theorem extracted_formal_statement_7.{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)) (k : ℕ+) (x' : E)
  (this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k))
  (t1 : f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x')
  (h_original_goal :
    f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) +
          inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - x') +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2 ≤
      f x' + inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - x') +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2) :
  f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x' := 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)
k : ℕ+
x' : E
this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k)
t1 : f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x'
⊢ f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) +
        inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - x') +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2 ≤
    f x' + inner (f' (Nesterov.y f f' γ x₀ k)) (Nesterov.x f f' γ x₀ ↑k - x') +
      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2

Tactic: rw [add_le_add_iff_right, add_le_add_iff_right]

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)
 k : ℕ+
 x' : E
 this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k)
 t1 : f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x'
 ⊢ f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x']

Relevant lemmas:
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