import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_35.{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)
  (update1 : Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) = γ k • Nesterov.v f f' γ x₀ (↑k - 1)) :
  ↑(Nesterov.l f f' γ x₀) / 2 *
      (‖γ k • (Nesterov.v f f' γ x₀ (↑k - 1) - xm)‖ ^ 2 - ‖γ k • (Nesterov.v f f' γ x₀ ↑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 -
        ‖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) := 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
update1 : Nesterov.y f f' γ x₀ k - (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) = γ k • Nesterov.v f f' γ x₀ (↑k - 1)
⊢ ↑(Nesterov.l f f' γ x₀) / 2 *
      (‖γ k • (Nesterov.v f f' γ x₀ (↑k - 1) - xm)‖ ^ 2 - ‖γ k • (Nesterov.v f f' γ x₀ ↑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 -
        ‖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)

Tactic:
rw [smul_sub, smul_sub, update1, alg.update3 k]

Tactic state after: