import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_4.{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)
  (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)
  (h4 :
    f (Nesterov.x f f' γ x₀ ↑1) - f xm - (1 - γ 1) * (f (Nesterov.x f f' γ x₀ (↑1 - 1)) - f xm) ≤
      ↑(Nesterov.l f f' γ x₀) * γ 1 ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑1 - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑1 - xm‖ ^ 2))
  (h_original_goal :
    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) :
  f (Nesterov.x f f' γ x₀ 1) - f xm ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 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
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
h4 :
  f (Nesterov.x f f' γ x₀ ↑1) - f xm - (1 - γ 1) * (f (Nesterov.x f f' γ x₀ (↑1 - 1)) - f xm) ≤
    ↑(Nesterov.l f f' γ x₀) * γ 1 ^ 2 / 2 *
      (‖Nesterov.v f f' γ x₀ (↑1 - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑1 - 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 ≤
    (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

Tactic:
rw [alg.initial1, sub_self, zero_mul, sub_zero] at h4
    rw [alg.initial1, sub_self, zero_div, zero_mul, zero_add]
    simp
    simp only [PNat.one_coe, Real.rpow_two, one_pow, mul_one, le_refl, tsub_eq_zero_of_le] at h4
    rw [← le_sub_iff_add_le, ← mul_sub]

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
 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
 h4 :
   f (Nesterov.x f f' γ x₀ 1) - f xm ≤
     ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
 ⊢ f (Nesterov.x f f' γ x₀ 1) - f xm ≤
     ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)]