theorem extracted_formal_statement_97.{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} (n : ℕ+)
  (this :
    IsMinOn
        (fun x =>
          Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
            (ADMM.x₁ E₂ F ↑n, x, ADMM.y E₁ E₂ n.natPred))
        univ (ADMM.x₂ E₁ F ↑n) ↔
      0 ∈
        SubderivAt
          (fun x =>
            Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
              (ADMM.x₁ E₂ F ↑n, x, ADMM.y E₁ E₂ n.natPred))
          (ADMM.x₂ E₁ F ↑n))
  (h :
    IsMinOn
      (fun «x₂» =>
        Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
          (ADMM.x₁ E₂ F (n.natPred + 1), «x₂», ADMM.y E₁ E₂ n.natPred))
      univ (ADMM.x₂ E₁ F (n.natPred + 1))) :
  n.natPred + 1 = ↑n := 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+
this :
  IsMinOn
      (fun x =>
        Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
          (ADMM.x₁ E₂ F ↑n, x, ADMM.y E₁ E₂ n.natPred))
      univ (ADMM.x₂ E₁ F ↑n) ↔
    0 ∈
      SubderivAt
        (fun x =>
          Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
            (ADMM.x₁ E₂ F ↑n, x, ADMM.y E₁ E₂ n.natPred))
        (ADMM.x₂ E₁ F ↑n)
h :
  IsMinOn
    (fun «x₂» =>
      Augmented_Lagrangian_Function E₁ E₂ F ADMM.toOptProblem (ADMM.ρ E₁ E₂ F)
        (ADMM.x₁ E₂ F (n.natPred + 1), «x₂», ADMM.y E₁ E₂ n.natPred))
    univ (ADMM.x₂ E₁ F (n.natPred + 1))
⊢ n.natPred + 1 = ↑n

Tactic: apply PNat.natPred_add_one

State after:

Relevant lemmas:
PNat.natPred_add_one : ∀ (n : ℕ+), n.natPred + 1 = ↑n