theorem extracted_formal_statement_118.{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_second_fix_stepsize f h f' x0] (n : ℕ+)
  (h_original_goal :
    (1 + (↑↑n + 1) - 2) * (1 + (↑↑n + 1)) ^ 2 / (1 + (↑↑n + 1)) / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / 2 ^ 2 ≤
      (1 + ↑↑n) ^ 2 / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / 2 ^ 2) :
  (1 + (↑↑n + 1) - 2) * (1 + (↑↑n + 1)) ^ 2 / (1 + (↑↑n + 1)) / ↑(Nesterov_second_fix_stepsize.l f h f' x0) ≤
    (1 + ↑↑n) ^ 2 / ↑(Nesterov_second_fix_stepsize.l f h f' x0) := 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_second_fix_stepsize f h f' x0
n : ℕ+
⊢ (1 + (↑↑n + 1) - 2) * (1 + (↑↑n + 1)) ^ 2 / (1 + (↑↑n + 1)) / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / 2 ^ 2 ≤
    (1 + ↑↑n) ^ 2 / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / 2 ^ 2

Tactic: apply div_le_div_of_nonneg_right _ (by positivity)

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_second_fix_stepsize f h f' x0
 n : ℕ+
 ⊢ (1 + (↑↑n + 1) - 2) * (1 + (↑↑n + 1)) ^ 2 / (1 + (↑↑n + 1)) / ↑(Nesterov_second_fix_stepsize.l f h f' x0) ≤
     (1 + ↑↑n) ^ 2 / ↑(Nesterov_second_fix_stepsize.l f h f' x0)]

Relevant lemmas:
div_le_div_of_nonneg_right : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_3 : PartialOrder G₀]
  [inst_4 : ZeroLEOneClass G₀] [inst_5 : PosMulReflectLT G₀] {a b c : G₀} [inst_6 : MulPosMono G₀],
  a ≤ b → 0 ≤ c → a / c ≤ b / c