theorem extracted_formal_statement_42.{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)
  (hi : ‖subgradient_method.g f x₀ i‖ ≤ ↑(subgradient_method.G f x₀))
  (h_original_goal :
    ‖subgradient_method.g f x₀ i‖ ^ 2 * subgradient_method.a f x₀ i ^ 2 ≤
      ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2) :
  ‖subgradient_method.g f x₀ i‖ ^ 2 ≤ ↑(subgradient_method.G f x₀) ^ 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
hi : ‖subgradient_method.g f x₀ i‖ ≤ ↑(subgradient_method.G f x₀)
⊢ ‖subgradient_method.g f x₀ i‖ ^ 2 * subgradient_method.a f x₀ i ^ 2 ≤
    ↑(subgradient_method.G f x₀) ^ 2 * subgradient_method.a f x₀ i ^ 2

Tactic: apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)

State after:
[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
 hi : ‖subgradient_method.g f x₀ i‖ ≤ ↑(subgradient_method.G f x₀)
 ⊢ ‖subgradient_method.g f x₀ i‖ ^ 2 ≤ ↑(subgradient_method.G f x₀) ^ 2]

Relevant lemmas:
mul_le_mul_of_nonneg_right : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_2 : Zero α] [inst_3 : Preorder α]
  [inst_4 : MulPosMono α], b ≤ c → 0 ≤ a → b * a ≤ c * a
sq_nonneg : ∀ {α : Type u} [inst : Semiring α] [inst_2 : LinearOrder α] [inst_3 : IsRightCancelAdd α]
  [inst_4 : ZeroLEOneClass α] [inst_5 : ExistsAddOfLE α] [inst_6 : PosMulMono α] [inst_7 : AddLeftStrictMono α] (a : α),
  0 ≤ a ^ 2