(ignored inputs)COMMENT R3 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,?z)) -> +(+(?x,?y),?z), +(?x,?y) -> +(?y,?x) ] Apply Direct Methods... Inner CPs: [ +(?x_4,?y) = +(+(?x_4,0),?y), +(?x_4,?x_1) = +(+(?x_4,?x_1),0), +(?x_4,s(+(?x_2,?y_2))) = +(+(?x_4,s(?x_2)),?y_2), +(?x_4,s(+(?x_3,?y_3))) = +(+(?x_4,?x_3),s(?y_3)), +(?x_4,+(?y_5,?x_5)) = +(+(?x_4,?x_5),?y_5), +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ] Outer CPs: [ 0 = 0, s(?y_3) = s(+(0,?y_3)), +(?y_4,?z_4) = +(+(0,?y_4),?z_4), ?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_4,?z_4))) = +(+(s(?x_2),?y_4),?z_4), s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)), s(+(?x_3,?y_3)) = +(s(?y_3),?x_3), +(+(?x_4,?y_4),?z_4) = +(+(?y_4,?z_4),?x_4) ] not Overlay, check Termination... unknown/not Terminating unknown Knuth&Bendix Linear unknown Development Closed unknown Strongly Closed unknown Upside-Parallel-Closed/Outside-Closed (inner) Parallel CPs: (not computed) unknown Toyama (Parallel CPs) Simultaneous CPs: [ +(+(?x_6,?x),?y) = +(?x_6,+(?y,?x)), +(+(?x,?y_5),?z_5) = +(+(?y_5,?z_5),?x), s(+(?x,?y_4)) = +(s(?y_4),?x), s(+(?x_3,?y)) = +(?y,s(?x_3)), ?x = +(0,?x), ?y = +(?y,0), +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),s(+(?y,?y_6))) = +(?x_1,+(+(?x,?y),s(?y_6))), +(+(?x_1,?x),s(+(?x_5,?z))) = +(?x_1,+(+(?x,s(?x_5)),?z)), +(+(?x_1,?x),?y) = +(?x_1,+(+(?x,?y),0)), +(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)), +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))), +(?x,+(?z,?y)) = +(+(?x,?y),?z), +(?x,s(+(?y,?y_5))) = +(+(?x,?y),s(?y_5)), +(?x,s(+(?x_4,?z))) = +(+(?x,s(?x_4)),?z), +(?x,?y) = +(+(?x,?y),0), +(?x,?z) = +(+(?x,0),?z), +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)), +(+(?y,?z),?x) = +(+(?x,?y),?z), s(+(?x_3,+(?y,?z))) = +(+(s(?x_3),?y),?z), +(?y,?z) = +(+(0,?y),?z), +(+(?z,?y),?x) = +(+(?x,?y),?z), +(s(+(?y,?y_5)),?x) = +(+(?x,?y),s(?y_5)), +(s(+(?x_4,?z)),?x) = +(+(?x,s(?x_4)),?z), +(?y,?x) = +(+(?x,?y),0), +(?z,?x) = +(+(?x,0),?z), +(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)), s(+(?x_3,+(?z,?y))) = +(+(s(?x_3),?y),?z), s(+(?x_3,s(+(?y,?y_8)))) = +(+(s(?x_3),?y),s(?y_8)), s(+(?x_3,s(+(?x_7,?z)))) = +(+(s(?x_3),s(?x_7)),?z), s(+(?x_3,?y)) = +(+(s(?x_3),?y),0), s(+(?x_3,?z)) = +(+(s(?x_3),0),?z), s(+(?x_3,+(+(?y,?y_4),?z_4))) = +(+(s(?x_3),?y),+(?y_4,?z_4)), +(?z,?y) = +(+(0,?y),?z), s(+(?y,?y_5)) = +(+(0,?y),s(?y_5)), s(+(?x_4,?z)) = +(+(0,s(?x_4)),?z), ?y = +(+(0,?y),0), ?z = +(+(0,0),?z), +(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)), +(+(?x_5,?x),s(?y)) = +(?x_5,s(+(?x,?y))), +(s(?y),?x) = s(+(?x,?y)), s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)), s(?y) = s(+(0,?y)), +(+(?x_5,s(?x)),?y) = +(?x_5,s(+(?x,?y))), +(?y,s(?x)) = s(+(?x,?y)), +(+(s(?x),?y_4),?z_4) = s(+(?x,+(?y_4,?z_4))), s(+(s(?x),?y_3)) = s(+(?x,s(?y_3))), s(?x) = s(+(?x,0)), +(+(?x_5,?x),0) = +(?x_5,?x), +(0,?x) = ?x, s(+(?x_2,0)) = s(?x_2), +(+(?x_5,0),?y) = +(?x_5,?y), +(?y,0) = ?y, +(+(0,?y_4),?z_4) = +(?y_4,?z_4), s(+(0,?y_3)) = s(?y_3), 0 = 0 ] unknown Okui (Simultaneous CPs) unknown Strongly Depth-Preserving & Non-E-Overlapping check Locally Decreasing Diagrams by Rule Labelling... Critical Pair <+(?x_4,?y), +(+(?x_4,0),?y)> by Rules <0, 4> preceded by [(+,2)] joinable by a reduction of rules <[], [([(+,1)],1)]> Critical Pair <+(?x_4,?x_1), +(+(?x_4,?x_1),0)> by Rules <1, 4> preceded by [(+,2)] joinable by a reduction of rules <[], [([],1)]> Critical Pair <+(?x_4,s(+(?x_2,?y_2))), +(+(?x_4,s(?x_2)),?y_2)> by Rules <2, 4> preceded by [(+,2)] joinable by a reduction of rules <[([],3),([(s,1)],4)], [([(+,1)],3),([],2)]> Critical Pair <+(?x_4,s(+(?x_3,?y_3))), +(+(?x_4,?x_3),s(?y_3))> by Rules <3, 4> preceded by [(+,2)] joinable by a reduction of rules <[([],3),([(s,1)],4)], [([],3)]> Critical Pair <+(?x_4,+(?y_5,?x_5)), +(+(?x_4,?x_5),?y_5)> by Rules <5, 4> preceded by [(+,2)] joinable by a reduction of rules <[([(+,2)],5),([],4)], []> joinable by a reduction of rules <[([],4),([(+,1)],5)], [([],5),([],4)]> Critical Pair <+(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?y,?z))> by Rules <4, 4> preceded by [(+,2)] joinable by a reduction of rules <[([],4),([(+,1)],4)], [([],4)]> Critical Pair <0, 0> by Rules <1, 0> preceded by [] joinable by a reduction of rules <[], []> Critical Pair by Rules <3, 0> preceded by [] joinable by a reduction of rules <[([(s,1)],0)], []> Critical Pair <+(+(0,?y_4),?z_4), +(?y_4,?z_4)> by Rules <4, 0> preceded by [] joinable by a reduction of rules <[([(+,1)],0)], []> Critical Pair <+(?y_5,0), ?y_5> by Rules <5, 0> preceded by [] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <2, 1> preceded by [] joinable by a reduction of rules <[([(s,1)],1)], []> Critical Pair <+(0,?x_5), ?x_5> by Rules <5, 1> preceded by [] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <3, 2> preceded by [] joinable by a reduction of rules <[([(s,1)],2)], [([(s,1)],3)]> Critical Pair <+(+(s(?x_2),?y_4),?z_4), s(+(?x_2,+(?y_4,?z_4)))> by Rules <4, 2> preceded by [] joinable by a reduction of rules <[([(+,1)],2),([],2)], [([(s,1)],4)]> Critical Pair <+(?y_5,s(?x_2)), s(+(?x_2,?y_5))> by Rules <5, 2> preceded by [] joinable by a reduction of rules <[([],3)], [([(s,1)],5)]> Critical Pair <+(s(?y_3),?x_5), s(+(?x_5,?y_3))> by Rules <5, 3> preceded by [] joinable by a reduction of rules <[([],2)], [([(s,1)],5)]> Critical Pair <+(+(?y_4,?z_4),?x_5), +(+(?x_5,?y_4),?z_4)> by Rules <5, 4> preceded by [] joinable by a reduction of rules <[([],5),([],4)], []> unknown Diagram Decreasing check by Reduction-Preserving Completion... STEP: 1 (parallel) S: [ +(0,?y) -> ?y, +(?x,0) -> ?x, +(s(?x),?y) -> s(+(?x,?y)), +(?x,s(?y)) -> s(+(?x,?y)) ] P: [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z), +(?x,?y) -> +(?y,?x) ] S: terminating CP(S,S): <0, 0> --> <0, 0> => yes --> => yes --> => yes --> => yes PCP_in(symP,S): CP(S,symP): <+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes <+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes <+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes --> => yes <+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes <+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes <+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes --> => yes --> => yes <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> => yes <+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> => yes --> => yes <+(?x_1,s(+(?x,?y))), +(+(?x_1,?x),s(?y))> --> => yes --> => yes <+(s(+(?x,?y)),?z_1), +(?x,+(s(?y),?z_1))> --> => yes --> => yes S: [ +(0,?y) -> ?y, +(?x,0) -> ?x, +(s(?x),?y) -> s(+(?x,?y)), +(?x,s(?y)) -> s(+(?x,?y)) ] P: [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z), +(?x,?y) -> +(?y,?x) ] Success Reduction-Preserving Completion Direct Methods: CR Final result: CR /local-scratch/hzankl/2012/cops2012/125.trs: Success(CR) (23 msec.)