theorem extracted_formal_statement_7.{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} (n : ℕ+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (h_original_goal :
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
          aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
        bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
      aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
    aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      (bb • (OptProblem.A₁ E₂) yy + bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) := 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
    aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))

Tactic: rw [smul_add]

State after:
[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+
 xx : E₁
 a✝³ : xx ∈ univ
 yy : E₁
 a✝² : yy ∈ univ
 aa bb : ℝ
 a✝¹ : 0 ≤ aa
 a✝ : 0 ≤ bb
 abh : aa + bb = 1
 ⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
         aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
       bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
     aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
       (bb • (OptProblem.A₁ E₂) yy + bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))]

Relevant lemmas:
smul_add : ∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_9 : DistribSMul M A] (a : M) (b₁ b₂ : A),
  a • (b₁ + b₂) = a • b₁ + a • b₂