+-comm : (x y : !$\mathbb{N}$!) !$\rightarrow$! x + y !$\equiv$! y + x
+-comm zero y rewrite (+zero {y}) = refl
+-comm (suc x) y = let open !$\equiv$!-Reasoning in
  begin
  suc (x + y) !$\equiv$!!$\langle$!!$\rangle$!
  suc (x + y) !$\equiv$!!$\langle$! cong suc (+-comm x y) !$\rangle$!
  suc (y + x) !$\equiv$!!$\langle$! sym (+-suc {y} {x}) !$\rangle$!
  y + suc x !$\blacksquare$!

-- +-suc : {x y : !$\mathbb{N}$!} !$\rightarrow$! x + suc y !$\equiv$! suc (x + y)
-- +-suc {zero} {y} = refl
-- +-suc {suc x} {y} = cong suc (+-suc {x} {y})