theorem extracted_formal_statement_152.{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_fix_stepsize f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) (k : ℕ)
  (h1 :
    f (Nesterov_first_fix_stepsize.x f h f' x0 (k + 1)) + h (Nesterov_first_fix_stepsize.x f h f' x0 (k + 1)) - f xm -
        h xm =
      f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm)
  (h2 :
    Nesterov_first_fix_stepsize.γ f h f' x0 k ^ 2 / (2 * Nesterov_first_fix_stepsize.t f h f' x0 k) * ‖x0 - xm‖ ^ 2 =
      Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2) :
  f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm ≤
    Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2 := 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_fix_stepsize f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
h1 :
  f (Nesterov_first_fix_stepsize.x f h f' x0 (k + 1)) + h (Nesterov_first_fix_stepsize.x f h f' x0 (k + 1)) - f xm -
      h xm =
    f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm
h2 :
  Nesterov_first_fix_stepsize.γ f h f' x0 k ^ 2 / (2 * Nesterov_first_fix_stepsize.t f h f' x0 k) * ‖x0 - xm‖ ^ 2 =
    Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2
⊢ f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm ≤
    Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2

Tactic: apply Nesterov_first_converge minφ

State after:

Relevant lemmas:
Nesterov_first_converge : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {f h : E → ℝ} {f' : E → E} {x0 : E} {alg : Nesterov_first f h f' x0} {xm : E},
  IsMinOn (f + h) univ xm →
    ∀ (k : ℕ),
      f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm ≤
        Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2