theorem extracted_formal_statement_16.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} [p : Nesterov_first_fix_stepsize f h f' x0] (n : ℕ+)
  (h_original_goal :
    ↑↑n * (2 + ↑↑n) ^ 2 / ((2 + ↑↑n) * ↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2) ≤
      (2 + (↑↑n - 1)) ^ 2 / (↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2)) :
  0 < ↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2 := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f✝ h✝ : E → ℝ
f'✝ : E → E
x0✝ : E
f h : E → ℝ
f' : E → E
x0 : E
p : Nesterov_first_fix_stepsize f h f' x0
n : ℕ+
⊢ ↑↑n * (2 + ↑↑n) ^ 2 / ((2 + ↑↑n) * ↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2) ≤
    (2 + (↑↑n - 1)) ^ 2 / (↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2)

Tactic: rw [mul_assoc, ← div_div, div_le_div_iff_of_pos_right, pow_two, ← mul_assoc, mul_div_assoc]
    rw [div_self]
    ring_nf
    simp
    linarith

State after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f✝ h✝ : E → ℝ
 f'✝ : E → E
 x0✝ : E
 f h : E → ℝ
 f' : E → E
 x0 : E
 p : Nesterov_first_fix_stepsize f h f' x0
 n : ℕ+
 ⊢ 0 < ↑(Nesterov_first_fix_stepsize.l f h f' x0) * 2 ^ 2]

Relevant lemmas:
div_self : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a / a = 1
pow_two : ∀ {M : Type u_2} [inst : Monoid M] (a : M), a ^ 2 = a * a
div_div : ∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a / b / c = a / (b * c)
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
div_le_div_iff_of_pos_right : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_3 : LinearOrder G₀]
  [inst_4 : ZeroLEOneClass G₀] {a b c : G₀} [inst_5 : PosMulStrictMono G₀] [inst_6 : MulPosStrictMono G₀],
  0 < c → (a / c ≤ b / c ↔ a ≤ b)
mul_div_assoc : ∀ {G : Type u_1} [inst : DivInvMonoid G] (a b c : G), a * b / c = a * (b / c)