb hb'
  · classical exact ite_eq_left_iff.2 fun ha' => False.elim <| ha' rfl

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
p : PMF α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet (s ∩ p.support)
⊢ p.toMeasure s = p.toOuterMeasure s

Tactic:
← restrict_toMeasure_support,

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
p : PMF α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet (s ∩ p.support)
⊢ (p.toMeasure.restrict p.support) s = p.toOuterMeasure s
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: case refine_2
a r : ℝ
ha : 0 < a
hr : 0 < r
x : ℝ
this : IsProbabilityMeasure (gammaMeasure a r)
⊢ AEStronglyMeasurable (gammaPDFReal a r) (volume.restrict (Iic x))

Tactic:
fun_prop

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

Goal state before:
pp: p : ℝ
⊢ Measurable (geometricPMFReal p)

Tactic:
fun_prop

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

Goal state before:
pp: case refine_1.hf
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_f : Integrable f ((traj κ a) x₀)
i_f' : Integrable (fun x ↦ ∫ (y : (n : ℕ) → X n), f y ∂(traj κ b) x) (Measure.map (frestrictLe b) ((traj κ a) x₀))
t : Set ((i : ↥(Iic b)) → X ↑i)
mt : MeasurableSet t
⊢ Measurable (frestrictLe b)

Tactic:
fun_prop

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

Goal state before:
pp: case refine_1
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_f : Integrable f ((traj κ a) x₀)
i_f' : Integrable (fun x ↦ ∫ (y : (n : ℕ) → X n), f y ∂(traj κ b) x) (Measure.map (frestrictLe b) ((traj κ a) x₀))
t : Set ((i : ↥(Iic b)) → X ↑i)
mt : MeasurableSet t
⊢ AEMeasurable (frestrictLe b) ((traj κ a) x₀)

Tactic:
fun_prop

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

Goal state before:
pp: case refine_2
t r : ℝ
ht : 0 < t
hr : 0 < r
x : ℝ
this : IsProbabilityMeasure (paretoMeasure t r)
⊢ AEStronglyMeasurable (paretoPDFReal t r) (volume.restrict (Iic x))

Tactic:
fun_prop

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

Goal state before:
pp: case hφ
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b c : ℕ
hab : a ≤ b
hbc : b ≤ c
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_cf : Integrable ((traj κ a) x₀)[f | ↑piLE c] ((traj κ a) x₀)
mcf : StronglyMeasurable ((traj κ a) x₀)[f | ↑piLE c]
x : (n : ℕ) → X n
h1 : ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x = ((traj κ a) x₀)[f | ↑piLE b] x
h2 :
  ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x =
    ∫ (y : (n : ℕ) → X n), ((traj κ a) x₀)[f | ↑piLE c] y ∂(traj κ b) (frestrictLe b x)
⊢ AEMeasurable (frestrictLe c) ((traj κ b) (frestrictLe b x))

Tactic:
fun_prop

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

Goal state before:
pp: case hfm
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b c : ℕ
hab : a ≤ b
hbc : b ≤ c
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_cf : Integrable ((traj κ a) x₀)[f | ↑piLE c] ((traj κ a) x₀)
mcf : StronglyMeasurable ((traj κ a) x₀)[f | ↑piLE c]
x : (n : ℕ) → X n
h1 : ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x = ((traj κ a) x₀)[f | ↑piLE b] x
h2 :
  ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x =
    ∫ (y : (n : ℕ) → X n), ((traj κ a) x₀)[f | ↑piLE c] y ∂(traj κ b) (frestrictLe b x)
⊢ AEStronglyMeasurable (fun y ↦ ((traj κ a) x₀)[f | ↑piLE c] (updateFinset x (Iic c) y))
    (Measure.map (frestrictLe c) ((traj κ b) (frestrictLe b x)))

Tactic:
fun_prop

Goal state after:
case hfm
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b c : ℕ
hab : a ≤ b
hbc : b ≤ c
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_cf : Integrable ((traj κ a) x₀)[f | ↑piLE c] ((traj κ a) x₀)
mcf : StronglyMeasurable ((traj κ a) x₀)[f | ↑piLE c]
x : (n : ℕ) → X n
h1 : ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x = ((traj κ a) x₀)[f | ↑piLE b] x
h2 :
  ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x =
    ∫ (y : (n : ℕ) → X n), ((traj κ a) x₀)[f | ↑piLE c] y ∂(traj κ b) (frestrictLe b x)
⊢ AEStronglyMeasurable (fun y ↦ ((traj κ a) x₀)[f | ↑piLE c] (updateFinset x (Iic c) y))
    (Measure.map (frestrictLe c) ((traj κ b) (frestrictLe b x)))
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: case hf
X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b c : ℕ
hab : a ≤ b
hbc : b ≤ c
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_cf : Integrable ((traj κ a) x₀)[f | ↑piLE c] ((traj κ a) x₀)
mcf : StronglyMeasurable ((traj κ a) x₀)[f | ↑piLE c]
x : (n : ℕ) → X n
h1 : ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x = ((traj κ a) x₀)[f | ↑piLE b] x
h2 :
  ((traj κ a) x₀)[((traj κ a) x₀)[f | ↑piLE c] | ↑piLE b] x =
    ∫ (y : (n : ℕ) → X n), ((traj κ a) x₀)[f | ↑piLE c] y ∂(traj κ b) (frestrictLe b x)
⊢ Measurable (frestrictLe c)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝ : IsProbabilityMeasure μ₀
⊢ AEMeasurable (⇑(piUnique fun i ↦ X ↑i).symm) μ₀

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝ : IsProbabilityMeasure μ₀
a : ℕ
⊢ Measurable (frestrictLe a)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝ : IsProbabilityMeasure μ₀
a : ℕ
⊢ Measurable fun x ↦ (frestrictLe a x, x (a + 1))

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝ : IsProbabilityMeasure μ₀
a : ℕ
x₀ : (i : ↥(Iic 0)) → X ↑i
⊢ Measurable fun x ↦ (frestrictLe a x, x (a + 1))

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝² : IsProbabilityMeasure μ₀
a : ℕ
inst✝¹ : StandardBorelSpace (X (a + 1))
inst✝ : Nonempty (X (a + 1))
⊢ Measurable (frestrictLe a)

Tactic:
fun_prop

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
μ₀ : Measure (X 0)
inst✝² : IsProbabilityMeasure μ₀
a : ℕ
inst✝¹ : StandardBorelSpace (X (a + 1))
inst✝ : Nonempty (X (a + 1))
⊢ Measurable fun x ↦ x (a + 1)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: r : ℝ≥0
t : ℝ
⊢ AEStronglyMeasurable (fun x ↦ cexp (↑(inner ℝ x t) * I)) Po(ℝ, r)

Tactic:
fun_prop

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
hXmeas : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
⊢ AEStronglyMeasurable id (Measure.map X μ)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: r₁ r₂ : ℝ≥0
⊢ Measurable ⇑(Nat.castAddMonoidHom ℝ)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : AddMonoidWithOne R
mR : MeasurableSpace R
inst✝ : MeasurableAdd₂ R
r₁ r₂ : ℝ≥0
⊢ Measurable ⇑(Nat.castAddMonoidHom R)

Tactic:
fun_prop

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
hXmeas : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
h : ¬μ {ω | X ω = 1} = ⊤
⊢ AEStronglyMeasurable id (μ (X ⁻¹' {0}) • Measure.dirac 0)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: r : ℝ≥0
⊢ Measurable (poissonPMFReal r)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
X : Ω → 𝓧
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
h : HasCondDistrib Y X κ P
f : 𝓨 → 𝓩
hf : Measurable f
⊢ AEMeasurable (Prod.map id f) (Measure.map (fun ω ↦ (X ω, Y ω)) P)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
X : Ω → 𝓧
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
h : HasCondDistrib Y X κ P
f : 𝓨 → 𝓩
hf : Measurable f
⊢ AEMeasurable (fun ω ↦ (X ω, Y ω)) P

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
f : 𝓩 → 𝓧
hf : Measurable f
Z : Ω → 𝓩
h : HasCondDistrib Y Z (κ.comap f hf) P
⊢ AEMeasurable (Prod.map f id) (Measure.map (fun a ↦ (Z a, Y a)) P)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
f : 𝓩 → 𝓧
hf : Measurable f
Z : Ω → 𝓩
h : HasCondDistrib Y Z (κ.comap f hf) P
⊢ AEMeasurable (fun a ↦ (Z a, Y a)) P

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasLawExists

Goal state before:
pp: ι : Type v
𝓧 : ι → Type u
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
⊢ ∀ (i : ι), Measurable ((fun i ↦ Function.eval i) i)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasLawExists

Goal state before:
pp: ι : Type v
𝓧 : ι → Type u
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
⊢ ∀ (i : ι), Measurable (Function.eval i)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
f : 𝓩 → 𝓧
hf : Measurable f
Z : Ω → 𝓩
h : HasCondDistrib Y Z (κ.comap f hf) P
s : Set (𝓧 × 𝓨)
hs : MeasurableSet s
⊢ Measurable (Prod.map f id)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝¹ : SFinite P
inst✝ : IsSFiniteKernel κ
f : 𝓩 → 𝓧
hf : Measurable f
Z : Ω → 𝓩
h : HasCondDistrib Y Z (κ.comap f hf) P
⊢ AEMeasurable Z P

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
X : Ω → 𝓧
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝² : SFinite P
inst✝¹ : IsSFiniteKernel κ
Z : Ω → 𝓩
η : Kernel (𝓧 × 𝓨) 𝓩
inst✝ : IsMarkovKernel η
h : HasCondDistrib (fun a ↦ (Y a, Z a)) X (κ ⊗ₖ η) P
hZ : AEMeasurable Z P
hY : AEMeasurable Y P
⊢ AEMeasurable (fun ω ↦ ((fun a ↦ (X a, Y a)) ω, Z ω)) P

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Probability.HasCondDistrib

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
𝓩 : Type u_4
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
m𝓩 : MeasurableSpace 𝓩
P : Measure Ω
X : Ω → 𝓧
Y : Ω → 𝓨
κ : Kernel 𝓧 𝓨
inst✝² : SFinite P
inst✝¹ : IsSFiniteKernel κ
Z : Ω → 𝓩
η : Kernel (𝓧 × 𝓨) 𝓩
inst✝ : IsMarkovKernel η
h : HasCondDistrib (fun a ↦ (Y a, Z a)) X (κ ⊗ₖ η) P
hZ : AEMeasurable Z P
hY : AEMeasurable Y P
⊢ AEMeasurable (⇑MeasurableEquiv.prodAssoc.symm) (Measure.map (fun ω ↦ (X ω, (fun a ↦ (Y a, Z a)) ω)) P)

Tactic:
fun_prop

Goal state after: