YES
(ignored inputs)COMMENT from Example 1 of \cite{OO03}
Rewrite Rules:
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     *(?x,s(?y)) -> +(?x,*(?x,?y)),
     *(s(?x),?y) -> +(*(?x,?y),?y),
     *(?x,?y) -> *(?y,?x),
     sq(?x) -> *(?x,?x),
     sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x))) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,+(?x_1,+(?y_1,?z_1))) = +(+(?x,+(?x_1,?y_1)),?z_1),
     +(?x,+(?y_2,?x_2)) = +(+(?x,?x_2),?y_2),
     +(?x,+(?x_3,s(?y_3))) = +(+(?x,s(?x_3)),?y_3),
     +(?x,+(s(?x_4),?y_4)) = +(+(?x,?x_4),s(?y_4)),
     +(+(+(?x,?y),?z),?z_1) = +(?x,+(+(?y,?z),?z_1)),
     +(+(?y_2,?x_2),?z_1) = +(?x_2,+(?y_2,?z_1)),
     +(+(?x_3,s(?y_3)),?z_1) = +(s(?x_3),+(?y_3,?z_1)),
     +(+(s(?x_4),?y_4),?z_1) = +(?x_4,+(s(?y_4),?z_1)),
     +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ +(+(+(?x_1,?y_1),?y),?z) = +(?x_1,+(?y_1,+(?y,?z))),
     +(+(?x,?y),?z) = +(+(?y,?z),?x),
     +(+(s(?x_3),?y),?z) = +(?x_3,s(+(?y,?z))),
     +(?x_1,+(?y_1,?z_1)) = +(?z_1,+(?x_1,?y_1)),
     +(?x_1,+(?y_1,s(?y_4))) = +(s(+(?x_1,?y_1)),?y_4),
     +(?y_2,s(?x_3)) = +(?x_3,s(?y_2)),
     +(s(?y_4),?x_2) = +(s(?x_2),?y_4),
     +(?x_3,s(s(?y_4))) = +(s(s(?x_3)),?y_4),
     +(s(?x_6),*(s(?x_6),?y_5)) = +(*(?x_6,s(?y_5)),s(?y_5)),
     +(?x_5,*(?x_5,?y_5)) = *(s(?y_5),?x_5),
     +(*(?x_6,?y_6),?y_6) = *(?y_6,s(?x_6)),
     *(s(?x_9),s(?x_9)) = +(*(?x_9,?x_9),s(+(?x_9,?x_9))) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ +(?x_1,+(?y_1,+(+(?y,?y_2),?z_2))) = +(+(+(?x_1,?y_1),?y),+(?y_2,?z_2)),
     +(?x_1,+(?y_1,+(?x_3,+(?y_3,?z)))) = +(+(+(?x_1,?y_1),+(?x_3,?y_3)),?z),
     +(?x_1,+(?y_1,+(?z,?y))) = +(+(+(?x_1,?y_1),?y),?z),
     +(?x_1,+(?y_1,+(?x_5,s(?z)))) = +(+(+(?x_1,?y_1),s(?x_5)),?z),
     +(?x_1,+(?y_1,+(s(?y),?y_6))) = +(+(+(?x_1,?y_1),?y),s(?y_6)),
     +(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)),
     +(+(?x_2,+(?y_2,?z)),?x) = +(+(?x,+(?x_2,?y_2)),?z),
     +(+(?z,?y),?x) = +(+(?x,?y),?z),
     +(+(?x_4,s(?z)),?x) = +(+(?x,s(?x_4)),?z),
     +(+(s(?y),?y_5),?x) = +(+(?x,?y),s(?y_5)),
     +(?x_3,s(+(+(?y,?y_4),?z_4))) = +(+(s(?x_3),?y),+(?y_4,?z_4)),
     +(?x_3,s(+(?x_5,+(?y_5,?z)))) = +(+(s(?x_3),+(?x_5,?y_5)),?z),
     +(?x_3,s(+(?z,?y))) = +(+(s(?x_3),?y),?z),
     +(?x_3,s(+(?x_7,s(?z)))) = +(+(s(?x_3),s(?x_7)),?z),
     +(?x_3,s(+(s(?y),?y_8))) = +(+(s(?x_3),?y),s(?y_8)),
     +(?x_1,+(?y_1,+(?y,?z))) = +(+(+(?x_1,?y_1),?y),?z),
     +(+(?y,?z),?x) = +(+(?x,?y),?z),
     +(?x_3,s(+(?y,?z))) = +(+(s(?x_3),?y),?z),
     +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
     +(?x,+(?x_2,+(?y_2,?z))) = +(+(?x,+(?x_2,?y_2)),?z),
     +(?x,+(?z,?y)) = +(+(?x,?y),?z),
     +(?x,+(?x_4,s(?z))) = +(+(?x,s(?x_4)),?z),
     +(?x,+(s(?y),?y_5)) = +(+(?x,?y),s(?y_5)),
     +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
     +(+(?x_1,?x),+(?x_3,+(?y_3,?z))) = +(?x_1,+(+(?x,+(?x_3,?y_3)),?z)),
     +(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)),
     +(+(?x_1,?x),+(?x_5,s(?z))) = +(?x_1,+(+(?x,s(?x_5)),?z)),
     +(+(?x_1,?x),+(s(?y),?y_6)) = +(?x_1,+(+(?x,?y),s(?y_6))),
     +(?x,+(+(+(?y,?y_3),?z_3),?z_2)) = +(+(+(?x,?y),+(?y_3,?z_3)),?z_2),
     +(?x,+(+(?x_4,+(?y_4,?z)),?z_2)) = +(+(+(?x,+(?x_4,?y_4)),?z),?z_2),
     +(?x,+(+(?z,?y),?z_2)) = +(+(+(?x,?y),?z),?z_2),
     +(?x,+(+(?x_6,s(?z)),?z_2)) = +(+(+(?x,s(?x_6)),?z),?z_2),
     +(?x,+(+(s(?y),?y_7),?z_2)) = +(+(+(?x,?y),s(?y_7)),?z_2),
     +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)),
     +(?x,+(+(?y,?z),?z_2)) = +(+(+(?x,?y),?z),?z_2),
     +(+(+(?x_2,+(?y_2,?y)),?y_1),?z_1) = +(+(?x_2,?y_2),+(?y,+(?y_1,?z_1))),
     +(+(+(+(?x,?y_3),?z_3),?y_1),?z_1) = +(?x,+(+(?y_3,?z_3),+(?y_1,?z_1))),
     +(+(+(?y,?x),?y_1),?z_1) = +(?x,+(?y,+(?y_1,?z_1))),
     +(+(+(?x_5,s(?y)),?y_1),?z_1) = +(s(?x_5),+(?y,+(?y_1,?z_1))),
     +(+(+(s(?x),?y_6),?y_1),?z_1) = +(?x,+(s(?y_6),+(?y_1,?z_1))),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,+(+(?x,?y_2),?z_2)) = +(?x,+(+(?y_2,?z_2),?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?z,+(?x_4,s(?y))) = +(s(?x_4),+(?y,?z)),
     +(?z,+(s(?x),?y_5)) = +(?x,+(s(?y_5),?z)),
     +(s(+(?x_5,+(?y_5,?y))),?y_4) = +(+(?x_5,?y_5),+(?y,s(?y_4))),
     +(s(+(+(?x,?y_6),?z_6)),?y_4) = +(?x,+(+(?y_6,?z_6),s(?y_4))),
     +(s(+(?y,?x)),?y_4) = +(?x,+(?y,s(?y_4))),
     +(s(+(?x_8,s(?y))),?y_4) = +(s(?x_8),+(?y,s(?y_4))),
     +(s(+(s(?x),?y_9)),?y_4) = +(?x,+(s(?y_9),s(?y_4))),
     +(+(+(?x,?y),?y_1),?z_1) = +(?x,+(?y,+(?y_1,?z_1))),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(s(+(?x,?y)),?y_4) = +(?x,+(?y,s(?y_4))),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(+(+(?x,?y_2),?z_2),?z) = +(?x,+(+(?y_2,?z_2),?z)),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(+(?x_4,s(?y)),?z) = +(s(?x_4),+(?y,?z)),
     +(+(s(?x),?y_5),?z) = +(?x,+(s(?y_5),?z)),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(+(+(?x,?y_3),?z_3),+(?z,?z_1)) = +(+(?x,+(+(?y_3,?z_3),?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x_5,s(?y)),+(?z,?z_1)) = +(+(s(?x_5),+(?y,?z)),?z_1),
     +(+(s(?x),?y_6),+(?z,?z_1)) = +(+(?x,+(s(?y_6),?z)),?z_1),
     +(+(?x_2,+(?x_3,+(?y_3,?y))),?z) = +(?x_2,+(+(?x_3,?y_3),+(?y,?z))),
     +(+(?x_2,+(+(?x,?y_4),?z_4)),?z) = +(?x_2,+(?x,+(+(?y_4,?z_4),?z))),
     +(+(?x_2,+(?y,?x)),?z) = +(?x_2,+(?x,+(?y,?z))),
     +(+(?x_2,+(?x_6,s(?y))),?z) = +(?x_2,+(s(?x_6),+(?y,?z))),
     +(+(?x_2,+(s(?x),?y_7)),?z) = +(?x_2,+(?x,+(s(?y_7),?z))),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x_2,+(?x,?y)),?z) = +(?x_2,+(?x,+(?y,?z))),
     +(+(?x,?y_1),?z_1) = +(+(?y_1,?z_1),?x),
     +(?x_2,+(?y_2,?y)) = +(?y,+(?x_2,?y_2)),
     +(?x_3,s(?y)) = +(?y,s(?x_3)),
     +(s(?x),?y_4) = +(s(?y_4),?x),
     +(+(?x_2,?x),?y) = +(?x_2,+(?y,?x)),
     +(?x,+(?y,?z_3)) = +(+(?y,?x),?z_3),
     +(+(s(?x),?y_1),?z_1) = +(?x,s(+(?y_1,?z_1))),
     +(s(s(?x)),?y_4) = +(?x,s(s(?y_4))),
     +(+(?x_2,s(?x)),?y) = +(?x_2,+(?x,s(?y))),
     +(s(?x),+(?y,?z_3)) = +(+(?x,s(?y)),?z_3),
     +(?x_2,+(?y_2,s(?y))) = +(s(+(?x_2,?y_2)),?y),
     +(?x_4,s(s(?y))) = +(s(s(?x_4)),?y),
     +(+(?x_2,?x),s(?y)) = +(?x_2,+(s(?x),?y)),
     +(?x,+(s(?y),?z_3)) = +(+(s(?x),?y),?z_3),
     +(*(?x_6,s(?y)),s(?y)) = +(s(?x_6),*(s(?x_6),?y)),
     *(s(?y),?x) = +(?x,*(?x,?y)),
     +(s(?x),*(s(?x),?y_6)) = +(*(?x,s(?y_6)),s(?y_6)),
     *(?y,s(?x)) = +(*(?x,?y),?y),
     +(?x,*(?x,?y_6)) = *(s(?y_6),?x),
     +(*(?x_7,?y),?y) = *(?y,s(?x_7)),
     +(*(?x_9,?x_9),s(+(?x_9,?x_9))) = *(s(?x_9),s(?x_9)),
     *(s(?x),s(?x)) = +(*(?x,?x),s(+(?x,?x))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <+(?x,+(?x_1,+(?y_1,?z_1))), +(+(?x,+(?x_1,?y_1)),?z_1)> by Rules <1, 0> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],0)], [([],1)]>
Critical Pair <+(?x,+(?y_2,?x_2)), +(+(?x,?x_2),?y_2)> by Rules <2, 0> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],2)], [([],1)]>
Critical Pair <+(?x,+(?x_3,s(?y_3))), +(+(?x,s(?x_3)),?y_3)> by Rules <3, 0> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],4)], [([],1)]>
Critical Pair <+(?x,+(s(?x_4),?y_4)), +(+(?x,?x_4),s(?y_4))> by Rules <4, 0> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],3)], [([],1)]>
Critical Pair <+(+(+(?x,?y),?z),?z_1), +(?x,+(+(?y,?z),?z_1))> by Rules <0, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],1)], [([],0)]>
Critical Pair <+(+(?y_2,?x_2),?z_1), +(?x_2,+(?y_2,?z_1))> by Rules <2, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],2)], [([],0)]>
Critical Pair <+(+(?x_3,s(?y_3)),?z_1), +(s(?x_3),+(?y_3,?z_1))> by Rules <3, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],4)], [([],0)]>
Critical Pair <+(+(s(?x_4),?y_4),?z_1), +(?x_4,+(s(?y_4),?z_1))> by Rules <4, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],3)], [([],0)]>
Critical Pair <+(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?y,?z))> by Rules <0, 0> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],1)], [([],1)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <1, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],0)], [([],0)]>
Critical Pair <+(?x_1,+(?y_1,+(?y,?z))), +(+(+(?x_1,?y_1),?y),?z)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([],0)], [([],1)]>
Critical Pair <+(+(?y,?z),?x_2), +(+(?x_2,?y),?z)> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2)], [([],1)]>
Critical Pair <+(?x_3,s(+(?y,?z))), +(+(s(?x_3),?y),?z)> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],4)], [([],1)]>
Critical Pair <+(?y_2,+(?x_1,?y_1)), +(?x_1,+(?y_1,?y_2))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2)], [([],0)]>
Critical Pair <+(s(+(?x_1,?y_1)),?y_4), +(?x_1,+(?y_1,s(?y_4)))> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([],0)]>
Critical Pair <+(?x_3,s(?y_3)), +(?y_3,s(?x_3))> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([],4)], [([],2)]>
 joinable by a reduction of rules <[([],2)], [([],4)]>
Critical Pair <+(s(?x_4),?y_4), +(s(?y_4),?x_4)> by Rules <4, 2> preceded by []
 joinable by a reduction of rules <[([],3)], [([],2)]>
 joinable by a reduction of rules <[([],2)], [([],3)]>
Critical Pair <+(s(s(?x_3)),?y_4), +(?x_3,s(s(?y_4)))> by Rules <4, 3> preceded by []
 joinable by a reduction of rules <[([],3)], [([],4)]>
Critical Pair <+(*(?x_6,s(?y_5)),s(?y_5)), +(s(?x_6),*(s(?x_6),?y_5))> by Rules <6, 5> preceded by []
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(+,2)],4),([],0)], [([(+,2)],6),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(+,2)],4)], [([(+,2)],6),([],0),([(+,1)],3),([],1)]>
 joinable by a reduction of rules <[([(+,1)],7),([(+,1)],6),([],1),([(+,2)],4)], [([(+,2)],7),([(+,2)],5),([],0),([],2)]>
 joinable by a reduction of rules <[([(+,1)],7),([(+,1)],6),([],1),([],2)], [([(+,2)],7),([(+,2)],5),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],4)], [([(+,2)],6),([(+,2)],2),([],0),([],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],4)], [([(+,2)],6),([],2),([],1),([(+,2)],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],4)], [([(+,2)],6),([],0),([(+,1)],2),([],1)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],4)], [([],2),([(+,1)],6),([],1),([(+,2)],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],2)], [([(+,2)],6),([],2),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([(+,2)],2)], [([],2),([(+,1)],6),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],2),([],1),([],2)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([(+,1)],3)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([(+,1)],3)], [([(+,2)],6),([],2),([],1),([],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([(+,1)],3)], [([(+,2)],6),([],0),([],2),([],0)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([(+,1)],3)], [([],2),([(+,1)],6),([],1),([],2)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([(+,1)],2)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([],2)], [([(+,2)],6),([],2),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([(+,1)],5),([],2),([],0),([],2)], [([],2),([(+,1)],6),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(+,2)],2),([],0)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([],2),([],1)], [([(+,2)],6),([],2),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([],2),([],1)], [([],2),([(+,1)],6),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([(+,1)],3)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],2)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([(+,1)],3)], [([(+,2)],6),([],2),([],1),([],2)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([(+,1)],3)], [([(+,2)],6),([],0),([],2),([],0)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([(+,1)],3)], [([],2),([(+,1)],6),([],1),([],2)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([(+,1)],2)], [([(+,2)],6),([(+,2)],2),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([],2)], [([(+,2)],6),([],2),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],2),([(+,2)],5),([],0),([],2)], [([],2),([(+,1)],6),([],1),([(+,2)],4)]>
Critical Pair <*(s(?y_5),?x_7), +(?x_7,*(?x_7,?y_5))> by Rules <7, 5> preceded by []
 joinable by a reduction of rules <[([],7),([],5)], []>
 joinable by a reduction of rules <[([],6),([(+,1)],7)], [([],2)]>
 joinable by a reduction of rules <[([],6),([],2)], [([(+,2)],7)]>
 joinable by a reduction of rules <[([],6)], [([(+,2)],7),([],2)]>
 joinable by a reduction of rules <[([],6)], [([],2),([(+,1)],7)]>
Critical Pair <*(?y_7,s(?x_6)), +(*(?x_6,?y_7),?y_7)> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[([],7),([],6)], []>
 joinable by a reduction of rules <[([],5),([(+,2)],7)], [([],2)]>
 joinable by a reduction of rules <[([],5),([],2)], [([(+,1)],7)]>
 joinable by a reduction of rules <[([],5)], [([(+,1)],7),([],2)]>
 joinable by a reduction of rules <[([],5)], [([],2),([(+,2)],7)]>
Critical Pair <+(*(?x_9,?x_9),s(+(?x_9,?x_9))), *(s(?x_9),s(?x_9))> by Rules <9, 8> preceded by []
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],3),([(+,1)],2)], [([],5),([(+,2)],6),([],0)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],3),([],2)], [([],6),([(+,1)],5),([],1)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],2),([(+,1)],4)], [([],5),([(+,2)],6),([],0)]>
 joinable by a reduction of rules <[([],4),([],0),([],2),([(+,2)],3)], [([],6),([(+,1)],5),([],1)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],3)], [([],6),([(+,1)],5),([],1),([],2)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],3)], [([],5),([(+,2)],6),([],0),([(+,1)],2)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],2)], [([],5),([(+,2)],6),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([],4),([],0),([],2)], [([],6),([(+,1)],5),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],4),([],2),([],1),([(+,2)],4)], [([],6),([(+,1)],5),([],1),([(+,2)],2)]>
 joinable by a reduction of rules <[([],4),([],2),([],1),([(+,2)],4)], [([],5),([(+,2)],6),([],0),([],2)]>
 joinable by a reduction of rules <[([],4),([],2),([],1),([(+,2)],2)], [([],6),([(+,1)],5),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],4),([],2),([],1),([],2)], [([],5),([(+,2)],6),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([],4),([],0),([(+,1)],2),([],1)], [([],6),([(+,1)],5),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],4),([],0),([],2),([],0)], [([],5),([(+,2)],6),([],0),([(+,1)],3)]>
 joinable by a reduction of rules <[([],2),([],3),([],1),([(+,2)],4)], [([],6),([(+,1)],5),([],1),([(+,2)],2)]>
 joinable by a reduction of rules <[([],2),([],3),([],1),([(+,2)],4)], [([],5),([(+,2)],6),([],0),([],2)]>
 joinable by a reduction of rules <[([],2),([],3),([],1),([(+,2)],2)], [([],6),([(+,1)],5),([],1),([(+,2)],4)]>
 joinable by a reduction of rules <[([],2),([],3),([],1),([],2)], [([],5),([(+,2)],6),([],0),([(+,1)],3)]>
unknown Diagram Decreasing
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x_1,?y_1),?z_1) -> +(?x_1,+(?y_1,?z_1)),
     +(?x_2,?y_2) -> +(?y_2,?x_2),
     +(s(?x_3),?y_3) -> +(?x_3,s(?y_3)),
     +(?x_4,s(?y_4)) -> +(s(?x_4),?y_4),
     *(?x_5,s(?y_5)) -> +(?x_5,*(?x_5,?y_5)),
     *(s(?x_6),?y_6) -> +(*(?x_6,?y_6),?y_6),
     *(?x_7,?y_7) -> *(?y_7,?x_7),
     sq(?x_8) -> *(?x_8,?x_8),
     sq(s(?x_9)) -> +(*(?x_9,?x_9),s(+(?x_9,?x_9))) ]
Sort Assignment:
 * : 31*31=>31
 + : 31*31=>31
 s : 31=>31
 sq : 31=>31
non-linear variables: {?x_5,?y_6,?x_8,?x_9}
non-linear types: {31}
types leq non-linear types: {31}
rules applicable to terms of non-linear types:
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x_1,?y_1),?z_1) -> +(?x_1,+(?y_1,?z_1)),
     +(?x_2,?y_2) -> +(?y_2,?x_2),
     +(s(?x_3),?y_3) -> +(?x_3,s(?y_3)),
     +(?x_4,s(?y_4)) -> +(s(?x_4),?y_4),
     *(?x_5,s(?y_5)) -> +(?x_5,*(?x_5,?y_5)),
     *(s(?x_6),?y_6) -> +(*(?x_6,?y_6),?y_6),
     *(?x_7,?y_7) -> *(?y_7,?x_7),
     sq(?x_8) -> *(?x_8,?x_8),
     sq(s(?x_9)) -> +(*(?x_9,?x_9),s(+(?x_9,?x_9))) ]
NLR:
0: {}
1: {}
2: {}
3: {}
4: {}
5: {0,1,2,3,4,5,6,7,8,9}
6: {0,1,2,3,4,5,6,7,8,9}
7: {}
8: {0,1,2,3,4,5,6,7,8,9}
9: {0,1,2,3,4,5,6,7,8,9}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x_1,?y_1),?z_1) -> +(?x_1,+(?y_1,?z_1)),
     +(?x_2,?y_2) -> +(?y_2,?x_2),
     +(s(?x_3),?y_3) -> +(?x_3,s(?y_3)),
     +(?x_4,s(?y_4)) -> +(s(?x_4),?y_4),
     *(?x_5,s(?y_5)) -> +(?x_5,*(?x_5,?y_5)),
     *(s(?x_6),?y_6) -> +(*(?x_6,?y_6),?y_6),
     *(?x_7,?y_7) -> *(?y_7,?x_7),
     sq(?x_8) -> *(?x_8,?x_8),
     sq(s(?x_9)) -> +(*(?x_9,?x_9),s(+(?x_9,?x_9))) ]
Sort Assignment:
 * : 31*31=>31
 + : 31*31=>31
 s : 31=>31
 sq : 31=>31
non-linear variables: {?x_5,?y_6,?x_8,?x_9}
non-linear types: {31}
types leq non-linear types: {31}
rules applicable to terms of non-linear types:
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x_1,?y_1),?z_1) -> +(?x_1,+(?y_1,?z_1)),
     +(?x_2,?y_2) -> +(?y_2,?x_2),
     +(s(?x_3),?y_3) -> +(?x_3,s(?y_3)),
     +(?x_4,s(?y_4)) -> +(s(?x_4),?y_4),
     *(?x_5,s(?y_5)) -> +(?x_5,*(?x_5,?y_5)),
     *(s(?x_6),?y_6) -> +(*(?x_6,?y_6),?y_6),
     *(?x_7,?y_7) -> *(?y_7,?x_7),
     sq(?x_8) -> *(?x_8,?x_8),
     sq(s(?x_9)) -> +(*(?x_9,?x_9),s(+(?x_9,?x_9))) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 20 rules by 3 steps unfolding
strenghten *(?x_13,?y_13) and *(?y_13,?x_13)
strenghten +(?x_17,?y_17) and +(?y_17,?x_17)
strenghten +(?x_3,s(?x_18)) and +(?x_18,s(?x_3))
strenghten +(?x_3,s(?x_19)) and +(s(?x_3),?x_19)
strenghten +(s(?x_16),?y_4) and +(?x_16,s(?y_4))
strenghten +(s(?x_17),?y_4) and +(s(?y_4),?x_17)
strenghten +(?x_7,*(?x_7,?y_5)) and *(s(?y_5),?x_7)
strenghten +(?x_12,*(?x_12,?y_5)) and *(?x_12,s(?y_5))
strenghten +(*(?x_6,?x_14),?x_14) and *(?x_14,s(?x_6))
strenghten +(*(?x_6,?x_15),?x_15) and *(s(?x_6),?x_15)
strenghten +(?x_1,+(?y_1,?x_18)) and +(?x_18,+(?x_1,?y_1))
strenghten +(?x_1,+(?y_1,?x_19)) and +(+(?x_1,?y_1),?x_19)
strenghten +(?x_2,+(?y_2,?z_1)) and +(+(?y_2,?x_2),?z_1)
strenghten +(?x_3,s(s(?y_4))) and +(s(s(?x_3)),?y_4)
strenghten +(+(?x,?x_2),?y_2) and +(?x,+(?y_2,?x_2))
strenghten +(+(?x,?x_16),?y_16) and +(?x,+(?x_16,?y_16))
strenghten +(+(?x_2,?y),?z) and +(+(?y,?z),?x_2)
strenghten +(?x_1,+(?y_1,s(?y_4))) and +(s(+(?x_1,?y_1)),?y_4)
strenghten +(?x_4,+(s(?y_4),?z_1)) and +(+(s(?x_4),?y_4),?z_1)
strenghten +(+(?x,?x_4),s(?y_4)) and +(?x,+(s(?x_4),?y_4))
strenghten +(+(?x,s(?x_3)),?y_3) and +(?x,+(?x_3,s(?y_3)))
strenghten +(+(s(?x_3),?y),?z) and +(?x_3,s(+(?y,?z)))
strenghten +(s(?x_3),+(?y_3,?z_1)) and +(+(?x_3,s(?y_3)),?z_1)
strenghten +(*(?x_9,?x_9),s(+(?x_9,?x_9))) and *(s(?x_9),s(?x_9))
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly) (failure)
unknown Non-Confluence
Check relative termination:
   [ +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     *(?x,s(?y)) -> +(?x,*(?x,?y)),
     *(s(?x),?y) -> +(*(?x,?y),?y),
     sq(?x) -> *(?x,?x),
     sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x))) ]
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     *(?x,?y) -> *(?y,?x) ]
Polynomial Interpretation:
  *:= (2)*x1+(2)*x1*x2+(2)*x2
  +:= (1)*x1+(1)*x2
  s:= (4)+(1)*x1
  sq:= (9)+(8)*x1+(12)*x1*x1
retract *(?x,s(?y)) -> +(?x,*(?x,?y))
retract *(s(?x),?y) -> +(*(?x,?y),?y)
retract sq(?x) -> *(?x,?x)
retract sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x)))
retract *(?x,s(?y)) -> +(?x,*(?x,?y))
retract *(s(?x),?y) -> +(*(?x,?y),?y)
retract sq(?x) -> *(?x,?x)
retract sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x)))
unknown relative termination
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ *(?x,s(?y)) -> +(?x,*(?x,?y)),
     *(s(?x),?y) -> +(*(?x,?y),?y),
     sq(?x) -> *(?x,?x),
     sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x))) ]
P:
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     *(?x,?y) -> *(?y,?x) ]
S: terminating
CP(S,S):
<+(*(?x_1,s(?y)),s(?y)), +(s(?x_1),*(s(?x_1),?y))> --> <+(+(?x_1,*(?x_1,?y)),s(?y)), +(s(?x_1),+(*(?x_1,?y),?y))> => no
<+(*(?x_3,?x_3),s(+(?x_3,?x_3))), *(s(?x_3),s(?x_3))> --> <+(*(?x_3,?x_3),s(+(?x_3,?x_3))), +(s(?x_3),+(*(?x_3,?x_3),?x_3))> => no
PCP_in(symP,S):
CP(S,symP):
<+(?x,*(?x,?y)), *(s(?y),?x)> --> <+(?x,*(?x,?y)), +(*(?y,?x),?x)> => no
<+(*(?x,?y),?y), *(?y,s(?x))> --> <+(*(?x,?y),?y), +(?y,*(?y,?x))> => no
check joinability condition:
check modulo joinability of +(+(?x_1,*(?x_1,?y)),s(?y)) and +(s(?x_1),+(*(?x_1,?y),?y)): joinable by {1,2,4,5}
check modulo joinability of +(*(?x_3,?x_3),s(+(?x_3,?x_3))) and +(s(?x_3),+(*(?x_3,?x_3),?x_3)): joinable by {1,2,4}
check modulo joinability of +(?x,*(?x,?y)) and +(*(?y,?x),?x): joinable by {3}
check modulo joinability of +(*(?x,?y),?y) and +(?y,*(?y,?x)): joinable by {3}
success
P':
   [ +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y,?x) -> +(?x,?y) ]
Check relative termination:
   [ *(?x,s(?y)) -> +(?x,*(?x,?y)),
     *(s(?x),?y) -> +(*(?x,?y),?y),
     sq(?x) -> *(?x,?x),
     sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x))) ]
   [ +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?y,?x) -> +(?x,?y) ]
Polynomial Interpretation:
  *:= (2)*x1+(2)*x1*x2+(2)*x2
  +:= (1)*x1+(1)*x2
  s:= (4)+(1)*x1
  sq:= (15)+(9)*x1+(8)*x1*x1
retract *(?x,s(?y)) -> +(?x,*(?x,?y))
retract *(s(?x),?y) -> +(*(?x,?y),?y)
retract sq(?x) -> *(?x,?x)
Polynomial Interpretation:
  *:= (2)*x1*x1+(1)*x2
  +:= (1)*x1+(1)*x2
  s:= (1)*x1
  sq:= (7)+(8)*x1+(12)*x1*x1
relatively terminating
S/P': relatively terminating
S:
   [ *(?x,s(?y)) -> +(?x,*(?x,?y)),
     *(s(?x),?y) -> +(*(?x,?y),?y),
     sq(?x) -> *(?x,?x),
     sq(s(?x)) -> +(*(?x,?x),s(+(?x,?x))) ]
P:
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     +(s(?x),?y) -> +(?x,s(?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     *(?x,?y) -> *(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
60.trs: Success(CR)
(19608 msec.)