theorem extracted_formal_statement_8.{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} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  let φ := fun x => f x + h x;
  (∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)) →
    (∀ (x : E),
        f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
          f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
            proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖Gt x‖ ^ 2) →
      (∀ (x : E),
          f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
            f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
              proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2) →
        (∀ (x z : E), f x + inner (f' x) (z - x) ≤ f z) →
          (∀ (x z : E),
              h (x - proximal_gradient_method.t f h f' x₀ • Gt x) +
                  inner (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x) ≤
                h z) →
            (∀ (x z : E),
                h (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
                  h z - inner (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x)) →
              (∀ (x z : E),
                  φ (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
                    φ z + inner (Gt x) (x - z) - proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2) →
                ∀ (p q : E),
                  inner q p * 2 - proximal_gradient_method.t f h f' x₀ * ‖p‖ ^ 2 =
                    proximal_gradient_method.t f h f' x₀ * inner q p * (proximal_gradient_method.t f h f' x₀)⁻¹ * 2 -
                      (proximal_gradient_method.t f h f' x₀)⁻¹ *
                        (‖proximal_gradient_method.t f h f' x₀‖ * ‖proximal_gradient_method.t f h f' x₀‖ *
                          ‖p‖ ^ 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+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
hG : ∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)
fieq1 :
  ∀ (x : E),
    f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
        proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖Gt x‖ ^ 2
fieq2 :
  ∀ (x : E),
    f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      f x - proximal_gradient_method.t f h f' x₀ * inner (f' x) (Gt x) +
        proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2
fieq3 : ∀ (x z : E), f x + inner (f' x) (z - x) ≤ f z
hieq1 :
  ∀ (x z : E),
    h (x - proximal_gradient_method.t f h f' x₀ • Gt x) +
        inner (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      h z
hieq2 :
  ∀ (x z : E),
    h (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      h z - inner (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x)
univieq :
  ∀ (x z : E),
    φ (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤
      φ z + inner (Gt x) (x - z) - proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2
x p q : E
⊢ inner q p * 2 - proximal_gradient_method.t f h f' x₀ * ‖p‖ ^ 2 =
    proximal_gradient_method.t f h f' x₀ * inner q p * (proximal_gradient_method.t f h f' x₀)⁻¹ * 2 -
      (proximal_gradient_method.t f h f' x₀)⁻¹ *
        (‖proximal_gradient_method.t f h f' x₀‖ * ‖proximal_gradient_method.t f h f' x₀‖ * ‖p‖ ^ 2)

Tactic: simp
    rw [mul_comm _ (inner q p), mul_assoc _ alg.t, mul_inv_cancel₀, ← mul_assoc]
    rw [← mul_assoc, inv_mul_cancel₀]
    simp
    repeat linarith [alg.tpos]

State after:

Relevant lemmas:
mul_inv_cancel₀ : ∀ {G₀ : Type u} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a * a⁻¹ = 1
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
inv_mul_cancel₀ : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a⁻¹ * a = 1