-- File: optlib/Optlib/Algorithm/ADMM/Theroem_converge.lean
Theorem statement:
theorem extracted_formal_statement_210.{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] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n → |Φ (Nat.succ 0)| + |Φ n.succ| < r + r := sorry

Tactic state before:
case hab
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
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ |Φ (Nat.succ 0)| + |Φ n.succ| < r + r

Tactic: apply add_lt_add (hr (Nat.succ 0)) (hr (Nat.succ n))

Tactic state after:

Premises used:
add_lt_add : ∀ {α : Type u_1} [inst : Add α] [inst_10 : Preorder α] [inst_11 : AddLeftStrictMono α]
  [inst_12 : AddRightStrictMono α] {a b c d : α}, a < b → c < d → a + c < b + d