theorem extracted_formal_statement_139.{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} {alg : Nesterov_second_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    2 ^ 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((1 + (↑k + 1)) ^ 2 * 2) ≤
      2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2) :
  2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 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
alg : Nesterov_second_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ 2 ^ 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((1 + (↑k + 1)) ^ 2 * 2) ≤
    2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2

Tactic: rw [pow_two, add_comm]

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
 alg : Nesterov_second_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 2 * 2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / ((↑k + 1 + 1) ^ 2 * 2) ≤
     2 * ↑(Nesterov_second_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2]

Relevant lemmas:
pow_two : ∀ {M : Type u_2} [inst : Monoid M] (a : M), a ^ 2 = a * a
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a