theorem extracted_formal_statement_170.{w, v, u} {Z : Type w} {ι : Type v} {f_1 : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n_1 : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f_1 I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f_1 I z| ≤ b) (n : ℕ)
  (ih :
    ∀ (c : ι → ℝ),
      (∃ C, ∀ (I : ι), |c I| ≤ C) →
        (∫ (x : Fin n → Ω × Ω),
            (fun ω => ⨆ I, ∑ i, (f_1 I (X (ω i).1) - f_1 I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
          ∫ (x : Fin n → Ω × Ω),
            (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f_1 I (X (ω i).1) - f_1 I (X (ω i).2)) + c I)
              x ∂Measure.pi fun x => μ.prod μ)
  (c : ι → ℝ) (C : ℝ) (hC : ∀ (I : ι), |c I| ≤ C) (x : (Ω × Ω) × (Fin n → Ω × Ω)) (f : ι)
  (h_original_goal :
    ∑ i, (f_1 f (X (Fin.snoc x.2 x.1 i).1) - f_1 f (X (Fin.snoc x.2 x.1 i).2)) + c f =
      ∑ i, (f_1 f (X (x.2 i).1) - f_1 f (X (x.2 i).2)) + (f_1 f (X x.1.1) - f_1 f (X x.1.2) + c f)) :
  ∑ i, (f_1 f (X (Fin.snoc x.2 x.1 i.castSucc).1) - f_1 f (X (Fin.snoc x.2 x.1 i.castSucc).2)) +
      (f_1 f (X (Fin.snoc x.2 x.1 (Fin.last n)).1) - f_1 f (X (Fin.snoc x.2 x.1 (Fin.last n)).2) + c f) =
    ∑ i, (f_1 f (X (x.2 i).1) - f_1 f (X (x.2 i).2)) + (f_1 f (X x.1.1) - f_1 f (X x.1.2) + c f) := sorry

State before:
case h.h.h.h
Z : Type w
ι : Type v
f✝ : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n✝ : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f✝ I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f✝ I z| ≤ b
n : ℕ
ih :
  ∀ (c : ι → ℝ),
    (∃ C, ∀ (I : ι), |c I| ≤ C) →
      (∫ (x : Fin n → Ω × Ω),
          (fun ω => ⨆ I, ∑ i, (f✝ I (X (ω i).1) - f✝ I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
        ∫ (x : Fin n → Ω × Ω),
          (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f✝ I (X (ω i).1) - f✝ I (X (ω i).2)) + c I)
            x ∂Measure.pi fun x => μ.prod μ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
x✝ : (Ω × Ω) × (Fin n → Ω × Ω)
f : ι
⊢ ∑ i, (f✝ f (X (Fin.snoc x✝.2 x✝.1 i).1) - f✝ f (X (Fin.snoc x✝.2 x✝.1 i).2)) + c f =
    ∑ i, (f✝ f (X (x✝.2 i).1) - f✝ f (X (x✝.2 i).2)) + (f✝ f (X x✝.1.1) - f✝ f (X x✝.1.2) + c f)

Tactic: rw [Fin.sum_univ_castSucc, add_assoc]

State after:
[case h.h.h.h
 Z : Type w
 ι : Type v
 f✝ : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 n✝ : ℕ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f✝ I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f✝ I z| ≤ b
 n : ℕ
 ih :
   ∀ (c : ι → ℝ),
     (∃ C, ∀ (I : ι), |c I| ≤ C) →
       (∫ (x : Fin n → Ω × Ω),
           (fun ω => ⨆ I, ∑ i, (f✝ I (X (ω i).1) - f✝ I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
         ∫ (x : Fin n → Ω × Ω),
           (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f✝ I (X (ω i).1) - f✝ I (X (ω i).2)) + c I)
             x ∂Measure.pi fun x => μ.prod μ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
 x✝ : (Ω × Ω) × (Fin n → Ω × Ω)
 f : ι
 ⊢ ∑ i, (f✝ f (X (Fin.snoc x✝.2 x✝.1 i.castSucc).1) - f✝ f (X (Fin.snoc x✝.2 x✝.1 i.castSucc).2)) +
       (f✝ f (X (Fin.snoc x✝.2 x✝.1 (Fin.last n)).1) - f✝ f (X (Fin.snoc x✝.2 x✝.1 (Fin.last n)).2) + c f) =
     ∑ i, (f✝ f (X (x✝.2 i).1) - f✝ f (X (x✝.2 i).2)) + (f✝ f (X x✝.1.1) - f✝ f (X x✝.1.2) + c f)]

Relevant lemmas:
Fin.sum_univ_castSucc : ∀ {β : Type u_2} [inst : AddCommMonoid β] {n : ℕ} (f : Fin (n + 1) → β),
  ∑ i, f i = ∑ i, f i.castSucc + f (Fin.last n)
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)