-- Define a simple function to increment a natural number
def incr : Nat → Nat
  | 0 => 1
  | n + 1 => incr n + 1

-- Prove that incrementing twice is equivalent to adding 2
theorem incr_twice_eq_add_two (n : Nat) : incr (incr n) = n + 2 := by
  induction n with
  | zero => simp [incr]
  | succ n' ih => simp [incr, ih]
