theorem extracted_formal_statement_14.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (min : IsMinOn f univ xm)
  (con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2)
  (h5 :
    ∀ (k : ℕ+),
      1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
        1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2)
  (h6 :
    ∀ (k : ℕ+),
      1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
        1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
          ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2)
  (h4 :
    f (Nesterov.x f f' γ x₀ ↑1) - f xm ≤
      ↑(Nesterov.l f f' γ x₀) * 1 ^ 2 / 2 *
        (‖Nesterov.v f f' γ x₀ (↑1 - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑1 - xm‖ ^ 2)) :
  f (Nesterov.x f f' γ x₀ 1) - f xm + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2 := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
min : IsMinOn f univ xm
con : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2
h5 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ (k - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2
h6 :
  ∀ (k : ℕ+),
    1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 ≤
      1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) +
        ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2
h4 :
  f (Nesterov.x f f' γ x₀ ↑1) - f xm ≤
    ↑(Nesterov.l f f' γ x₀) * 1 ^ 2 / 2 *
      (‖Nesterov.v f f' γ x₀ (↑1 - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑1 - xm‖ ^ 2)
⊢ f (Nesterov.x f f' γ x₀ 1) - f xm + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2 ≤
    ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 0 - xm‖ ^ 2

Tactic: simp only [PNat.one_coe, Real.rpow_two, one_pow, mul_one, le_refl, tsub_eq_zero_of_le] at h4
    rw [← le_sub_iff_add_le, ← mul_sub]
    exact h4

State after:

Relevant lemmas:
le_refl : ∀ {α : Type u_1} [inst : Preorder α] (a : α), a ≤ a
one_pow : ∀ {M : Type u_2} [inst : Monoid M] (n : ℕ), 1 ^ n = 1
Real.rpow_two : ∀ (x : ℝ), x ^ 2 = x ^ 2
mul_sub : ∀ {α : Type u} [inst : NonUnitalNonAssocRing α] (a b c : α), a * (b - c) = a * b - a * c
le_sub_iff_add_le : ∀ {α : Type u} [inst : AddGroup α] [inst_3 : LE α] [inst_4 : AddRightMono α] {a b c : α},
  a ≤ c - b ↔ a + b ≤ c
PNat.one_coe : ↑1 = 1
mul_one : ∀ {M : Type u} [inst : MulOneClass M] (a : M), a * 1 = a
tsub_eq_zero_of_le : ∀ {α : Type u_1} [inst : AddCommMonoid α] [inst_3 : PartialOrder α]
  [inst_4 : CanonicallyOrderedAdd α] [inst_5 : Sub α] [inst_6 : OrderedSub α] {a b : α}, a ≤ b → a - b = 0