theorem extracted_formal_statement_80 : 0 < 2 ^ 2 := sorry

State before:
case hb
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
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h4 :
  ∀ (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) * γ k ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)
h5 :
  ∀ (k : ℕ+),
    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 - 1) ^ 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
h6 :
  ∀ (k : ℕ+),
    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 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h7 :
  1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 ≤
    (1 - γ 1) / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 0) - f xm) +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2
k : ℕ+
h8 :
  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 ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2
h9 : 1 / (2 / (↑↑k + 1)) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) ≤ ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2
h10 :
  ↑(Nesterov.l f f' γ x₀) / 2 * ‖x₀ - xm‖ ^ 2 / (1 / (2 / (↑↑k + 1)) ^ 2) =
    2 * ↑(Nesterov.l f f' γ x₀) / (↑↑k + 1) ^ 2 * ‖x₀ - xm‖ ^ 2
⊢ 0 < 2 ^ 2

Tactic: simp only [gt_iff_lt, zero_lt_two, pow_pos]

State after:

Relevant lemmas:
gt_iff_lt : ∀ {α : Type u} [inst : LT α] {x y : α}, x > y ↔ y < x
pow_pos : ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] [inst_1 : PartialOrder M₀] {a : M₀} [inst_2 : ZeroLEOneClass M₀]
  [inst_3 : PosMulStrictMono M₀], 0 < a → ∀ (n : ℕ), 0 < a ^ n
zero_lt_two : ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_1 : PartialOrder α] [inst_2 : ZeroLEOneClass α]
  [inst_3 : NeZero 1] [inst_4 : AddLeftMono α], 0 < 2