theorem extracted_formal_statement_291.{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)) (hn₁ : φ (n₁ + 1) ≥ 1) : φ (n₁ + 1) - 1 < φ (n₂ + 1) - 1 := sorry

State before:
case a.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_kkt
n₁ n₂ : ℕ
h : n₁ < n₂
h₁ : φ (n₁ + 1) < φ (n₂ + 1)
hn₁ : φ (n₁ + 1) ≥ 1
⊢ φ (n₁ + 1) - 1 < φ (n₂ + 1) - 1

Tactic: apply Nat.sub_lt_sub_right hn₁ h₁

State after:

Relevant lemmas:
Nat.sub_lt_sub_right : ∀ {a b c : ℕ}, c ≤ a → a < b → a - c < b - c