theorem extracted_formal_statement_2.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {γ : ℕ+ → ℝ} {alg : Nesterov f f' γ x₀}
  (hfun : ConvexOn ℝ univ f) (hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)) (k : ℕ+) (x' : E)
  (this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k)) :
  f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x' := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
γ : ℕ+ → ℝ
alg : Nesterov f f' γ x₀
hfun : ConvexOn ℝ univ f
hg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)
k : ℕ+
x' : E
this : f' (Nesterov.y f f' γ x₀ k) = ↑(Nesterov.l f f' γ x₀) • (Nesterov.y f f' γ x₀ k - Nesterov.x f f' γ x₀ ↑k)
⊢ f (Nesterov.y f f' γ x₀ k) + inner (f' (Nesterov.y f f' γ x₀ k)) (x' - Nesterov.y f f' γ x₀ k) ≤ f x'

Tactic: exact Convex_first_order_condition' (alg.diff (alg.y k)) hfun (by trivial) x' (by trivial)

State after:

Relevant lemmas:
Convex_first_order_condition' : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E]
  [inst_4 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {s : Set E} {x : E},
  HasGradientAt f (f' x) x → ConvexOn ℝ s f → x ∈ s → ∀ y ∈ s, f x + inner (f' x) (y - x) ≤ f y