theorem extracted_formal_statement_290.{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] ⦃n₁ n₂ : ℕ⦄
  (h : n₁ < n₂) (h₁ : φ (n₁ + 1) < φ (n₂ + 1)) : φ (n₁ + 1) ≥ n₁ + 1 := 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_kkt
n₁ n₂ : ℕ
h : n₁ < n₂
h₁ : φ (n₁ + 1) < φ (n₂ + 1)
⊢ φ (n₁ + 1) ≥ n₁ + 1

Tactic: apply StrictMono.id_le (hphi_StrictMono fullrank₁ fullrank₂)

State after:

Relevant lemmas:
StrictMono.id_le : ∀ {β : Type u_2} [inst : LinearOrder β] [inst_10 : WellFoundedLT β] {f : β → β},
  StrictMono f → id ≤ f