YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x),
     *(0,?y) -> 0,
     *(s(?x),?y) -> +(*(?x,?y),?y),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     *(?x,?y) -> *(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_8) = +(?x,+(0,?z_8)),
     +(s(+(?x_1,?y_1)),?z_8) = +(?x_1,+(s(?y_1),?z_8)),
     +(?y_2,?z_8) = +(0,+(?y_2,?z_8)),
     +(s(+(?x_3,?y_3)),?z_8) = +(s(?x_3),+(?y_3,?z_8)),
     +(+(?y_9,?x_9),?z_8) = +(?x_9,+(?y_9,?z_8)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     +(?x_8,?y_8) = +(?x_8,+(?y_8,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(+(?x_8,?y_8),?y_1)) = +(?x_8,+(?y_8,s(?y_1))),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1),
     ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     0 = 0,
     0 = +(*(?x_7,0),0),
     0 = *(0,?x_4),
     +(*(0,?y_5),0) = 0,
     +(*(s(?x_7),?y_5),s(?x_7)) = +(*(?x_7,s(?y_5)),s(?y_5)),
     +(*(?x_5,?y_5),?x_5) = *(s(?y_5),?x_5),
     0 = *(?y_6,0),
     +(*(?x_7,?y_7),?y_7) = *(?y_7,s(?x_7)),
     +(?x_8,+(?y_8,?z_8)) = +(?z_8,+(?x_8,?y_8)) ]
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:
   [ 0 = 0,
     s(+(?x_3,0)) = s(?x_3),
     +(?x_8,+(?y_8,0)) = +(?x_8,?y_8),
     +(0,?x) = ?x,
     +(?x,+(0,?z_9)) = +(?x,?z_9),
     s(?y) = s(+(0,?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     +(?x_8,+(?y_8,s(?y))) = s(+(+(?x_8,?y_8),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x,+(s(?y),?z_9)) = +(s(+(?x,?y)),?z_9),
     s(+(0,?y_2)) = s(?y_2),
     +(?y,0) = ?y,
     +(0,+(?y,?z_9)) = +(?y,?z_9),
     s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(s(?x),+(?y,?z_9)) = +(s(+(?x,?y)),?z_9),
     +(*(?x_7,0),0) = 0,
     *(0,?x) = 0,
     0 = +(*(0,?y),0),
     +(*(?x_7,s(?y)),s(?y)) = +(*(s(?x_7),?y),s(?x_7)),
     *(s(?y),?x) = +(*(?x,?y),?x),
     +(*(0,?y_6),0) = 0,
     *(?y,0) = 0,
     0 = +(*(?x,0),0),
     +(*(s(?x),?y_6),s(?x)) = +(*(?x,s(?y_6)),s(?y_6)),
     *(?y,s(?x)) = +(*(?x,?y),?y),
     +(?x_1,+(?y_1,?y)) = +(+(?x_1,?y_1),+(?y,0)),
     ?x = +(?x,+(0,0)),
     s(+(?x,?y_3)) = +(?x,+(s(?y_3),0)),
     ?y = +(0,+(?y,0)),
     s(+(?x_5,?y)) = +(s(?x_5),+(?y,0)),
     +(?y,?x) = +(?x,+(?y,0)),
     s(+(+(?x_3,+(?y_3,?y)),?y_2)) = +(+(?x_3,?y_3),+(?y,s(?y_2))),
     s(+(?x,?y_2)) = +(?x,+(0,s(?y_2))),
     s(+(s(+(?x,?y_5)),?y_2)) = +(?x,+(s(?y_5),s(?y_2))),
     s(+(?y,?y_2)) = +(0,+(?y,s(?y_2))),
     s(+(s(+(?x_7,?y)),?y_2)) = +(s(?x_7),+(?y,s(?y_2))),
     s(+(+(?y,?x),?y_2)) = +(?x,+(?y,s(?y_2))),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?x) = +(?x,+(0,?z)),
     +(?z,s(+(?x,?y_3))) = +(?x,+(s(?y_3),?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,s(+(?x_5,?y))) = +(s(?x_5),+(?y,?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?x,+(?y,0)),
     s(+(+(?x,?y),?y_2)) = +(?x,+(?y,s(?y_2))),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?x,?z) = +(?x,+(0,?z)),
     +(s(+(?x,?y_3)),?z) = +(?x,+(s(?y_3),?z)),
     +(?y,?z) = +(0,+(?y,?z)),
     +(s(+(?x_5,?y)),?z) = +(s(?x_5),+(?y,?z)),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(?x,+(?z,?z_1)) = +(+(?x,+(0,?z)),?z_1),
     +(s(+(?x,?y_4)),+(?z,?z_1)) = +(+(?x,+(s(?y_4),?z)),?z_1),
     +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1),
     +(s(+(?x_6,?y)),+(?z,?z_1)) = +(+(s(?x_6),+(?y,?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?y = +(?y,0),
     s(+(?x_4,?y)) = +(?y,s(?x_4)),
     +(?x_9,+(?y_9,?y)) = +(?y,+(?x_9,?y_9)),
     +(?x,+(?y,?z_10)) = +(+(?y,?x),?z_10),
     0 = *(0,?x),
     +(*(?x,?y_6),?x) = *(s(?y_6),?x),
     0 = *(?y,0),
     +(*(?x_8,?y),?y) = *(?y,s(?x_8)) ]
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,?z_8), +(?x,+(0,?z_8))> by Rules <0, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],2)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_8), +(?x_1,+(s(?y_1),?z_8))> by Rules <1, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],8)], [([(+,2)],3),([],1)]>
Critical Pair <+(?y_2,?z_8), +(0,+(?y_2,?z_8))> by Rules <2, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(s(+(?x_3,?y_3)),?z_8), +(s(?x_3),+(?y_3,?z_8))> by Rules <3, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],8)], [([],3)]>
Critical Pair <+(+(?y_9,?x_9),?z_8), +(?x_9,+(?y_9,?z_8))> by Rules <9, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],9),([],8)], []>
 joinable by a reduction of rules <[([],8),([(+,2)],9)], [([],9),([],8)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <8, 8> preceded by [(+,1)]
 joinable by a reduction of rules <[([],8),([(+,2)],8)], [([],8)]>
Critical Pair <0, 0> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <s(+(?x_3,0)), s(?x_3)> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(?x_8,+(?y_8,0)), +(?x_8,?y_8)> by Rules <8, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
Critical Pair <+(0,?x_9), ?x_9> by Rules <9, 0> preceded by []
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <s(?y_1), s(+(0,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], [([(s,1)],2)]>
Critical Pair <s(+(?x_3,s(?y_1))), s(+(s(?x_3),?y_1))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([(s,1)],1)], [([(s,1)],3)]>
Critical Pair <+(?x_8,+(?y_8,s(?y_1))), s(+(+(?x_8,?y_8),?y_1))> by Rules <8, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],8)]>
Critical Pair <+(s(?y_1),?x_9), s(+(?x_9,?y_1))> by Rules <9, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([(s,1)],9)]>
Critical Pair <+(?y_9,0), ?y_9> by Rules <9, 2> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_9,s(?x_3)), s(+(?x_3,?y_9))> by Rules <9, 3> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],9)]>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <+(*(?x_7,0),0), 0> by Rules <7, 4> preceded by []
 joinable by a reduction of rules <[([(+,1)],4),([],2)], []>
 joinable by a reduction of rules <[([(+,1)],4),([],0)], []>
 joinable by a reduction of rules <[([],0),([],4)], []>
Critical Pair <*(0,?x_10), 0> by Rules <10, 4> preceded by []
 joinable by a reduction of rules <[([],6)], []>
Critical Pair <0, +(*(0,?y_5),0)> by Rules <6, 5> preceded by []
 joinable by a reduction of rules <[], [([(+,1)],6),([],2)]>
 joinable by a reduction of rules <[], [([(+,1)],6),([],0)]>
 joinable by a reduction of rules <[], [([],0),([],6)]>
Critical Pair <+(*(?x_7,s(?y_5)),s(?y_5)), +(*(s(?x_7),?y_5),s(?x_7))> by Rules <7, 5> preceded by []
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([(+,1)],7),([],9),([],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([(+,1)],7),([],1),([(s,1)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([],9),([(+,2)],7),([],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([],9),([],3),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([],1),([(s,1),(+,1)],7),([(s,1)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([(+,1)],9),([],1),([(s,1)],8)], [([],1),([(s,1)],9),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],9),([(+,2)],3)], [([(+,1)],7),([],8),([(+,2)],1)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],1),([(+,2),(s,1)],9)], [([(+,1)],7),([],8),([(+,2)],1)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([(+,1)],7),([],9),([],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([(+,1)],7),([],1),([(s,1)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([],9),([(+,2)],7),([],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([],9),([],3),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([],1),([(s,1),(+,1)],7),([(s,1)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1),(+,1)],9),([(s,1)],8)], [([],1),([(s,1)],9),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],8),([(s,1),(+,2)],9)], [([(+,1)],7),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],8),([(s,1),(+,2)],9)], [([],1),([(s,1),(+,1)],7),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([(+,1)],7),([],9),([],3)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([(+,1)],7),([],1),([(s,1)],9)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([],9),([(+,2)],7),([],3)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([],9),([],3),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([],1),([(s,1),(+,1)],7),([(s,1)],9)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1),(+,1)],9),([(s,1)],8)], [([],1),([(s,1)],9),([(s,1),(+,2)],7)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],8),([(s,1),(+,2)],9)], [([(+,1)],7),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],8),([(s,1),(+,2)],9)], [([],1),([(s,1),(+,1)],7),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],9),([],3)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],9),([],3)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],9),([],3)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],1)], [([(+,1)],7),([],8),([(+,2)],9),([(+,2)],3)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],1)], [([(+,1)],7),([],8),([(+,2)],1),([(+,2),(s,1)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],9)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],9)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],9)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],8)], [([(+,1)],7),([],1),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],8)], [([],1),([(s,1),(+,1)],7),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([],9),([(+,2)],5),([],3)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([(+,2)],5),([],3)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([(+,2)],5),([],3)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([],3),([(s,1),(+,2)],5)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([],3),([(s,1),(+,2)],5)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],9),([],3),([(s,1),(+,2)],5)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],9)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],9)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],9)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],8)], [([(+,1)],7),([],1),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],8)], [([],1),([(s,1),(+,1)],7),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([],1),([(s,1)],9),([(s,1),(+,2)],5)], [([(+,1)],7),([(+,1)],9),([],1),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1)],9),([(s,1),(+,2)],5)], [([(+,1)],7),([],1),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([],1),([(s,1)],9),([(s,1),(+,2)],5)], [([],1),([(s,1),(+,1)],7),([(s,1),(+,1)],9),([(s,1)],8)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],1),([],1)], [([(+,1)],7),([],1),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],8),([(+,2)],1),([],1)], [([],1),([(s,1),(+,1)],7),([(s,1)],8),([(s,1),(+,2)],9)]>
 joinable by a reduction of rules <[([(+,1)],5),([],1),([(s,1)],8),([(s,1),(+,2)],9)], [([(+,1)],7),([],8),([(+,2)],1),([],1)]>
 joinable by a reduction of rules <[([],1),([(s,1),(+,1)],5),([(s,1)],8),([(s,1),(+,2)],9)], [([(+,1)],7),([],8),([(+,2)],1),([],1)]>
Critical Pair <*(s(?y_5),?x_10), +(*(?x_10,?y_5),?x_10)> by Rules <10, 5> preceded by []
 joinable by a reduction of rules <[([],7)], [([(+,1)],10)]>
Critical Pair <*(?y_10,0), 0> by Rules <10, 6> preceded by []
 joinable by a reduction of rules <[([],4)], []>
Critical Pair <*(?y_10,s(?x_7)), +(*(?x_7,?y_10),?y_10)> by Rules <10, 7> preceded by []
 joinable by a reduction of rules <[([],5)], [([(+,1)],10)]>
Critical Pair <+(?y_9,+(?x_8,?y_8)), +(?x_8,+(?y_8,?y_9))> by Rules <9, 8> preceded by []
 joinable by a reduction of rules <[([],9),([],8)], []>
unknown Diagram Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     *(?x_4,0) -> 0,
     *(?x_5,s(?y_5)) -> +(*(?x_5,?y_5),?x_5),
     *(0,?y_6) -> 0,
     *(s(?x_7),?y_7) -> +(*(?x_7,?y_7),?y_7),
     +(+(?x_8,?y_8),?z_8) -> +(?x_8,+(?y_8,?z_8)),
     +(?x_9,?y_9) -> +(?y_9,?x_9),
     *(?x_10,?y_10) -> *(?y_10,?x_10) ]
Sort Assignment:
 * : 29*29=>29
 + : 29*29=>29
 0 : =>29
 s : 29=>29
non-linear variables: {?x_5,?y_7}
non-linear types: {29}
types leq non-linear types: {29}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     *(?x_4,0) -> 0,
     *(?x_5,s(?y_5)) -> +(*(?x_5,?y_5),?x_5),
     *(0,?y_6) -> 0,
     *(s(?x_7),?y_7) -> +(*(?x_7,?y_7),?y_7),
     +(+(?x_8,?y_8),?z_8) -> +(?x_8,+(?y_8,?z_8)),
     +(?x_9,?y_9) -> +(?y_9,?x_9),
     *(?x_10,?y_10) -> *(?y_10,?x_10) ]
NLR:
0: {}
1: {}
2: {}
3: {}
4: {}
5: {0,1,2,3,4,5,6,7,8,9,10}
6: {}
7: {0,1,2,3,4,5,6,7,8,9,10}
8: {}
9: {}
10: {}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     *(?x_4,0) -> 0,
     *(?x_5,s(?y_5)) -> +(*(?x_5,?y_5),?x_5),
     *(0,?y_6) -> 0,
     *(s(?x_7),?y_7) -> +(*(?x_7,?y_7),?y_7),
     +(+(?x_8,?y_8),?z_8) -> +(?x_8,+(?y_8,?z_8)),
     +(?x_9,?y_9) -> +(?y_9,?x_9),
     *(?x_10,?y_10) -> *(?y_10,?x_10) ]
Sort Assignment:
 * : 29*29=>29
 + : 29*29=>29
 0 : =>29
 s : 29=>29
non-linear variables: {?x_5,?y_7}
non-linear types: {29}
types leq non-linear types: {29}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     *(?x_4,0) -> 0,
     *(?x_5,s(?y_5)) -> +(*(?x_5,?y_5),?x_5),
     *(0,?y_6) -> 0,
     *(s(?x_7),?y_7) -> +(*(?x_7,?y_7),?y_7),
     +(+(?x_8,?y_8),?z_8) -> +(?x_8,+(?y_8,?z_8)),
     +(?x_9,?y_9) -> +(?y_9,?x_9),
     *(?x_10,?y_10) -> *(?y_10,?x_10) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 21 rules by 3 steps unfolding
strenghten *(?x_13,0) and 0
strenghten *(0,?x_10) and 0
strenghten +(?x_17,0) and ?x_17
strenghten +(0,?x_9) and ?x_9
strenghten *(?x_20,?y_20) and *(?y_20,?x_20)
strenghten s(+(?x_3,0)) and s(?x_3)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten +(*(?x_7,0),0) and 0
strenghten +(*(0,?y_5),0) and 0
strenghten s(+(?x_3,?y_9)) and +(?y_9,s(?x_3))
strenghten s(+(?x_9,?y_1)) and +(s(?y_1),?x_9)
strenghten +(?x,+(0,?z_8)) and +(?x,?z_8)
strenghten +(?x_8,+(?y_8,0)) and +(?x_8,?y_8)
strenghten +(0,+(?x_17,?z_8)) and +(?x_17,?z_8)
strenghten +(*(?x_7,?y_10),?y_10) and *(?y_10,s(?x_7))
strenghten +(*(?x_7,?y_19),?y_19) and *(s(?x_7),?y_19)
strenghten +(*(?x_10,?y_5),?x_10) and *(s(?y_5),?x_10)
strenghten +(*(?x_19,?y_5),?x_19) and *(?x_19,s(?y_5))
strenghten +(?x_8,+(?y_8,?y_9)) and +(?y_9,+(?x_8,?y_8))
strenghten +(?x_9,+(?y_9,?z_8)) and +(+(?y_9,?x_9),?z_8)
strenghten s(+(s(?x_3),?y_1)) and s(+(?x_3,s(?y_1)))
strenghten +(?x_1,+(s(?y_1),?z_8)) and +(s(+(?x_1,?y_1)),?z_8)
strenghten +(s(?x_3),+(?y_3,?z_8)) and +(s(+(?x_3,?y_3)),?z_8)
strenghten s(+(+(?x_8,?y_8),?y_1)) and +(?x_8,+(?y_8,s(?y_1)))
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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     *(?x,0) -> 0,
     *(?x,s(?y)) -> +(*(?x,?y),?x),
     *(0,?y) -> 0,
     *(s(?x),?y) -> +(*(?x,?y),?y) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     *(?x,?y) -> *(?y,?x) ]
Polynomial Interpretation:
  *:= (1)*x1+(1)*x1*x2+(1)*x2
  +:= (1)*x1+(1)*x2
  0:= (11)
  s:= (4)+(1)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
retract *(s(?x),?y) -> +(*(?x,?y),?y)
Polynomial Interpretation:
  *:= (1)+(2)*x1+(2)*x1*x1+(2)*x2+(2)*x2*x2
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (14)
  s:= (4)+(1)*x1
retract +(?x,0) -> ?x
retract +(?x,s(?y)) -> s(+(?x,?y))
retract +(0,?y) -> ?y
retract +(s(?x),?y) -> s(+(?x,?y))
retract *(?x,s(?y)) -> +(*(?x,?y),?x)
retract *(s(?x),?y) -> +(*(?x,?y),?y)
Polynomial Interpretation:
  *:= (3)+(2)*x1+(2)*x2
  +:= (1)+(1)*x1+(1)*x2
  0:= 0
  s:= (15)*x1
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Final result: CR
169.trs: Success(CR)
(5321 msec.)