theorem extracted_formal_statement_3.{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))
  (this : f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ) :
  f (subgradient_method.x f x₀ 0) - f xm ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀) := 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)
this : f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
⊢ f (subgradient_method.x f x₀ 0) - f xm ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀)

Tactic: apply le_trans this _
    rw [← inner_neg_right, neg_sub, alg.initial]
    apply le_trans (real_inner_le_norm _ _) _
    rw [mul_comm]
    apply mul_le_mul_of_nonneg_left h₀' (norm_nonneg _)

State after:

Relevant lemmas:
inner_neg_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_2 : SeminormedAddCommGroup E]
  [inst_3 : InnerProductSpace 𝕜 E] (x y : E), ⟪x, -y⟫_𝕜 = -⟪x, y⟫_𝕜
le_trans : ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b ≤ c → a ≤ c
mul_le_mul_of_nonneg_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_2 : Zero α] [inst_3 : Preorder α]
  [inst_4 : PosMulMono α], b ≤ c → 0 ≤ a → a * b ≤ a * c
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
neg_sub : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), -(a - b) = b - a
norm_nonneg : ∀ {E : Type u_5} [inst : SeminormedAddGroup E] (a : E), 0 ≤ ‖a‖
real_inner_le_norm : ∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_2 : InnerProductSpace ℝ F] (x y : F),
  ⟪x, y⟫_ℝ ≤ ‖x‖ * ‖y‖