theorem extracted_formal_statement_56.{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)) (min : IsMinOn f univ xm) (k : ℕ+)
  (h4 :
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2)
  (con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2) (this_1 : γ k ^ 2 > 0)
  (this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))) : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 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)
min : IsMinOn f univ xm
k : ℕ+
h4 :
  1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
    (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
con : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
this✝ : γ k ^ 2 > 0
this : f xm ≤ f (Nesterov.x f f' γ x₀ (↑k - 1))
⊢ (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2

Tactic: exact con

State after:

Relevant lemmas: