theorem extracted_formal_statement_0.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (k : ℕ+)
  (h_original_goal :
    f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
          f (proximal_gradient_method.xm f h f' x₀) -
        h (proximal_gradient_method.xm f h f' x₀) ≤
      1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2) :
  2 * ↑↑k * proximal_gradient_method.t f h f' x₀ *
      (f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
          f (proximal_gradient_method.xm f h f' x₀) -
        h (proximal_gradient_method.xm f h f' x₀)) ≤
    ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2 := sorry

State before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
⊢ f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
        f (proximal_gradient_method.xm f h f' x₀) -
      h (proximal_gradient_method.xm f h f' x₀) ≤
    1 / (2 * ↑↑k * proximal_gradient_method.t f h f' x₀) * ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2

Tactic: rw [mul_comm, mul_one_div, le_div_iff₀, mul_comm]

State after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 2 * ↑↑k * proximal_gradient_method.t f h f' x₀ *
       (f (proximal_gradient_method.x f h f' x₀ ↑k) + h (proximal_gradient_method.x f h f' x₀ ↑k) -
           f (proximal_gradient_method.xm f h f' x₀) -
         h (proximal_gradient_method.xm f h f' x₀)) ≤
     ‖x₀ - proximal_gradient_method.xm f h f' x₀‖ ^ 2,
 E : Type u_1
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : CompleteSpace E
 inst✝ : ProperSpace E
 x₀ : E
 f : E → ℝ
 f' : E → E
 h : E → ℝ
 alg : proximal_gradient_method f h f' x₀
 k : ℕ+
 ⊢ 0 < 2 * ↑↑k * proximal_gradient_method.t f h f' x₀]

Relevant lemmas:
le_div_iff₀ : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_4 : PartialOrder G₀] [inst_5 : ZeroLEOneClass G₀]
  [inst_6 : PosMulReflectLT G₀] {a b c : G₀} [inst_7 : MulPosMono G₀], 0 < c → (a ≤ b / c ↔ a * c ≤ b)
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_one_div : ∀ {G : Type u_3} [inst : DivInvMonoid G] (x y : G), x * (1 / y) = x / y