import Mathlib
import Optlib


open Set Real

open Set Real PNat

theorem extracted_formal_statement_4.{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)
  (z : E) (k : ℕ)
  (h_original_goal :
    1 / Nesterov_first.t f h f' x0 k *
        (Nesterov_first.t f h f' x0 k * h z -
          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))) =
      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))) :
  inner
      (f' (Nesterov_first.y f h f' x0 k) -
        (Nesterov_first.t f h f' x0 k)⁻¹ • (Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1)))
      (z - Nesterov_first.x f h f' x0 (k + 1)) =
    inner
      (f' (Nesterov_first.y f h f' x0 k) +
        (Nesterov_first.t f h f' x0 k)⁻¹ • (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)) := 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
z : E
k : ℕ
⊢ 1 / Nesterov_first.t f h f' x0 k *
      (Nesterov_first.t f h f' x0 k * h z -
        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))) =
    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))

Tactic:
rw [sub_eq_add_neg, ← inner_neg_left, mul_add, ← mul_assoc, one_div_mul_cancel]
    simp
    rw [← real_inner_smul_left, smul_sub, inv_smul_smul₀]

Tactic state after:
[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
 z : E
 k : ℕ
 ⊢ inner
       (f' (Nesterov_first.y f h f' x0 k) -
         (Nesterov_first.t f h f' x0 k)⁻¹ • (Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1)))
       (z - Nesterov_first.x f h f' x0 (k + 1)) =
     inner
       (f' (Nesterov_first.y f h f' x0 k) +
         (Nesterov_first.t f h f' x0 k)⁻¹ • (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)),
 case ha
 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
 z : E
 k : ℕ
 ⊢ Nesterov_first.t f h f' x0 k ≠ 0,
 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
 z : E
 k : ℕ
 ⊢ Nesterov_first.t f h f' x0 k ≠ 0]