theorem extracted_formal_statement_6.{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') :
  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 + (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) +
        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 := 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 + (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) +
        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 [inner_add_right, ← add_assoc]

State after:

Relevant lemmas:
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
inner_add_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_3 : SeminormedAddCommGroup E]
  [inst_4 : InnerProductSpace 𝕜 E] (x y z : E), inner x (y + z) = inner x y + inner x z