awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
a b : ℕ
hab : a ≤ b
inl : IsSFiniteKernel (partialTraj (X := X) κ a a)
inl₁ :
  ∀ (k : ℕ) (hak : a ≤ k),
    IsSFiniteKernel (partialTraj (X := X) κ a k) →
      IsSFiniteKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))
⊢ IsSFiniteKernel (partialTraj (X := X) κ a b)

Tactic:
induction b, hab using Nat.le_induction with
    | base => rw [partialTraj_self]; infer_instance
    | succ k hak => rw [partialTraj_succ_of_le hak]; infer_instance

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
a b : ℕ
hba : b ≤ a
inr :
  IsSFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)
⊢ IsSFiniteKernel (partialTraj (X := X) κ a b)

Tactic:
rw [partialTraj_le hba]

Premises used: ProbabilityTheory.Kernel.partialTraj_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
a b : ℕ
hba : b ≤ a
⊢ IsSFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
inl : ∀ (hab : a ≤ b), IsFiniteKernel (partialTraj (X := X) κ a b)
inr : ∀ (hba : b ≤ a), IsFiniteKernel (partialTraj (X := X) κ a b)
⊢ IsFiniteKernel (partialTraj (X := X) κ a b)

Tactic:
obtain hab | hba := le_total a b

Premises used: le_total
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
inl :
  IsFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id
⊢ IsFiniteKernel (partialTraj (X := X) κ a a)

Tactic:
rw [partialTraj_self]

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
⊢ IsFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsFiniteKernel (partialTraj (X := X) κ a k)
inl :
  IsFiniteKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))
⊢ IsFiniteKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))

Tactic:
rw [partialTraj_succ_of_le hak]

Premises used: ProbabilityTheory.Kernel.partialTraj_succ_of_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsFiniteKernel (partialTraj (X := X) κ a k)
⊢ IsFiniteKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
hab : a ≤ b
inl : IsFiniteKernel (partialTraj (X := X) κ a a)
inl₁ :
  ∀ (k : ℕ) (hak : a ≤ k),
    IsFiniteKernel (partialTraj (X := X) κ a k) →
      IsFiniteKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))
⊢ IsFiniteKernel (partialTraj (X := X) κ a b)

Tactic:
induction b, hab using Nat.le_induction with
    | base => rw [partialTraj_self]; infer_instance
    | succ k hak => rw [partialTraj_succ_of_le hak]; infer_instance

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
hba : b ≤ a
inr :
  IsFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)
⊢ IsFiniteKernel (partialTraj (X := X) κ a b)

Tactic:
rw [partialTraj_le hba]

Premises used: ProbabilityTheory.Kernel.partialTraj_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsFiniteKernel (κ n)
a b : ℕ
hba : b ≤ a
⊢ IsFiniteKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
inl : ∀ (hab : a ≤ b), IsZeroOrMarkovKernel (partialTraj (X := X) κ a b)
inr : ∀ (hba : b ≤ a), IsZeroOrMarkovKernel (partialTraj (X := X) κ a b)
⊢ IsZeroOrMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
obtain hab | hba := le_total a b

Premises used: le_total
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
inl :
  IsZeroOrMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id
⊢ IsZeroOrMarkovKernel (partialTraj (X := X) κ a a)

Tactic:
rw [partialTraj_self]

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
⊢ IsZeroOrMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsZeroOrMarkovKernel (partialTraj (X := X) κ a k)
inl :
  IsZeroOrMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))
⊢ IsZeroOrMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))

Tactic:
rw [partialTraj_succ_of_le hak]

Premises used: ProbabilityTheory.Kernel.partialTraj_succ_of_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsZeroOrMarkovKernel (partialTraj (X := X) κ a k)
⊢ IsZeroOrMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
inl : IsZeroOrMarkovKernel (partialTraj (X := X) κ a a)
inl₁ :
  ∀ (k : ℕ) (hak : a ≤ k),
    IsZeroOrMarkovKernel (partialTraj (X := X) κ a k) →
      IsZeroOrMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))
⊢ IsZeroOrMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
induction b, hab using Nat.le_induction with
    | base => rw [partialTraj_self]; infer_instance
    | succ k hak => rw [partialTraj_succ_of_le hak]; infer_instance

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
hba : b ≤ a
inr :
  IsZeroOrMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)
⊢ IsZeroOrMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
rw [partialTraj_le hba]

Premises used: ProbabilityTheory.Kernel.partialTraj_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsZeroOrMarkovKernel (κ n)
a b : ℕ
hba : b ≤ a
⊢ IsZeroOrMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
inl : ∀ (hab : a ≤ b), IsMarkovKernel (partialTraj (X := X) κ a b)
inr : ∀ (hba : b ≤ a), IsMarkovKernel (partialTraj (X := X) κ a b)
⊢ IsMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
obtain hab | hba := le_total a b

Premises used: le_total
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
inl :
  IsMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id
⊢ IsMarkovKernel (partialTraj (X := X) κ a a)

Tactic:
rw [partialTraj_self]

Premises used: ProbabilityTheory.Kernel.partialTraj_self
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
⊢ IsMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) Kernel.id

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsMarkovKernel (partialTraj (X := X) κ a k)
inl :
  IsMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))
⊢ IsMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))

Tactic:
rw [partialTraj_succ_of_le hak]

Premises used: ProbabilityTheory.Kernel.partialTraj_succ_of_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsMarkovKernel (partialTraj (X := X) κ a k)
⊢ IsMarkovKernel (β := (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Ioc k (k + 1))) ↦ X (↑i : ℕ)))
    ((κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))

Tactic:
have := IsMarkovKernel.map (κ k) (piSingleton k).measurable
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsMarkovKernel (partialTraj (X := X) κ a k)
this :
  IsMarkovKernel (β := (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Ioc k (k + 1))) ↦ X (↑i : ℕ)))
    ((κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
⊢ IsMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))

Tactic:
IsMarkovKernel.map _ measurable_IicProdIoc

Premises used: Set.Mem, setOf, measurable_IicProdIoc, ProbabilityTheory.Kernel.IsMarkovKernel.map
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b k : ℕ
hak : a ≤ k
h : IsMarkovKernel (partialTraj (X := X) κ a k)
this :
  IsMarkovKernel (β := (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Ioc k (k + 1))) ↦ X (↑i : ℕ)))
    ((κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
⊢ IsMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (mβ :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic (k + 1))) ↦ X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a k)
      (_root_.IicProdIoc (X := X) k (k + 1)))

Tactic:
exact IsMarkovKernel.map _ measurable_IicProdIoc

Premises used: measurable_IicProdIoc, ProbabilityTheory.Kernel.IsMarkovKernel.map
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
inl : IsMarkovKernel (partialTraj (X := X) κ a a)
inl₁ :
  ∀ (k : ℕ) (hak : a ≤ k),
    IsMarkovKernel (partialTraj (X := X) κ a k) →
      IsMarkovKernel (β := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (partialTraj (X := X) κ a (k + 1))
⊢ IsMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
induction b, hab using Nat.le_induction with
    | base => rw [partialTraj_self]; infer_instance
    | succ k hak =>
      rw [partialTraj_succ_of_le hak]
      have := IsMarkovKernel.map (κ k) (piSingleton k).measurable
      exact IsMarkovKernel.map _ measurable_IicProdIoc

Premises used: measurable_IicProdIoc, ProbabilityTheory.Kernel.partialTraj_self, ProbabilityTheory.Kernel.IsMarkovKernel.map
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hba : b ≤ a
inr :
  IsMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)
⊢ IsMarkovKernel (partialTraj (X := X) κ a b)

Tactic:
rw [partialTraj_le hba]

Premises used: ProbabilityTheory.Kernel.partialTraj_le
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a_1 b_1 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hba : b ≤ a
⊢ IsMarkovKernel.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) (mα :=
    MeasurableSpace.pi (X := fun (i : ↥(Iic a)) ↦ X (↑i : ℕ))) (deterministic (frestrictLe₂ (π := X) hba) ⋯)

Tactic:
infer_instance
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
a : ℕ
rw :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (a + 1))) → X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a a)
      (_root_.IicProdIoc (X := X) a (a + 1)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)))
      (_root_.IicProdIoc (X := X) a (a + 1)))
rw₁ :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (a + 1))) → X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)) ∘ₖ
        Kernel.id)
      (_root_.IicProdIoc (X := X) a (a + 1)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)))
      (_root_.IicProdIoc (X := X) a (a + 1)))
rw₂ rw₃ :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (a + 1))) → X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)))
      (_root_.IicProdIoc (X := X) a (a + 1)))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)))
      (_root_.IicProdIoc (X := X) a (a + 1)))
⊢ Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (a + 1))) → X (↑i : ℕ))) (partialTraj (X := X) κ a (a + 1))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (a + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ a).map (⇑(piSingleton (X := X) a) : (a_1 : X (a + 1)) → (i : ↥(Ioc a (a + 1))) → X (↑i : ℕ)))
      (_root_.IicProdIoc (X := X) a (a + 1)))

Tactic:
rw [partialTraj_succ_of_le le_rfl, partialTraj_self, comp_id]

Premises used: ProbabilityTheory.Kernel.partialTraj_succ_of_le, le_rfl, ProbabilityTheory.Kernel.partialTraj_self, ProbabilityTheory.Kernel.comp_id
-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
hab : a ≤ b
rw :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (b + 1))) → X (↑i : ℕ))) (partialTraj (X := X) κ a (b + 1))
    (map (β := ((i : ↥(Iic b)) → X (↑i : ℕ)) × ((i : ↥(Ioc b (b + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
        (Kernel.id ×ₖ (κ b).map (⇑(piSingleton (X := X) b) : (a : X (b + 1)) → (i : ↥(Ioc b (b + 1))) → X (↑i : ℕ)))
        (_root_.IicProdIoc (X := X) b (b + 1)) ∘ₖ
      partialTraj (X := X) κ a b)
rw₁ :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (b + 1))) → X (↑i : ℕ))) (partialTraj (X := X) κ a (b + 1))
    (map (β := ((i : ↥(Iic b)) → X (↑i : ℕ)) × ((i : ↥(Ioc b (b + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ b).map (⇑(piSingleton (X := X) b) : (a : X (b + 1)) → (i : ↥(Ioc b (b + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a b)
      (_root_.IicProdIoc (X := X) b (b + 1)))
rw₂ rw₃ :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (b + 1))) → X (↑i : ℕ)))
    (map (β := ((i : ↥(Iic b)) → X (↑i : ℕ)) × ((i : ↥(Ioc b (b + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ b).map (⇑(piSingleton (X := X) b) : (a : X (b + 1)) → (i : ↥(Ioc b (b + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a b)
      (_root_.IicProdIoc (X := X) b (b + 1)))
    (map (β := ((i : ↥(Iic b)) → X (↑i : ℕ)) × ((i : ↥(Ioc b (b + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (κ b).map (⇑(piSingleton (X := X) b) : (a : X (b + 1)) → (i : ↥(Ioc b (b + 1))) → X (↑i : ℕ)) ∘ₖ
        partialTraj (X := X) κ a b)
      (_root_.IicProdIoc (X := X) b (b + 1)))
⊢ Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (b + 1))) → X (↑i : ℕ))) (partialTraj (X := X) κ a (b + 1))
    (partialTraj (X := X) κ b (b + 1) ∘ₖ partialTraj (X := X) κ a b)

Tactic:
rw [partialTraj_succ_self, ← map_comp, partialTraj_succ_of_le hab]

Premises used: ProbabilityTheory.Kernel.partialTraj_succ_of_le, ProbabilityTheory.Kernel.partialTraj_succ_self, ProbabilityTheory.Kernel.map_comp