theorem extracted_formal_statement_76.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_first f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm)
  (h1 :
    ∀ (k : ℕ),
      Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
          Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
        SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1)))
  (hieq1 :
    ∀ (z : E) (k : ℕ),
      Nesterov_first.t f h f' x0 k * h (Nesterov_first.x f h f' x0 (k + 1)) +
          inner
            (Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
              Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
            (z - Nesterov_first.x f h f' x0 (k + 1)) ≤
        Nesterov_first.t f h f' x0 k * h z)
  (hieq2 :
    ∀ (z : E) (k : ℕ),
      h (Nesterov_first.x f h f' x0 (k + 1)) ≤
        h z +
          inner
            (f' (Nesterov_first.y f h f' x0 k) +
              (1 / Nesterov_first.t f h f' x0 k) • (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k))
            (z - Nesterov_first.x f h f' x0 (k + 1)))
  (fieq1 :
    ∀ (k : ℕ) (x y : E), f x ≤ f y + inner (f' y) (x - y) + 1 / (2 * Nesterov_first.t f h f' x0 k) * ‖x - y‖ ^ 2) :
  let φ := fun z => f z + h z;
  (∀ (z : E) (k : ℕ),
      φ (Nesterov_first.x f h f' x0 (k + 1)) ≤
        φ z +
            1 / Nesterov_first.t f h f' x0 k *
              inner (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)
                (z - Nesterov_first.x f h f' x0 (k + 1)) +
          1 / (2 * Nesterov_first.t f h f' x0 k) *
            ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2) →
    (∀ (k : ℕ),
        φ (Nesterov_first.x f h f' x0 (k + 1)) - φ xm -
            (1 - Nesterov_first.γ f h f' x0 k) * (φ (Nesterov_first.x f h f' x0 k) - φ xm) ≤
          1 / Nesterov_first.t f h f' x0 k *
              inner (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)
                ((1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k + Nesterov_first.γ f h f' x0 k • xm -
                  Nesterov_first.x f h f' x0 (k + 1)) +
            1 / (2 * Nesterov_first.t f h f' x0 k) *
              ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2) →
      let v := fun k =>
        Nesterov_first.x f h f' x0 (↑k - 1) +
          (1 / Nesterov_first.γ f h f' x0 (↑k - 1)) •
            (Nesterov_first.x f h f' x0 ↑k - Nesterov_first.x f h f' x0 (↑k - 1));
      (∀ (k : ℕ+),
          Nesterov_first.y f h f' x0 ↑k =
            (1 - Nesterov_first.γ f h f' x0 ↑k) • Nesterov_first.x f h f' x0 ↑k + Nesterov_first.γ f h f' x0 ↑k • v k) →
        ∀ (k : ℕ),
          (∀ (a b : E),
              1 / Nesterov_first.t f h f' x0 k * inner (a - b) (-a) +
                  1 / (2 * Nesterov_first.t f h f' x0 k) * ‖a - b‖ ^ 2 =
                1 / (2 * Nesterov_first.t f h f' x0 k) * (‖b‖ ^ 2 - ‖a‖ ^ 2)) →
            let a :=
              Nesterov_first.x f h f' x0 (k + 1) - (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k -
                Nesterov_first.γ f h f' x0 k • xm;
            let b :=
              Nesterov_first.y f h f' x0 k - (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k -
                Nesterov_first.γ f h f' x0 k • xm;
            Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k = a - b →
              (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k + Nesterov_first.γ f h f' x0 k • xm -
                  Nesterov_first.x f h f' x0 (k + 1) =
                Nesterov_first.γ f h f' x0 k • xm -
                  (Nesterov_first.x f h f' x0 (k + 1) -
                    (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k) := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_first f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
h1 :
  ∀ (k : ℕ),
    Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
        Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈
      SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1))
hieq1 :
  ∀ (z : E) (k : ℕ),
    Nesterov_first.t f h f' x0 k * h (Nesterov_first.x f h f' x0 (k + 1)) +
        inner
          (Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -
            Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
          (z - Nesterov_first.x f h f' x0 (k + 1)) ≤
      Nesterov_first.t f h f' x0 k * h z
hieq2 :
  ∀ (z : E) (k : ℕ),
    h (Nesterov_first.x f h f' x0 (k + 1)) ≤
      h z +
        inner
          (f' (Nesterov_first.y f h f' x0 k) +
            (1 / Nesterov_first.t f h f' x0 k) • (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k))
          (z - Nesterov_first.x f h f' x0 (k + 1))
fieq1 : ∀ (k : ℕ) (x y : E), f x ≤ f y + inner (f' y) (x - y) + 1 / (2 * Nesterov_first.t f h f' x0 k) * ‖x - y‖ ^ 2
φ : E → ℝ := fun z => f z + h z
φieq2 :
  ∀ (z : E) (k : ℕ),
    φ (Nesterov_first.x f h f' x0 (k + 1)) ≤
      φ z +
          1 / Nesterov_first.t f h f' x0 k *
            inner (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)
              (z - Nesterov_first.x f h f' x0 (k + 1)) +
        1 / (2 * Nesterov_first.t f h f' x0 k) * ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2
φieq3 :
  ∀ (k : ℕ),
    φ (Nesterov_first.x f h f' x0 (k + 1)) - φ xm -
        (1 - Nesterov_first.γ f h f' x0 k) * (φ (Nesterov_first.x f h f' x0 k) - φ xm) ≤
      1 / Nesterov_first.t f h f' x0 k *
          inner (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)
            ((1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k + Nesterov_first.γ f h f' x0 k • xm -
              Nesterov_first.x f h f' x0 (k + 1)) +
        1 / (2 * Nesterov_first.t f h f' x0 k) * ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2
v : ℕ+ → E :=
  fun k =>
    Nesterov_first.x f h f' x0 (↑k - 1) +
      (1 / Nesterov_first.γ f h f' x0 (↑k - 1)) • (Nesterov_first.x f h f' x0 ↑k - Nesterov_first.x f h f' x0 (↑k - 1))
eq :
  ∀ (k : ℕ+),
    Nesterov_first.y f h f' x0 ↑k =
      (1 - Nesterov_first.γ f h f' x0 ↑k) • Nesterov_first.x f h f' x0 ↑k + Nesterov_first.γ f h f' x0 ↑k • v k
k : ℕ
aux :
  ∀ (a b : E),
    1 / Nesterov_first.t f h f' x0 k * inner (a - b) (-a) + 1 / (2 * Nesterov_first.t f h f' x0 k) * ‖a - b‖ ^ 2 =
      1 / (2 * Nesterov_first.t f h f' x0 k) * (‖b‖ ^ 2 - ‖a‖ ^ 2)
a : E :=
  Nesterov_first.x f h f' x0 (k + 1) - (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k -
    Nesterov_first.γ f h f' x0 k • xm
b : E :=
  Nesterov_first.y f h f' x0 k - (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k -
    Nesterov_first.γ f h f' x0 k • xm
h7 : Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k = a - b
⊢ (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k + Nesterov_first.γ f h f' x0 k • xm -
      Nesterov_first.x f h f' x0 (k + 1) =
    Nesterov_first.γ f h f' x0 k • xm -
      (Nesterov_first.x f h f' x0 (k + 1) - (1 - Nesterov_first.γ f h f' x0 k) • Nesterov_first.x f h f' x0 k)

Tactic: rw [← sub_add, sub_add_comm, sub_add_eq_add_sub]

State after:

Relevant lemmas:
sub_add_eq_add_sub : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a + c - b
sub_add_comm : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = c - b + a
sub_add : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a - (b - c)