theorem extracted_formal_statement_17.{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_first_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h_original_goal :
    Nesterov_first_fix_stepsize.γ f h f' x0 k ^ 2 / (2 * Nesterov_first_fix_stepsize.t f h f' x0 k) * ‖x0 - xm‖ ^ 2 ≤
      2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 * ‖x0 - xm‖ ^ 2) :
  2 * 2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0) / 2 ≤ 2 * ↑(Nesterov_first_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
alg : Nesterov_first_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
⊢ Nesterov_first_fix_stepsize.γ f h f' x0 k ^ 2 / (2 * Nesterov_first_fix_stepsize.t f h f' x0 k) * ‖x0 - xm‖ ^ 2 ≤
    2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2 * ‖x0 - xm‖ ^ 2

Tactic: apply mul_le_mul_of_nonneg_right
    rw [alg.γeq k, alg.teq k]
    field_simp
    rw [pow_two, add_comm]
    rw [mul_comm ((k + 2 : ℝ) ^ 2), ← div_div, div_le_div_iff_of_pos_right]

State after:
[case h
 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_first_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 2 * 2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0) / 2 ≤ 2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0),
 case h
 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_first_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 0 < (↑k + 2) ^ 2,
 case a0
 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_first_fix_stepsize f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 k : ℕ
 ⊢ 0 ≤ ‖x0 - xm‖ ^ 2]

Relevant lemmas:
pow_two : ∀ {M : Type u_2} [inst : Monoid M] (a : M), a ^ 2 = a * a
mul_le_mul_of_nonneg_right : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_3 : Zero α] [inst_4 : Preorder α]
  [inst_5 : MulPosMono α], b ≤ c → 0 ≤ a → b * a ≤ c * a
div_div : ∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b c : α), a / b / c = a / (b * c)
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
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)
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a