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)),
     *(k,0) -> 0,
     *(k,s(?y)) -> +(k,*(k,?y)),
     +(?x,?y) -> +(?y,?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_6) = +(?x,+(0,?z_6)),
     +(s(+(?x_1,?y_1)),?z_6) = +(?x_1,+(s(?y_1),?z_6)),
     +(?y_2,?z_6) = +(0,+(?y_2,?z_6)),
     +(s(+(?x_3,?y_3)),?z_6) = +(s(?x_3),+(?y_3,?z_6)),
     +(+(?y_5,?x_5),?z_6) = +(?x_5,+(?y_5,?z_6)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     ?x = +(0,?x),
     +(?x_6,?y_6) = +(?x_6,+(?y_6,0)),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1),
     s(+(+(?x_6,?y_6),?y_1)) = +(?x_6,+(?y_6,s(?y_1))),
     ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(?y_5,+(?x_6,?y_6)) = +(?x_6,+(?y_6,?y_5)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly 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),
     +(0,?x) = ?x,
     +(?x_6,+(?y_6,0)) = +(?x_6,?y_6),
     +(?x,+(0,?z_7)) = +(?x,?z_7),
     s(?y) = s(+(0,?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x_6,+(?y_6,s(?y))) = s(+(+(?x_6,?y_6),?y)),
     +(?x,+(s(?y),?z_7)) = +(s(+(?x,?y)),?z_7),
     s(+(0,?y_2)) = s(?y_2),
     +(?y,0) = ?y,
     +(0,+(?y,?z_7)) = +(?y,?z_7),
     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_7)) = +(s(+(?x,?y)),?z_7),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?y = +(?y,0),
     s(+(?x_4,?y)) = +(?y,s(?x_4)),
     +(?x_6,+(?y_6,?y)) = +(?y,+(?x_6,?y_6)),
     +(?x,+(?y,?z_7)) = +(+(?y,?x),?z_7),
     +(?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) ]
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_6), +(?x,+(0,?z_6))> by Rules <0, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],2)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_6), +(?x_1,+(s(?y_1),?z_6))> by Rules <1, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],7)], [([(+,2)],3),([],1)]>
Critical Pair <+(?y_2,?z_6), +(0,+(?y_2,?z_6))> by Rules <2, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(s(+(?x_3,?y_3)),?z_6), +(s(?x_3),+(?y_3,?z_6))> by Rules <3, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],7)], [([],3)]>
Critical Pair <+(+(?y_5,?x_5),?z_6), +(?x_5,+(?y_5,?z_6))> by Rules <6, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],6),([],7)], []>
 joinable by a reduction of rules <[([],7),([(+,2)],6)], [([],6),([],7)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <7, 7> preceded by [(+,1)]
 joinable by a reduction of rules <[([],7),([(+,2)],7)], [([],7)]>
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 <+(0,?x_5), ?x_5> by Rules <6, 0> preceded by []
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <+(?x_6,+(?y_6,0)), +(?x_6,?y_6)> by Rules <7, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
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 <+(s(?y_1),?x_5), s(+(?x_5,?y_1))> by Rules <6, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([(s,1)],6)]>
Critical Pair <+(?x_6,+(?y_6,s(?y_1))), s(+(+(?x_6,?y_6),?y_1))> by Rules <7, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],7)]>
Critical Pair <+(?y_5,0), ?y_5> by Rules <6, 2> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_5,s(?x_3)), s(+(?x_3,?y_5))> by Rules <6, 3> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],6)]>
Critical Pair <+(?x_6,+(?y_6,?z_6)), +(?z_6,+(?x_6,?y_6))> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[], [([],6),([],7)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 18 rules by 3 steps unfolding
strenghten +(?x_9,0) and ?x_9
strenghten +(0,?x_5) and ?x_5
strenghten +(?x_12,?y_12) and +(?y_12,?x_12)
strenghten +(?x_15,s(0)) and s(?x_15)
strenghten +(s(0),?x_15) and s(?x_15)
strenghten s(+(?x_3,0)) and s(?x_3)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten s(+(?x_3,?x_13)) and +(?x_13,s(?x_3))
strenghten s(+(?x_3,?x_14)) and +(s(?x_3),?x_14)
strenghten s(+(?x_5,?y_1)) and +(s(?y_1),?x_5)
strenghten s(+(?x_11,?y_1)) and +(?x_11,s(?y_1))
strenghten +(?x,+(0,?z_6)) and +(?x,?z_6)
strenghten +(?x_6,+(?y_6,0)) and +(?x_6,?y_6)
strenghten +(0,+(?x_9,?z_6)) and +(?x_9,?z_6)
strenghten s(+(?x_3,s(0))) and s(s(?x_3))
strenghten +(?x_5,+(?y_5,?z_6)) and +(+(?y_5,?x_5),?z_6)
strenghten +(?x_6,+(?y_6,?x_13)) and +(?x_13,+(?x_6,?y_6))
strenghten +(?x_6,+(?y_6,?x_14)) and +(+(?x_6,?y_6),?x_14)
strenghten +(?z_6,+(?x_6,?y_6)) and +(?x_6,+(?y_6,?z_6))
strenghten s(+(s(?x_3),?y_1)) and s(+(?x_3,s(?y_1)))
strenghten +(?x_6,+(?y_6,s(0))) and s(+(?x_6,?y_6))
strenghten +(?x_15,+(s(0),?z_6)) and +(s(?x_15),?z_6)
strenghten +(?x_1,+(s(?y_1),?z_6)) and +(s(+(?x_1,?y_1)),?z_6)
strenghten +(s(?x_3),+(?y_3,?z_6)) and +(s(+(?x_3,?y_3)),?z_6)
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)),
     *(k,0) -> 0,
     *(k,s(?y)) -> +(k,*(k,?y)) ]
   [ +(?x,?y) -> +(?y,?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Polynomial Interpretation:
  *:= (2)+(1)*x1+(1)*x2
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= (8)
  k:= 0
  s:= (8)+(8)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract *(k,0) -> 0
retract *(k,s(?y)) -> +(k,*(k,?y))
Polynomial Interpretation:
  *:= (2)*x1*x1+(1)*x2
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (10)
  k:= 0
  s:= (1)+(1)*x1
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Final result: CR
151.trs: Success(CR)
(7317 msec.)