theorem extracted_formal_statement_28.{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}
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ)
  (h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀))
  (h_original_goal :
    2 * (subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm)) ≤
      2 * (‖x₀ - xm‖ * (↑(subgradient_method.G f x₀) * subgradient_method.a f x₀ 0))) :
  subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm) ≤
    ‖x₀ - xm‖ * (↑(subgradient_method.G f x₀) * subgradient_method.a f x₀ 0) := 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
hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)
⊢ 2 * (subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm)) ≤
    2 * (‖x₀ - xm‖ * (↑(subgradient_method.G f x₀) * subgradient_method.a f x₀ 0))

Tactic: apply (mul_le_mul_left two_pos).mpr

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
 hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
 h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)
 ⊢ subgradient_method.a f x₀ 0 * (f (subgradient_method.x f x₀ 0) - f xm) ≤
     ‖x₀ - xm‖ * (↑(subgradient_method.G f x₀) * subgradient_method.a f x₀ 0)]

Relevant lemmas:
mul_le_mul_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_2 : Zero α] [inst_3 : Preorder α]
  [inst_4 : PosMulMono α] [inst_5 : PosMulReflectLE α], 0 < a → (a * b ≤ a * c ↔ b ≤ c)
two_pos : ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_2 : PartialOrder α] [inst_3 : ZeroLEOneClass α]
  [inst_4 : NeZero 1] [inst_5 : AddLeftMono α], 0 < 2