theorem extracted_formal_statement_31.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  let A_x_sum := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  let ey' := ey (n + 1);
  let Ae1 := (OptProblem.A₁ E₂) (e₁ (n + 1));
  let Ae2 := (OptProblem.A₂ E₁) (e₂ (n + 1));
  ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1⟫_ℝ +
        ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
    ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum + ADMM.ρ E₁ E₂ F • A_x_sum, Ae1⟫_ℝ := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1⟫_ℝ +
        ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
    ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum + ADMM.ρ E₁ E₂ F • A_x_sum, Ae1⟫_ℝ

Tactic: rw [inner_add_left, add_assoc]
    rw [inner_smul_left A_x_sum Ae1 ρ, starRingEnd_eq_R, add_comm]
    ring

State after:

Relevant lemmas:
inner_smul_left : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_10 : SeminormedAddCommGroup E]
  [inst_11 : InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜), ⟪r • x, y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪x, y⟫_𝕜
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a
inner_add_left : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_10 : SeminormedAddCommGroup E]
  [inst_11 : InnerProductSpace 𝕜 E] (x y z : E), ⟪x + y, z⟫_𝕜 = ⟪x, z⟫_𝕜 + ⟪y, z⟫_𝕜