theorem extracted_formal_statement_8.{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) →
        ConvexOn ℝ univ h := sorry

State before:
case hfun.hf
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 h

Tactic: exact alg.convh

State after:

Relevant lemmas: