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}
  (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)) :
  f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ := 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)
⊢ f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ

Tactic: simp [hk₀]
    rw [add_comm]
    apply hk₀

State after:

Relevant lemmas:
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a