theorem extracted_formal_statement_3.{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) (k : ℕ) :
  let update2 := Nesterov_first.update2 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
f h : E → ℝ
f' : E → E
x0 : E
alg : Nesterov_first f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
k : ℕ
update2 : prox_prop (Nesterov_first.t f h f' x0 k • h)
  (Nesterov_first.y f h f' x0 k - Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))
  (Nesterov_first.x f h f' x0 (k + 1)) :=
  Nesterov_first.update2 k
⊢ ConvexOn ℝ univ h

Tactic: exact alg.convh

State after:

Relevant lemmas: