theorem extracted_formal_statement_54.{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 : γ k ^ 2 > 0)
  (h_original_goal :
    (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 ≤
      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) :
  (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
    1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) := 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
⊢ (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 ≤
    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

Tactic: simp only [Real.rpow_two, ge_iff_le, add_le_add_iff_right, gt_iff_lt, sub_pos, sub_neg]

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
 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
 ⊢ (1 - γ k) / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤
     1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm)]

Relevant lemmas:
sub_neg : ∀ {α : Type u} [inst : AddGroup α] [inst_3 : LT α] [inst_4 : AddRightStrictMono α] {a b : α},
  a - b < 0 ↔ a < b
gt_iff_lt : ∀ {α : Type u} [inst : LT α] {x y : α}, x > y ↔ y < x
sub_pos : ∀ {α : Type u} [inst : AddGroup α] [inst_3 : LT α] [inst_4 : AddRightStrictMono α] {a b : α},
  0 < a - b ↔ b < a
Real.rpow_two : ∀ (x : ℝ), x ^ 2 = x ^ 2
ge_iff_le : ∀ {α : Type u} [inst : LE α] {x y : α}, x ≥ y ↔ y ≤ x
add_le_add_iff_right : ∀ {α : Type u_1} [inst : Add α] [inst_3 : LE α] [inst_4 : AddRightMono α]
  [inst_5 : AddRightReflectLE α] (a : α) {b c : α}, b + a ≤ c + a ↔ b ≤ c