have : k + c + (k + c + 1 - k) = c + (k + c + 1) := by
  simp only [add_assoc, add_tsub_cancel_left, add_left_comm]
case mk.intro
k c : ℕ
hk : k < k + c + 1
⊢ k + c - (k + c + (k + c + 1 - ↑⟨k, hk⟩)) % (k + c + 1) = ↑⟨k, hk⟩
case mk.intro
k c : ℕ
hk : k < k + c + 1
this : k + c + (k + c + 1 - k) = c + (k + c + 1)
⊢ k + c - (k + c + (k + c + 1 - ↑⟨k, hk⟩)) % (k + c + 1) = ↑⟨k, hk⟩