-- File: optlib/Optlib/Algorithm/ADMM/Lemma.lean
Theorem statement:
theorem extracted_formal_statement_8.{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) :
  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

Tactic 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 + bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))

Tactic: rw [← add_assoc, add_assoc (aa • A₁ (xx)), add_comm (bb • A₁ (yy)), ← add_assoc]

Tactic state after:

Premises used:
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