import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_18.{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)
  (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 / γ k ^ 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)
  (h_original_goal : f (Nesterov.x f f' γ x₀ ↑k) - f xm ≤ 2 * ↑(Nesterov.l f f' γ x₀) / (↑↑k + 1) ^ 2 * ‖x₀ - xm‖ ^ 2) :
  0 < 1 / (2 / (↑↑k + 1)) ^ 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)
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 / γ k ^ 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
⊢ f (Nesterov.x f f' γ x₀ ↑k) - f xm ≤ 2 * ↑(Nesterov.l f f' γ x₀) / (↑↑k + 1) ^ 2 * ‖x₀ - xm‖ ^ 2

Tactic:
rw [hg k] at h9
    rw [← le_div_iff₀'] at h9
    ·
  rw [h10] at h9
  exact h9

Tactic state after:
[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 < 1 / (2 / (↑↑k + 1)) ^ 2]