import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_18.{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 (Nesterov_first.x 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)
                (Nesterov_first.x f h f' x0 k - 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 →
        φ (Nesterov_first.x f h f' x0 (k + 1)) - 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)
                  (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 →
          Nesterov_first.γ f h f' x0 k * (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 + 1)) - φ (Nesterov_first.x f h f' x0 k)) ≤
            Nesterov_first.γ 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)
                      (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) +
              (1 - Nesterov_first.γ 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)
                      (Nesterov_first.x f h f' x0 k - 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) := sorry

Tactic 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
k : ℕ
ieq1 :
  φ (Nesterov_first.x f h f' x0 (k + 1)) - φ (Nesterov_first.x 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)
          (Nesterov_first.x f h f' x0 k - 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
ieq2 :
  φ (Nesterov_first.x f h f' x0 (k + 1)) - 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)
          (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
⊢ Nesterov_first.γ f h f' x0 k * (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 + 1)) - φ (Nesterov_first.x f h f' x0 k)) ≤
    Nesterov_first.γ 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)
              (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) +
      (1 - Nesterov_first.γ 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)
              (Nesterov_first.x f h f' x0 k - 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)

Tactic:
apply add_le_add
    · rw [mul_le_mul_left]; exact ieq2; linarith [(alg.γbound k).1]
    · apply mul_le_mul_of_nonneg_left; exact ieq1; linarith [(alg.γbound k).2]

Tactic state after: