theorem extracted_formal_statement_38.{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) :
  ‖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

State before:
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‖

Tactic: rw [mul_inv_cancel₀ (by linarith), one_smul, sub_smul, one_smul, add_comm, sub_add]

State after:

Relevant lemmas:
mul_inv_cancel₀ : ∀ {G₀ : Type u} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a * a⁻¹ = 1
sub_add : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a - (b - c)
one_smul : ∀ (M : Type u_1) {α : Type u_5} [inst : Monoid M] [inst_3 : MulAction M α] (b : α), 1 • b = b
sub_smul : ∀ {R : Type u_1} {M : Type u_3} [inst : Ring R] [inst_3 : AddCommGroup M] [inst_4 : Module R M] (r s : R)
  (y : M), (r - s) • y = r • y - s • y
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a