Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case inl.refine_2
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ ((partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a (n + 1)).map (restrict₂ ⋯)) x₀ =
    (const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)) x₀

Tactic:
rw [const_apply, partialTraj_succ_of_le (by lia), map_const, prod_const_comp, id_comp,
        ← map_comp_right, this, map_comp_right, ← map_prod_map, hind, Kernel.map_id, map_apply,
        prod_apply, const_apply, const_apply, Measure.map_piSingleton,
        Measure.pi_prod_map_IocProdIoc]

Goal state after:
case inl.refine_2.hab
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ a ≤ n

case inl.refine_2.hbc
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ n ≤ n + 1

case inl.refine_2.hf
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (IocProdIoc a n (n + 1))

case inl.refine_2.hf
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (restrict₂ ⋯)

case inl.refine_2.hg
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable id

case inl.refine_2.hf
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (Prod.map (restrict₂ ⋯) id)

case inl.refine_2.hg
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (IocProdIoc a n (n + 1))

case inl.refine_2.hf
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (_root_.IicProdIoc n (n + 1))

case inl.refine_2.hg
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable (restrict₂ ⋯)

case inl.refine_2.hf
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
n : ℕ
hn : a.succ ≤ n
hind :
  (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a n).map (restrict₂ ⋯) =
    const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)
x₀ : (i : ↥(Iic a)) → X ↑i
this : restrict₂ ⋯ ∘ _root_.IicProdIoc n (n + 1) = IocProdIoc a n (n + 1) ∘ Prod.map (restrict₂ ⋯) id
⊢ Measurable ⇑(piSingleton n)