theorem extracted_formal_statement_3.{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)))
  (h_original_goal : (↑q + 2) * f (g (q + 3)) = (↑q.succ + 1) * f (g (q.succ + 2))) :
  ↑q + 2 = ↑q + 1 + 1 ∨ f (g (q + 3)) = 0 := 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))
⊢ (↑q + 2) * f (g (q + 3)) = (↑q.succ + 1) * f (g (q.succ + 2))

Tactic: simp

State after:
[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))
 ⊢ ↑q + 2 = ↑q + 1 + 1 ∨ f (g (q + 3)) = 0]

Relevant lemmas: