import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_1.{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))
  (h2 :
    ∀ (k : ℕ+) (x' : E),
      f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
        ↑(Nesterov.l f f' γ x₀) *
            inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x 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)
  (k : ℕ+)
  (this :
    f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) =
      γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1))))
  (h_original_goal :
    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) *
          inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
            ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x 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) :
  γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1))) ≤
    ↑(Nesterov.l f f' γ x₀) *
        inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
          ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x 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

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)
h2 :
  ∀ (k : ℕ+) (x' : E),
    f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
      ↑(Nesterov.l f f' γ x₀) *
          inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x 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
k : ℕ+
this :
  f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) =
    γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
      (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1)))
⊢ 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) *
        inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
          ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x 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:
rw [this]
    have lzero : 0 < k + (1 : ℝ) := Nat.cast_add_one_pos k
    have hz : γ k ≥ (0 : ℝ) := by to_theorem rw [hg k]apply div_nonneg (by norm_num);linarith

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)
 h2 :
   ∀ (k : ℕ+) (x' : E),
     f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤
       ↑(Nesterov.l f f' γ x₀) *
           inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x 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
 k : ℕ+
 this :
   f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) =
     γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
       (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1)))
 lzero : 0 < ↑↑k + 1
 hz : γ k ≥ 0
 ⊢ γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
       (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1))) ≤
     ↑(Nesterov.l f f' γ x₀) *
         inner (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k)
           ((1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x 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]