theorem extracted_formal_statement_116.{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 - Nesterov_second_fix_stepsize.γ f h f' x0 (↑n + 1)) * Nesterov_second_fix_stepsize.t f h f' x0 (↑n + 1) /
        Nesterov_second_fix_stepsize.γ f h f' x0 (↑n + 1) ^ 2 ≤
      Nesterov_second_fix_stepsize.t f h f' x0 ↑n / Nesterov_second_fix_stepsize.γ f h f' x0 ↑n ^ 2) :
  (1 - if ↑n + 1 = 0 then 1 / 2 else 2 / (1 + ↑(↑n + 1))) * (1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0)) /
      (if ↑n + 1 = 0 then 1 / 2 else 2 / (1 + ↑(↑n + 1))) ^ 2 ≤
    1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (if ↑n = 0 then 1 / 2 else 2 / (1 + ↑↑n)) ^ 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_second_fix_stepsize f h f' x0
n : ℕ+
⊢ (1 - Nesterov_second_fix_stepsize.γ f h f' x0 (↑n + 1)) * Nesterov_second_fix_stepsize.t f h f' x0 (↑n + 1) /
      Nesterov_second_fix_stepsize.γ f h f' x0 (↑n + 1) ^ 2 ≤
    Nesterov_second_fix_stepsize.t f h f' x0 ↑n / Nesterov_second_fix_stepsize.γ f h f' x0 ↑n ^ 2

Tactic: rw [p.teq n, p.teq (n + 1), p.γeq n, p.γeq (n + 1)]

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 - if ↑n + 1 = 0 then 1 / 2 else 2 / (1 + ↑(↑n + 1))) * (1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0)) /
       (if ↑n + 1 = 0 then 1 / 2 else 2 / (1 + ↑(↑n + 1))) ^ 2 ≤
     1 / ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (if ↑n = 0 then 1 / 2 else 2 / (1 + ↑↑n)) ^ 2]

Relevant lemmas: