theorem extracted_formal_statement_14.{u_2, u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_1 : IsProbabilityMeasure μ] {𝓧 : Type u_2} {m𝓧 : MeasurableSpace 𝓧} [hnonempty𝓧 : Nonempty 𝓧] {m : ℕ}
  {X' : Ω → 𝓧} (hX'' : Measurable X') {f' : (Fin m → 𝓧) → ℝ} {c' : Fin m → ℝ}
  (hfι : ∀ (i : Fin m) (x : Fin m → 𝓧) (x' : 𝓧), |f' x - f' (Function.update x i x')| ≤ c' i)
  (hf'' : StronglyMeasurable f') {t : ℝ} (ht : t ≥ 0) (ht' : t * ∑ i, c' i ^ 2 ≤ 1)
  (hX''' : ∀ (i : Fin m), Measurable fun ω => X' (ω i)) :
  let Y := fun k Xk =>
    ∫ (x : Fin m → Ω), (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else X' (ω i)) x ∂Measure.pi fun x => μ;
  ∀ (k : Fin m) (Xk : Fin ↑k → 𝓧) (x : 𝓧),
    Y k.succ (Fin.snoc Xk x) =
      ∫ (x_1 : Fin m → Ω),
        (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else if ↑i = ↑k then x else X' (ω i))
          x_1 ∂Measure.pi fun x => μ := sorry

State before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
𝓧 : Type u_2
m𝓧 : MeasurableSpace 𝓧
hnonempty𝓧 : Nonempty 𝓧
m : ℕ
X' : Ω → 𝓧
hX'' : Measurable X'
f' : (Fin m → 𝓧) → ℝ
c' : Fin m → ℝ
hfι : ∀ (i : Fin m) (x : Fin m → 𝓧) (x' : 𝓧), |f' x - f' (Function.update x i x')| ≤ c' i
hf'' : StronglyMeasurable f'
ε : ℝ
hε : ε ≥ 0
t : ℝ
ht : t ≥ 0
ht' : t * ∑ i, c' i ^ 2 ≤ 1
hX''' : ∀ (i : Fin m), Measurable fun ω => X' (ω i)
Y : (k : Fin m.succ) → (Fin ↑k → 𝓧) → ℝ :=
  fun k Xk =>
    ∫ (x : Fin m → Ω), (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else X' (ω i)) x ∂Measure.pi fun x => μ
A : (k : Fin m) → (Fin ↑k → 𝓧) → ℝ :=
  fun k Xk =>
    ⨅ x,
      ∫ (x_1 : Fin m → Ω),
        (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else if ↑i = ↑k then x else X' (ω i))
          x_1 ∂Measure.pi fun x => μ
B : (k : Fin m) → (Fin ↑k → 𝓧) → ℝ :=
  fun k Xk =>
    ⨆ x,
      ∫ (x_1 : Fin m → Ω),
        (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else if ↑i = ↑k then x else X' (ω i))
          x_1 ∂Measure.pi fun x => μ
k : Fin m
Xk : Fin ↑k → 𝓧
x : 𝓧
⊢ Y k.succ (Fin.snoc Xk x) =
    ∫ (x_1 : Fin m → Ω),
      (fun ω => f' fun i => if h : ↑i < ↑k then Xk ⟨↑i, h⟩ else if ↑i = ↑k then x else X' (ω i))
        x_1 ∂Measure.pi fun x => μ

Tactic: apply integral_congr_ae
    filter_upwards with ω
    apply congr rfl
    ext i
    if h : i.1 < k.1 then 
  have : i.1 < (Fin.succ k).1 := by dsimp; linarith
  rw [dif_pos this, dif_pos h]
  have : ⟨i.1, this⟩ = (⟨i.1, h⟩ : Fin k).castSucc := rfl
  rw [this]
  apply Fin.snoc_castSucc
else
  rw [dif_neg h]
  if h2 : i.1 = k.1 then 
    have : i.1 < (Fin.succ k).1 := by dsimp; linarith
    rw [dif_pos this, if_pos h2]
    have : ⟨i.1, this⟩ = Fin.last k := by
      apply Eq.symm
      apply Fin.eq_mk_iff_val_eq.mpr
      exact id (Eq.symm h2)
    rw [this]
    apply Fin.snoc_last
  else
    have : ¬(i.1 < (Fin.succ k).1) := by
      simp only [Fin.val_succ, not_lt]
      simp only [Fin.val_fin_lt, not_lt] at h
      apply Fin.val_add_one_le_of_lt
      exact lt_of_le_of_ne h fun a => h2 (congrArg Fin.val (id (Eq.symm a)))
    rw [dif_neg this, if_neg h2]

State after:

Relevant lemmas:
Fin.val_add_one_le_of_lt : ∀ {n : ℕ} {a b : Fin n}, a < b → ↑a + 1 ≤ ↑b
Fin.val_fin_lt : ∀ {n : ℕ} {a b : Fin n}, ↑a < ↑b ↔ a < b
lt_of_le_of_ne : ∀ {α : Type u_1} [inst : PartialOrder α] {a b : α}, a ≤ b → a ≠ b → a < b
Fin.snoc_castSucc : ∀ {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (Fin.last n)) (p : (i : Fin n) → α i.castSucc)
  (i : Fin n), Fin.snoc p x i.castSucc = p i
Fin.val_succ : ∀ {n : ℕ} (j : Fin n), ↑j.succ = ↑j + 1
congrArg : ∀ {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β), a₁ = a₂ → f a₁ = f a₂
MeasureTheory.integral_congr_ae : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G]
  [inst_2 : NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} {f g : α → G},
  f =ᶠ[ae μ] g → ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ
not_lt : ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, ¬a < b ↔ b ≤ a
dif_pos : ∀ {c : Prop} {h : Decidable c} (hc : c) {α : Sort u} {t : c → α} {e : ¬c → α}, dite c t e = t hc
if_neg : ∀ {c : Prop} {h : Decidable c}, ¬c → ∀ {α : Sort u} {t e : α}, (if c then t else e) = e
Fin.snoc_last : ∀ {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (Fin.last n)) (p : (i : Fin n) → α i.castSucc),
  Fin.snoc p x (Fin.last n) = x
congr : ∀ {α : Sort u} {β : Sort v} {f₁ f₂ : α → β} {a₁ a₂ : α}, f₁ = f₂ → a₁ = a₂ → f₁ a₁ = f₂ a₂
Eq.symm : ∀ {α : Sort u} {a b : α}, a = b → b = a
dif_neg : ∀ {c : Prop} {h : Decidable c} (hnc : ¬c) {α : Sort u} {t : c → α} {e : ¬c → α}, dite c t e = e hnc
if_pos : ∀ {c : Prop} {h : Decidable c}, c → ∀ {α : Sort u} {t e : α}, (if c then t else e) = t