theorem extracted_formal_statement_202.{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_kkt] (n : ℕ)
  (this : φ (n + 1) > 0) : ↑(φ₁ n) = φ (n + 1) := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝¹⁰ : NormedAddCommGroup E₁
inst✝⁹ : InnerProductSpace ℝ E₁
inst✝⁸ : FiniteDimensional ℝ E₁
inst✝⁷ : NormedAddCommGroup E₂
inst✝⁶ : InnerProductSpace ℝ E₂
inst✝⁵ : FiniteDimensional ℝ E₂
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : 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✝¹ inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
this : φ (n + 1) > 0
⊢ ↑(φ₁ n) = φ (n + 1)

Tactic: exact Nat.sub_one_add_one_eq_of_pos this

State after:

Relevant lemmas:
Nat.sub_one_add_one_eq_of_pos : ∀ {n : ℕ}, 0 < n → n - 1 + 1 = n