theorem extracted_formal_statement_6.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (k : ℕ)
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) (k₀ : ¬k = 0)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (this :
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
          2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
        ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2 =
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
          -(2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
        ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2)
  (inq₁ :
    ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 ≤
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2)
  (inq₂ :
    2 * subgradient_method.a f x₀ i *
        (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) ≤
      2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ) :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 ≤
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2 := sorry

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
k : ℕ
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
k₀ : ¬k = 0
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
i : ℕ
left✝ : i ≥ 0
hi₂ : i ≤ k
this :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
        2 * subgradient_method.a f x₀ i *
          (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2 =
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2
inq₁ :
  ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 ≤
    ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2
inq₂ :
  2 * subgradient_method.a f x₀ i *
      (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) ≤
    2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ
⊢ ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * ⟪subgradient_method.x f x₀ i - xm, subgradient_method.a f x₀ i • subgradient_method.g f x₀ i⟫_ℝ) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 ≤
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2

Tactic: rw [add_assoc, add_assoc]
    apply add_le_add_left
    apply add_le_add
    · apply neg_le_neg; apply inq₂
    · apply inq₁

State after:

Relevant lemmas:
neg_le_neg : ∀ {α : Type u} [inst : OrderedAddCommGroup α] {a b : α}, a ≤ b → -b ≤ -a
add_le_add : ∀ {α : Type u_1} [inst : Add α] [inst_2 : Preorder α] [inst_3 : AddLeftMono α] [inst_4 : AddRightMono α]
  {a b c d : α}, a ≤ b → c ≤ d → a + c ≤ b + d
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
add_le_add_left : ∀ {α : Type u_1} [inst : Add α] [inst_2 : LE α] [inst_3 : AddLeftMono α] {b c : α},
  b ≤ c → ∀ (a : α), a + b ≤ a + c