theorem extracted_formal_statement_2.{u_1} {E : Type u_1} {f : E → ℝ} {g : ℕ → E} (q : ℕ)
  (IH1 : ∑ k ∈ Finset.range (q + 1), f (g (k + 1)) ≥ (↑q + 1) * f (g (q + 2)))
  (mono : f (g (q + 2 + 1)) ≤ f (g (q + 2))) :
  ∑ k ∈ Finset.range (q.succ + 1), f (g (k + 1)) = ∑ k ∈ Finset.range (q + 1), f (g (k + 1)) + f (g (q + 2)) := sorry

State before:
E : Type u_1
f : E → ℝ
g : ℕ → E
q : ℕ
IH1 : ∑ k ∈ Finset.range (q + 1), f (g (k + 1)) ≥ (↑q + 1) * f (g (q + 2))
mono : f (g (q + 2 + 1)) ≤ f (g (q + 2))
⊢ ∑ k ∈ Finset.range (q.succ + 1), f (g (k + 1)) = ∑ k ∈ Finset.range (q + 1), f (g (k + 1)) + f (g (q + 2))

Tactic: rw [Finset.sum_range_succ (fun k ↦ f (g (k + 1))) q.succ]

State after:

Relevant lemmas:
Finset.sum_range_succ : ∀ {β : Type u_4} [inst : AddCommMonoid β] (f : ℕ → β) (n : ℕ),
  ∑ x ∈ Finset.range (n + 1), f x = ∑ x ∈ Finset.range n, f x + f n