import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_9.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1))
  (h2 :
    ∀ (k : ℕ+) (x' : E),
      f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
        ↑(Nesterov.l f f' γ x₀) *
            inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x f f' γ x₀ ↑k) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2)
  (h3 :
    ∀ (k : ℕ+),
      f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
        ↑(Nesterov.l f f' γ x₀) *
            inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
              ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x f f' γ x₀ ↑k) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2)
  (k : ℕ+) (hz : γ k ≥ 0) (hzs : γ k > 0)
  (h_original_goal :
    ↑(Nesterov.l f f' γ x₀) / 2 *
        (‖Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2 -
          ‖γ k •
                  (Nesterov.x f f' γ x₀ (↑k - 1) +
                    (1 / γ k) • (Nesterov.x f f' γ x₀ ↑k - Nesterov.x f f' γ x₀ (↑k - 1))) -
                γ k • xm‖ ^
            2) =
      ↑(Nesterov.l f f' γ x₀) / 2 *
        (‖Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2 -
          ‖Nesterov.x f f' γ x₀ ↑k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2)) :
  ‖γ k • Nesterov.x f f' γ x₀ (↑k - 1) + (γ k * (γ k)⁻¹) • (Nesterov.x f f' γ x₀ ↑k - Nesterov.x f f' γ x₀ (↑k - 1)) -
        γ k • xm‖ =
    ‖Nesterov.x f f' γ x₀ ↑k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
h2 :
  ∀ (k : ℕ+) (x' : E),
    f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
      ↑(Nesterov.l f f' γ x₀) *
          inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x f f' γ x₀ ↑k) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2
h3 :
  ∀ (k : ℕ+),
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) *
          inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
            ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x f f' γ x₀ ↑k) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2
k : ℕ+
hz : γ k ≥ 0
hzs : γ k > 0
⊢ ↑(Nesterov.l f f' γ x₀) / 2 *
      (‖Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2 -
        ‖γ k • (Nesterov.x f f' γ x₀ (↑k - 1) + (1 / γ k) • (Nesterov.x f f' γ x₀ ↑k - Nesterov.x f f' γ x₀ (↑k - 1))) -
              γ k • xm‖ ^
          2) =
    ↑(Nesterov.l f f' γ x₀) / 2 *
      (‖Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2 -
        ‖Nesterov.x f f' γ x₀ ↑k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖ ^ 2)

Tactic:
rw [smul_add, smul_smul]
    simp
    left

Tactic state after:
[case h
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm x₀ : E
 γ : ℕ+ → ℝ
 alg : Nesterov f f' γ x₀
 hfun : ConvexOn ℝ univ f
 hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
 h2 :
   ∀ (k : ℕ+) (x' : E),
     f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
       ↑(Nesterov.l f f' γ x₀) *
           inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x f f' γ x₀ ↑k) +
         ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2
 h3 :
   ∀ (k : ℕ+),
     f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
       ↑(Nesterov.l f f' γ x₀) *
           inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
             ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x f f' γ x₀ ↑k) +
         ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2
 k : ℕ+
 hz : γ k ≥ 0
 hzs : γ k > 0
 ⊢ ‖γ k • Nesterov.x f f' γ x₀ (↑k - 1) + (γ k * (γ k)⁻¹) • (Nesterov.x f f' γ x₀ ↑k - Nesterov.x f f' γ x₀ (↑k - 1)) -
         γ k • xm‖ =
     ‖Nesterov.x f f' γ x₀ ↑k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - γ k • xm‖]