theorem extracted_formal_statement_0.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {x0 : E} {f h : E → ℝ} {f' : E → E} {alg : Nesterov_second f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) :
  let φ := fun z => f z + h z;
  (∀ (z : E), φ z = f z + h z) →
    ∀ (k : ℕ+),
      prox_prop ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
          (Nesterov_second.y f h f' x0 (↑k - 1) -
            (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k))
          (Nesterov_second.y f h f' x0 ↑k) →
        Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
              Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k) ∈
            SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k) →
          Nesterov_second.y f h f' x0 (↑k - 1) -
                  (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) •
                    f' (Nesterov_second.z f h f' x0 k) -
                Nesterov_second.y f h f' x0 ↑k ∈
              SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
                (Nesterov_second.y f h f' x0 ↑k) ∧
            (fun x => Nesterov_second.γ f h f' x0 ↑k • x)
                (Nesterov_second.y f h f' x0 (↑k - 1) -
                    (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) •
                      f' (Nesterov_second.z f h f' x0 k) -
                  Nesterov_second.y f h f' x0 ↑k) =
              Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
                Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k) := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x0 : E
f h : E → ℝ
f' : E → E
alg : Nesterov_second f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
φ : E → ℝ := fun z => f z + h z
φdef : ∀ (z : E), φ z = f z + h z
k : ℕ+
h1 :
  prox_prop ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
    (Nesterov_second.y f h f' x0 (↑k - 1) -
      (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k))
    (Nesterov_second.y f h f' x0 ↑k)
⊢ Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
      Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k) ∈
    SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k)

Tactic: rw [prox_iff_subderiv] at h1
    have upd2 := @SubderivAt.pos_smul _ _ _ ((alg.t k / alg.γ k) • h) (alg.y k) (alg.γ k) (alg.γbound k).1
    rw [← smul_assoc, smul_eq_mul, mul_div, mul_comm, ← mul_div, div_self, mul_one] at upd2
    rw [upd2]
    use (alg.y (↑k - 1) - (alg.t ↑k / alg.γ ↑k) • f' (alg.z k) - alg.y ↑k)

State after:
[case h
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 x0 : E
 f h : E → ℝ
 f' : E → E
 alg : Nesterov_second f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 φ : E → ℝ := fun z => f z + h z
 φdef : ∀ (z : E), φ z = f z + h z
 k : ℕ+
 h1 :
   Nesterov_second.y f h f' x0 (↑k - 1) -
         (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
       Nesterov_second.y f h f' x0 ↑k ∈
     SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h) (Nesterov_second.y f h f' x0 ↑k)
 upd2 :
   SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k) =
     Nesterov_second.γ f h f' x0 ↑k •
       SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
         (Nesterov_second.y f h f' x0 ↑k)
 ⊢ Nesterov_second.y f h f' x0 (↑k - 1) -
           (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
         Nesterov_second.y f h f' x0 ↑k ∈
       SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
         (Nesterov_second.y f h f' x0 ↑k) ∧
     (fun x => Nesterov_second.γ f h f' x0 ↑k • x)
         (Nesterov_second.y f h f' x0 (↑k - 1) -
             (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
           Nesterov_second.y f h f' x0 ↑k) =
       Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
         Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k),
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 x0 : E
 f h : E → ℝ
 f' : E → E
 alg : Nesterov_second f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 φ : E → ℝ := fun z => f z + h z
 φdef : ∀ (z : E), φ z = f z + h z
 k : ℕ+
 h1 :
   Nesterov_second.y f h f' x0 (↑k - 1) -
         (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
       Nesterov_second.y f h f' x0 ↑k ∈
     SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h) (Nesterov_second.y f h f' x0 ↑k)
 upd2 :
   SubderivAt ((Nesterov_second.t f h f' x0 ↑k * (Nesterov_second.γ f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k)) • h)
       (Nesterov_second.y f h f' x0 ↑k) =
     Nesterov_second.γ f h f' x0 ↑k •
       SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
         (Nesterov_second.y f h f' x0 ↑k)
 ⊢ Nesterov_second.γ f h f' x0 ↑k ≠ 0,
 case hfun
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 x0 : E
 f h : E → ℝ
 f' : E → E
 alg : Nesterov_second f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 φ : E → ℝ := fun z => f z + h z
 φdef : ∀ (z : E), φ z = f z + h z
 k : ℕ+
 h1 :
   prox_prop ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
     (Nesterov_second.y f h f' x0 (↑k - 1) -
       (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k))
     (Nesterov_second.y f h f' x0 ↑k)
 ⊢ ConvexOn ℝ univ ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)]

Relevant lemmas:
smul_eq_mul : ∀ {α : Type u_9} [inst : Mul α] (a b : α), a • b = a * b
div_self : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a / a = 1
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_div : ∀ {G : Type u_3} [inst : DivInvMonoid G] (a b c : G), a * (b / c) = a * b / c
smul_assoc : ∀ {α : Type u_5} {M : Type u_9} {N : Type u_10} [inst : SMul M N] [inst_3 : SMul N α] [inst_4 : SMul M α]
  [inst_5 : IsScalarTower M N α] (x : M) (y : N) (z : α), (x • y) • z = x • y • z
mul_one : ∀ {M : Type u} [inst : MulOneClass M] (a : M), a * 1 = a
SubderivAt.pos_smul : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] {f : E → ℝ}
  {x : E} {c : ℝ}, 0 < c → SubderivAt (c • f) x = c • SubderivAt f x
prox_iff_subderiv : ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {x : E} (f : E → ℝ),
  ConvexOn ℝ univ f → ∀ (u : E), prox_prop f x u ↔ x - u ∈ SubderivAt f u