theorem extracted_formal_statement_11.{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] :
  let C := (Φ 0 + 1) ⊔ (Φ 1 + 1);
  ∀ (k : ℕ), Φ k < C → k = 0 → Φ 1 < C := sorry

State before:
case pos
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
C : ℝ := (Φ 0 + 1) ⊔ (Φ 1 + 1)
k : ℕ
h : Φ k < C
hh : k = 0
⊢ Φ 1 < C

Tactic: apply lt_max_iff.2
    right
    linarith

State after:

Relevant lemmas:
lt_max_iff : ∀ {α : Type u} [inst : LinearOrder α] {a b c : α}, a < b ⊔ c ↔ a < b ∨ a < c