theorem extracted_formal_statement_3.{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)) (k : ℕ+) (x' : E)
  (this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k))
  (t1 : f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x') :
  f (Nesterov.x f f' γ x₀ ↑k) ≤
    f (Nesterov.y f f' γ x₀ k) +
        inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - Nesterov.y 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 := sorry

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)
k : ℕ+
x' : E
this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k)
t1 : f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x'
⊢ f (Nesterov.x f f' γ x₀ ↑k) ≤
    f (Nesterov.y f f' γ x₀ k) +
        inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - Nesterov.y 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

Tactic: exact lipschitz_continuos_upper_bound' alg.diff alg.smooth (alg.y k) (alg.x k)

State after:

Relevant lemmas:
lipschitz_continuos_upper_bound' : ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {l : NNReal},
  (∀ (x₁ : E), HasGradientAt f (f' x₁) x₁) →
    LipschitzWith l f' → ∀ (x y : E), f y ≤ f x + inner (f' x) (y - x) + ↑l / 2 * ‖y - x‖ ^ 2