+-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})