-- Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel
Goal:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel.{u_1, u_2} 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_1 t : Set ℝ
ht : MeasurableSet t
iht :
  ∫⁻ (b : β) in s,
      (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
        t ∂(ν : (a : α) → Measure β) a =
    (((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞) (s ×ˢ t)
calc :
  ∫⁻ (b : β) in s,
      (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
        tᶜ ∂(ν : (a : α) → Measure β) a =
    ∫⁻ (b : β) in s,
      HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞) univ :
          ℝ≥0∞)
        ((((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞) t :
          ℝ≥0∞) ∂(ν : (a : α) → Measure β) a
calc₁ :
  ∫⁻ (b : β) in s,
      HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞) univ :
          ℝ≥0∞)
        ((((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞) t :
          ℝ≥0∞) ∂(ν : (a : α) → Measure β) a =
    ∫⁻ (b : β) in s,
        (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
          univ ∂(ν : (a : α) → Measure β) a -
      ∫⁻ (b : β) in s,
        (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
          t ∂(ν : (a : α) → Measure β) a
calc₂ :
  ∫⁻ (b : β) in s,
        (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
          univ ∂(ν : (a : α) → Measure β) a -
      ∫⁻ (b : β) in s,
        (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
          t ∂(ν : (a : α) → Measure β) a =
    HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞)
          (SProd.sprod (β := Set ℝ) s univ) :
        ℝ≥0∞)
      ((((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞) (s ×ˢ t) : ℝ≥0∞)
calc₃ :
  HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞)
          (SProd.sprod (β := Set ℝ) s univ) :
        ℝ≥0∞)
      ((((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞) (s ×ˢ t) : ℝ≥0∞) =
    (((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞) (s ×ˢ tᶜ)
⊢ ∫⁻ (b : β) in s,
      (((IsCondKernelCDF.toKernel f hf : (a : α × β) → Measure ℝ) (a, b) : Measure ℝ) : (a : Set ℝ) → ℝ≥0∞)
        tᶜ ∂(ν : (a : α) → Measure β) a =
    (((κ : (a : α) → Measure (β × ℝ)) a : Measure (β × ℝ)) : (a : Set (β × ℝ)) → ℝ≥0∞) (s ×ˢ tᶜ)

Tactic:
calc ∫⁻ b in s, hf.toKernel f (a, b) tᶜ ∂(ν a)
      = ∫⁻ b in s, hf.toKernel f (a, b) univ - hf.toKernel f (a, b) t ∂(ν a) := by
          congr with x; rw [measure_compl ht (measure_ne_top (hf.toKernel f (a, x)) _)]
    _ = ∫⁻ b in s, hf.toKernel f (a, b) univ ∂(ν a)
          - ∫⁻ b in s, hf.toKernel f (a, b) t ∂(ν a) := by
        rw [lintegral_sub]
        · exact (Kernel.measurable_coe (hf.toKernel f) ht).comp measurable_prodMk_left
        · rw [iht]
          exact measure_ne_top _ _
        · exact Eventually.of_forall fun a ↦ measure_mono (subset_univ _)
    _ = κ a (s ×ˢ univ) - κ a (s ×ˢ t) := by
        rw [setLIntegral_toKernel_univ hf a hs, iht]
    _ = κ a (s ×ˢ tᶜ) := by
        rw [← measure_diff _ (hs.prod ht).nullMeasurableSet (measure_ne_top _ _)]
        · rw [prod_diff_prod, compl_eq_univ_diff]
          simp only [diff_self, empty_prod, union_empty]
        · rw [prod_subset_prod_iff]
          exact Or.inl ⟨subset_rfl, subset_univ t⟩

Premises used: SProd.sprod, Compl.compl, Set.univ, DFunLike.coe