-- Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean
lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
      = κ a (s ×ˢ Iic x)

Proof state:
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≤ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑j)

Tactic:
refine measure_mono (Iic_subset_Iic.mpr ?_)

Result:
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ↑↑i ≤ ↑↑j

Proof state:
case e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (⋂ i, Iic ↑↑i)

Tactic:
congr with y : 1

Result:
case e_f.h.h.e_6.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ∈ Iic x ↔ y ∈ ⋂ i, Iic ↑↑i

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
⊢ Nonempty { r' // x < ↑r' }

Tactic:
obtain ⟨r, hrx⟩ := exists_rat_gt x

Result:
case intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
r : ℚ
hrx : x < ↑r
⊢ Nonempty { r' // x < ↑r' }

Proof state:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)

Tactic:
simp_rw [setLIntegral_stieltjesOfMeasurableRat_rat hf _ _ hs]

Result:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, (κ a) (s ×ˢ Iic ↑↑b) = (κ a) (s ×ˢ Iic x)

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q

Tactic:
exact (measurable_stieltjesOfMeasurableRat hf.measurable _).comp measurable_prod_mk_left

Result:
no goals

Proof state:
case e_f.h.h.e_6.h.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a✝ : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a✝).restrict s = 0
b : β
y : ℝ
h : y ≤ x
a : ℚ
ha : x < ↑a
⊢ x ≤ ↑a

Tactic:
exact mod_cast ha.le

Result:
no goals

Proof state:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (⋂ i, s ×ˢ Iic ↑↑i) = (κ a) (s ×ˢ Iic x)

Tactic:
rw [← prod_iInter]

Result:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (s ×ˢ ⋂ i, Iic ↑↑i) = (κ a) (s ×ˢ Iic x)

Proof state:
case neg.hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∃ i, (κ a) (s ×ˢ Iic ↑↑i) ≠ ⊤

Tactic:
exact ⟨h_nonempty.some, measure_ne_top _ _⟩

Result:
no goals