theorem extracted_formal_statement_7.{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)
  (z : E) (k : ℕ) :
  f (Nesterov_first.y f h f' x0 k) +
            inner (f' (Nesterov_first.y f h f' x0 k))
              (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k) +
          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 +
        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)) =
    f (Nesterov_first.y f h f' x0 k) + inner (f' (Nesterov_first.y f h f' x0 k)) (z - Nesterov_first.y f h f' x0 k) +
          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 +
      h z := 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
z : E
k : ℕ
⊢ f (Nesterov_first.y f h f' x0 k) +
            inner (f' (Nesterov_first.y f h f' x0 k))
              (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k) +
          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 +
        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)) =
    f (Nesterov_first.y f h f' x0 k) + inner (f' (Nesterov_first.y f h f' x0 k)) (z - Nesterov_first.y f h f' x0 k) +
          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 +
      h z

Tactic: rw [add_assoc _ _ (h z), add_right_comm, add_assoc _ _ (h z)]
    rw [add_right_cancel_iff, inner_add_left, ← add_assoc, add_assoc (f (alg.y k))]
    rw [← inner_add_right]
    simp
    rw [real_inner_smul_left]

State after:

Relevant lemmas:
add_right_cancel_iff : ∀ {G : Type u_1} [inst : Add G] [inst_3 : IsRightCancelAdd G] {a b c : G}, b + a = c + a ↔ b = c
add_right_comm : ∀ {G : Type u_3} [inst : AddCommSemigroup G] (a b c : G), a + b + c = a + c + b
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
real_inner_smul_left : ∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F] (x y : F)
  (r : ℝ), inner (r • x) y = r * inner x y
inner_add_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_3 : SeminormedAddCommGroup E]
  [inst_4 : InnerProductSpace 𝕜 E] (x y z : E), inner x (y + z) = inner x y + inner x z
inner_add_left : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_3 : SeminormedAddCommGroup E]
  [inst_4 : InnerProductSpace 𝕜 E] (x y z : E), inner (x + y) z = inner x z + inner y z