YES (ignored inputs)COMMENT from the collection of \cite{AT2012} Rewrite Rules: [ +(0,?y) -> ?y, +(?x,0) -> ?x, +(s(?x),?y) -> s(+(?x,?y)), +(?x,s(?y)) -> s(+(?x,?y)), +(?x,?y) -> +(?y,?x) ] Apply Direct Methods... Inner CPs: [ ] Outer CPs: [ 0 = 0, s(?y_3) = s(+(0,?y_3)), ?y = +(?y,0), s(?x_2) = s(+(?x_2,0)), ?x_1 = +(0,?x_1), s(+(?x_2,s(?y_3))) = s(+(s(?x_2),?y_3)), s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)), s(+(?x_3,?y_3)) = +(s(?y_3),?x_3) ] Overlay, check Innermost Termination... unknown Innermost Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Final result: CR 158.trs: Success(CR) (0 msec.)