theorem extracted_formal_statement_251.{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 A₂_x_diff := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n);
  let r_n_1 :=
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂;
  let υ_diff := υ (↑n + 1) - υ ↑n;
  let x_diff := ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n;
  let A₂T := ContinuousLinearMap.adjoint (OptProblem.A₂ E₁);
  ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ = M (n + 1) + ⟪υ_diff, A₂_x_diff⟫_ℝ →
    ⟪υ_diff, A₂_x_diff⟫_ℝ = ⟪A₂T υ_diff, x_diff⟫_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+
A₂_x_diff : F := (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n)
r_n : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂
r_n_1 : F :=
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (↑n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1)) - OptProblem.b E₁ E₂
υ_diff : F := υ (↑n + 1) - υ ↑n
y_diff : F := ADMM.y E₁ E₂ (↑n + 1) - ADMM.y E₁ E₂ ↑n
x_diff : E₂ := ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n
A₂T : F →L[ℝ] E₂ := ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)
h : ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ = M (n + 1) + ⟪υ_diff, A₂_x_diff⟫_ℝ
⊢ ⟪υ_diff, A₂_x_diff⟫_ℝ = ⟪A₂T υ_diff, x_diff⟫_ℝ

Tactic: rw [ContinuousLinearMap.adjoint_inner_left]

State after:

Relevant lemmas:
ContinuousLinearMap.adjoint_inner_left : ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜]
  [inst_10 : NormedAddCommGroup E] [inst_11 : NormedAddCommGroup F] [inst_12 : InnerProductSpace 𝕜 E]
  [inst_13 : InnerProductSpace 𝕜 F] [inst_14 : CompleteSpace E] [inst_15 : CompleteSpace F] (A : E →L[𝕜] F) (x : E)
  (y : F), ⟪(ContinuousLinearMap.adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜