Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inr.succ
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ (deterministic Prod.fst ⋯ ×ₖ
            (deterministic Prod.snd ⋯ ×ₖ ((κ k).map ⇑(piSingleton k)).comap (_root_.IicProdIoc a k) ?inr.succ.hf✝).map
              (IocProdIoc a k (k + 1)) ∘ₖ
          (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯))).map
      ⇑(MeasurableEquiv.IicProdIoc ⋯) =
    (Kernel.id ×ₖ (partialTraj κ a (k + 1)).map (restrict₂ ⋯)).map (_root_.IicProdIoc a (k + 1))

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable id

pp: case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (IocProdIoc a k (k + 1))

pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ MeasurableSpace ((i : ↥(Ioc a k.succ)) → X ↑i)

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (Prod.map id (IocProdIoc a k (k + 1)))

pp: case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable ⇑(MeasurableEquiv.IicProdIoc ⋯)

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable ⇑prodAssoc

pp: case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1)))

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (Prod.map (_root_.IicProdIoc a k) id)

pp: case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc k (k + 1))

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc a k)

pp: case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc k (k + 1))

pp: case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc a k)

Tactic:
map_apply_eq_iff_map_symm_apply_eq,

Goal state after:
case inr.succ
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ deterministic Prod.fst ⋯ ×ₖ
        (deterministic Prod.snd ⋯ ×ₖ ((κ k).map ⇑(piSingleton k)).comap (_root_.IicProdIoc a k) ?inr.succ.hf✝).map
          (IocProdIoc a k (k + 1)) ∘ₖ
      (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)) =
    ((Kernel.id ×ₖ (partialTraj κ a (k + 1)).map (restrict₂ ⋯)).map (_root_.IicProdIoc a (k + 1))).map
      ⇑(MeasurableEquiv.IicProdIoc ⋯).symm

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable id

case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (IocProdIoc a k (k + 1))

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (Prod.map id (IocProdIoc a k (k + 1)))

case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable ⇑(MeasurableEquiv.IicProdIoc ⋯)

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable ⇑prodAssoc

case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1)))

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (Prod.map (_root_.IicProdIoc a k) id)

case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc k (k + 1))

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc a k)

case inr.succ.hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc k (k + 1))

case inr.succ.hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)
this :
  _root_.IicProdIoc k (k + 1) ∘ Prod.map (_root_.IicProdIoc a k) id =
    (⇑(MeasurableEquiv.IicProdIoc ⋯) ∘ Prod.map id (IocProdIoc a k (k + 1))) ∘ ⇑prodAssoc
⊢ Measurable (_root_.IicProdIoc a k)