theorem extracted_formal_statement_6.{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),
                inner (Gt x) (x - z) + (h z - inner (Gt x) (proximal_gradient_method.t f h f' x₀ • Gt x)) +
                      (f z + proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2) =
                    inner (Gt x) (x - z) + (φ z - proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2) →
                  f z + h z + -proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2 =
                    φ z + -(proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 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)
x z : E
⊢ inner (Gt x) (x - z) + (h z - inner (Gt x) (proximal_gradient_method.t f h f' x₀ • Gt x)) +
      (f z + proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2) =
    inner (Gt x) (x - z) + (φ z - proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2)

Tactic: rw [add_assoc, add_left_cancel_iff]
    rw [inner_smul_right, real_inner_self_eq_norm_sq]
    rw [add_comm_sub, ← add_sub]
    have (a : ℝ) : alg.t / 2 * a - alg.t * a = -alg.t / 2 * a := by ring
    rw [this, sub_eq_add_neg, ← add_assoc, add_comm (h z) (f z)]

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+
 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)
 x z : E
 this :
   ∀ (a : ℝ),
     proximal_gradient_method.t f h f' x₀ / 2 * a - proximal_gradient_method.t f h f' x₀ * a =
       -proximal_gradient_method.t f h f' x₀ / 2 * a
 ⊢ f z + h z + -proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2 =
     φ z + -(proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2)]

Relevant lemmas:
sub_eq_add_neg : ∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
add_sub : ∀ {G : Type u_3} [inst : SubNegMonoid G] (a b c : G), a + (b - c) = a + b - c
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
real_inner_self_eq_norm_sq : ∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F]
  (x : F), inner x x = ‖x‖ ^ 2
add_comm_sub : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a + (c - b)
add_left_cancel_iff : ∀ {G : Type u_1} [inst : Add G] [inst_4 : IsLeftCancelAdd G] {a b c : G}, a + b = a + c ↔ b = c
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a
inner_smul_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_4 : SeminormedAddCommGroup E]
  [inst_5 : InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜), inner x (r • y) = r * inner x y