theorem extracted_formal_statement_5.{u_1} {E : Type u_1} {f : E → ℝ} {g : ℕ → E}
  (mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)) (n : ℕ)
  (h : (↑n + 1) * f (g (n.succ + 1)) / (↑n + 1 + 1) ≤ (∑ k ∈ Finset.range n.succ, f (g (k + 1))) / (↑n + 1 + 1))
  (h_original_goal :
    (∑ k ∈ Finset.range n.succ, f (g (k + 1))) / (↑n.succ + 1) + f (g (n.succ + 1)) / (↑n.succ + 1) ≥
      ↑n.succ * f (g (n.succ + 1)) / (↑n.succ + 1) + f (g (n.succ + 1)) / (↑n.succ + 1)) :
  (↑n + 1) * f (g (n + 1 + 1)) / (↑n + 1 + 1) ≤ (∑ k ∈ Finset.range (n + 1), f (g (k + 1))) / (↑n + 1 + 1) := sorry

State before:
E : Type u_1
f : E → ℝ
g : ℕ → E
mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)
n : ℕ
h : (↑n + 1) * f (g (n.succ + 1)) / (↑n + 1 + 1) ≤ (∑ k ∈ Finset.range n.succ, f (g (k + 1))) / (↑n + 1 + 1)
⊢ (∑ k ∈ Finset.range n.succ, f (g (k + 1))) / (↑n.succ + 1) + f (g (n.succ + 1)) / (↑n.succ + 1) ≥
    ↑n.succ * f (g (n.succ + 1)) / (↑n.succ + 1) + f (g (n.succ + 1)) / (↑n.succ + 1)

Tactic: simp

State after:
[E : Type u_1
 f : E → ℝ
 g : ℕ → E
 mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)
 n : ℕ
 h : (↑n + 1) * f (g (n.succ + 1)) / (↑n + 1 + 1) ≤ (∑ k ∈ Finset.range n.succ, f (g (k + 1))) / (↑n + 1 + 1)
 ⊢ (↑n + 1) * f (g (n + 1 + 1)) / (↑n + 1 + 1) ≤ (∑ k ∈ Finset.range (n + 1), f (g (k + 1))) / (↑n + 1 + 1)]

Relevant lemmas: