theorem extracted_formal_statement_420.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (n : ℕ) :
  let x := (OptProblem.A₁ E₂) (e₁ n);
  let xx := (OptProblem.A₂ E₁) (e₂ n);
  ‖x‖ = ‖x + xx - xx‖ := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
n : ℕ
x : F := (OptProblem.A₁ E₂) (e₁ n)
xx : F := (OptProblem.A₂ E₁) (e₂ n)
⊢ ‖x‖ = ‖x + xx - xx‖

Tactic: rw [← add_sub, sub_self, add_zero]

State after:

Relevant lemmas:
add_sub : ∀ {G : Type u_3} [inst : SubNegMonoid G] (a b c : G), a + (b - c) = a + b - c
sub_self : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a - a = 0
add_zero : ∀ {M : Type u} [inst : AddZeroClass M] (a : M), a + 0 = a