import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_34.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω' : Fin n → Ω) (i : ι) (this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)))
  (h_original_goal :
    (∫ (x : Fin n → Ω), -(↑n * (b + b)) ∂Measure.pi fun x => μ) ≤
      ∫ (x : Fin n → Ω), ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)) ∂Measure.pi fun x => μ) :
  (fun a => -(↑n * (b + b))) ≤ fun a => ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (a x_1)) := sorry

Tactic state before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω' : Fin n → Ω
i : ι
this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1))
⊢ (∫ (x : Fin n → Ω), -(↑n * (b + b)) ∂Measure.pi fun x => μ) ≤
    ∫ (x : Fin n → Ω), ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)) ∂Measure.pi fun x => μ

Tactic:
apply MeasureTheory.integral_mono
    · simp
    ·
  constructor
  · refine AEStronglyMeasurable.const_add ?hg.left.hf (∑ x : Fin n, f i (X (ω' x)))
    refine aestronglyMeasurable_iff_aemeasurable.mpr ?hg.left.hf.a
    refine AEMeasurable.neg ?hg.left.hf.a.hf
    have w : ∀ (x_1 : Fin n), (AEMeasurable (fun a ↦ f i (X (a x_1))) μⁿ) :=
      by
      intro x_1
      exact Integrable.aemeasurable (integrable_aux0 hf hf' i x_1)
    apply Measurable.aemeasurable
    apply measurable_pi_sum (hf i)
  · apply @MeasureTheory.hasFiniteIntegral_of_bounded _ _ _ _ _ _ _ (n * (b + b))
    filter_upwards
    intro a
    simp only [norm_eq_abs]
    calc
      _ ≤ |∑ x : Fin n, f i (X (ω' x))| + |∑ x_1 : Fin n, f i (X (a x_1))| := by
        exact abs_sub (∑ x : Fin n, f i (X (ω' x))) (∑ x_1 : Fin n, f i (X (a x_1)))
      _ ≤ (↑n * b) + (↑n * b) :=
        by
        have w : |∑ x : Fin n, f i (X (ω' x))| ≤ ↑n * b := by
          calc
            _ ≤ ∑ x : Fin n, |f i (X (ω' x))| := by
              exact Finset.abs_sum_le_sum_abs (fun i_1 ↦ f i (X (ω' i_1))) Finset.univ
            _ ≤ ∑ x : Fin n, b := by exact Finset.sum_le_sum fun i_1 a ↦ hf' i (X (ω' i_1))
            _ = ↑n * b := by simp
        have w' : |∑ x_1 : Fin n, f i (X (a x_1))| ≤ ↑n * b := by
          calc
            _ ≤ ∑ x_1 : Fin n, |f i (X (a x_1))| := by
              exact Finset.abs_sum_le_sum_abs (fun i_1 ↦ f i (X (a i_1))) Finset.univ
            _ ≤ ∑ x_1 : Fin n, b := by exact Finset.sum_le_sum fun i_1 s ↦ hf' i (X (a i_1))
            _ = ↑n * b := by simp
        exact add_le_add w w'
      _ = ↑n * (b + b) := by ring

Tactic state after:
[case h
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 X : Ω → Z
 inst_2 : Nonempty ι
 inst_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω' : Fin n → Ω
 i : ι
 this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1))
 ⊢ (fun a => -(↑n * (b + b))) ≤ fun a => ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (a x_1))]