theorem extracted_formal_statement_378.{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)]
  (h_add :
    Tendsto
      (fun n =>
        1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)
      atTop (𝓝 0))
  (h_add' :
    Tendsto
      (fun n =>
        1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 +
          ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
            ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)
      atTop (𝓝 0))
  (n : ℕ) :
  Φ (φ n) =
    1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 +
      ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 := sorry

State before:
case h
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)
h_add :
  Tendsto
    (fun n =>
      1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)
    atTop (𝓝 0)
h_add' :
  Tendsto
    (fun n =>
      1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 +
        ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
          ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)
    atTop (𝓝 0)
n : ℕ
⊢ Φ (φ n) =
    1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 +
      ((1 - ADMM.τ E₁ E₂ F) ⊔ (1 - 1 / ADMM.τ E₁ E₂ F)) * ADMM.ρ E₁ E₂ F *
        ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2

Tactic: rw [Φ, Ψ]

State after:

Relevant lemmas: