-- File: optlib/Optlib/Algorithm/ADMM/Lemma.lean
Theorem statement:
theorem extracted_formal_statement_156.{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)
  (split_ey : ey (n + 1) = ADMM.y E₁ E₂ (n + 1) - admm_kkt.y)
  (h_original_goal :
    ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1)) +
          (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y) +
        ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) =
      ADMM.y E₁ E₂ (n + 1) +
            (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
              ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
          ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
        admm_kkt.y) :
  ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
      (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y +
        (ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) -
          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)))) =
    ADMM.y E₁ E₂ (n + 1) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        (ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) -
          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
      admm_kkt.y := 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
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
e' : E₁ := e₁ (n + 1)
block : F :=
  -ey (n + 1) -
      ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
    ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
u' : E₁ :=
  -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
      (ADMM.y E₁ E₂ (n + 1) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)))
x_diff : E₁ := ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁
succ_n : ℕ+ := (n + 1).toPNat'
block₁ : F :=
  ADMM.y E₁ E₂ (n + 1) +
      ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
    ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
split_ey : ey (n + 1) = ADMM.y E₁ E₂ (n + 1) - admm_kkt.y
⊢ ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1)) +
        (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y) +
      ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) =
    ADMM.y E₁ E₂ (n + 1) +
          (((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₁ E₂) (e₁ (n + 1)) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • (OptProblem.A₂ E₁) (e₂ (n + 1))) +
        ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
      admm_kkt.y

Tactic: rw [add_assoc, ← smul_add, smul_sub]

Tactic 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
 Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
 e' : E₁ := e₁ (n + 1)
 block : F :=
   -ey (n + 1) -
       ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
     ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
 u' : E₁ :=
   -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
       (ADMM.y E₁ E₂ (n + 1) +
           ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
             ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
         ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)))
 x_diff : E₁ := ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁
 succ_n : ℕ+ := (n + 1).toPNat'
 block₁ : F :=
   ADMM.y E₁ E₂ (n + 1) +
       ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
     ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
 split_ey : ey (n + 1) = ADMM.y E₁ E₂ (n + 1) - admm_kkt.y
 ⊢ ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
       (ADMM.y E₁ E₂ (n + 1) - admm_kkt.y +
         (ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) -
           ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)))) =
     ADMM.y E₁ E₂ (n + 1) +
           ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
             ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) +
         (ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) -
           ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1))) -
       admm_kkt.y]

Premises used:
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
smul_add : ∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_10 : DistribSMul M A] (a : M) (b₁ b₂ : A),
  a • (b₁ + b₂) = a • b₁ + a • b₂
smul_sub : ∀ {M : Type u_1} {A : Type u_7} [inst : Monoid M] [inst_10 : AddGroup A] [inst_11 : DistribMulAction M A]
  (r : M) (x y : A), r • (x - y) = r • x - r • y