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)),
     sum(0) -> 0,
     sum(s(?x)) -> +(s(?x),sum(?x)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_5) = +(?x,+(0,?z_5)),
     +(s(+(?x_1,?y_1)),?z_5) = +(?x_1,+(s(?y_1),?z_5)),
     +(?y_2,?z_5) = +(0,+(?y_2,?z_5)),
     +(s(+(?x_3,?y_3)),?z_5) = +(s(?x_3),+(?y_3,?z_5)),
     +(+(?y_6,?x_6),?z_5) = +(?x_6,+(?y_6,?z_5)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     +(?x_5,?y_5) = +(?x_5,+(?y_5,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(+(?x_5,?y_5),?y_1)) = +(?x_5,+(?y_5,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)),
     +(?x_5,+(?y_5,?z_5)) = +(?z_5,+(?x_5,?y_5)) ]
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_5,+(?y_5,0)) = +(?x_5,?y_5),
     +(0,?x) = ?x,
     +(?x,+(0,?z_6)) = +(?x,?z_6),
     s(?y) = s(+(0,?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     +(?x_5,+(?y_5,s(?y))) = s(+(+(?x_5,?y_5),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x,+(s(?y),?z_6)) = +(s(+(?x,?y)),?z_6),
     s(+(0,?y_2)) = s(?y_2),
     +(?y,0) = ?y,
     +(0,+(?y,?z_6)) = +(?y,?z_6),
     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_6)) = +(s(+(?x,?y)),?z_6),
     +(?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_6,+(?y_6,?y)) = +(?y,+(?x_6,?y_6)),
     +(?x,+(?y,?z_7)) = +(+(?y,?x),?z_7) ]
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_5), +(?x,+(0,?z_5))> by Rules <0, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],2)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_5), +(?x_1,+(s(?y_1),?z_5))> by Rules <1, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],6)], [([(+,2)],3),([],1)]>
Critical Pair <+(?y_2,?z_5), +(0,+(?y_2,?z_5))> by Rules <2, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(s(+(?x_3,?y_3)),?z_5), +(s(?x_3),+(?y_3,?z_5))> by Rules <3, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],6)], [([],3)]>
Critical Pair <+(+(?y_6,?x_6),?z_5), +(?x_6,+(?y_6,?z_5))> by Rules <7, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],7),([],6)], []>
 joinable by a reduction of rules <[([],6),([(+,2)],7)], [([],7),([],6)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <6, 6> preceded by [(+,1)]
 joinable by a reduction of rules <[([],6),([(+,2)],6)], [([],6)]>
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_5,+(?y_5,0)), +(?x_5,?y_5)> by Rules <6, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
Critical Pair <+(0,?x_6), ?x_6> by Rules <7, 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_5,+(?y_5,s(?y_1))), s(+(+(?x_5,?y_5),?y_1))> by Rules <6, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],6)]>
Critical Pair <+(s(?y_1),?x_6), s(+(?x_6,?y_1))> by Rules <7, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([(s,1)],7)]>
Critical Pair <+(?y_6,0), ?y_6> by Rules <7, 2> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_6,s(?x_3)), s(+(?x_3,?y_6))> by Rules <7, 3> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],7)]>
Critical Pair <+(?y_6,+(?x_5,?y_5)), +(?x_5,+(?y_5,?y_6))> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[([],7),([],6)], []>
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)),
     sum(0) -> 0,
     sum(s(?x_4)) -> +(s(?x_4),sum(?x_4)),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_6,?x_6) ]
Sort Assignment:
 + : 21*21=>21
 0 : =>21
 s : 21=>21
 sum : 21=>21
non-linear variables: {?x_4}
non-linear types: {21}
types leq non-linear types: {21}
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)),
     sum(0) -> 0,
     sum(s(?x_4)) -> +(s(?x_4),sum(?x_4)),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_6,?x_6) ]
NLR:
0: {}
1: {}
2: {}
3: {}
4: {}
5: {0,1,2,3,4,5,6,7}
6: {}
7: {}
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)),
     sum(0) -> 0,
     sum(s(?x_4)) -> +(s(?x_4),sum(?x_4)),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_6,?x_6) ]
Sort Assignment:
 + : 21*21=>21
 0 : =>21
 s : 21=>21
 sum : 21=>21
non-linear variables: {?x_4}
non-linear types: {21}
types leq non-linear types: {21}
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)),
     sum(0) -> 0,
     sum(s(?x_4)) -> +(s(?x_4),sum(?x_4)),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_6,?x_6) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 18 rules by 3 steps unfolding
strenghten sum(0) and 0
strenghten +(?x_9,0) and ?x_9
strenghten +(0,?x_6) and ?x_6
strenghten +(sum(0),?x_11) and ?x_11
strenghten s(+(?x_3,0)) and s(?x_3)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten s(+(?x_3,sum(0))) and s(?x_3)
strenghten s(+(?x_3,?y_6)) and +(?y_6,s(?x_3))
strenghten s(+(?x_6,?y_1)) and +(s(?y_1),?x_6)
strenghten +(?x,+(0,?z_5)) and +(?x,?z_5)
strenghten +(?x_5,+(?y_5,0)) and +(?x_5,?y_5)
strenghten +(0,+(?x_9,?z_5)) and +(?x_9,?z_5)
strenghten +(?x_5,+(?y_5,sum(0))) and +(?x_5,?y_5)
strenghten +(?x_11,+(sum(0),?z_5)) and +(?x_11,?z_5)
strenghten +(?x_5,+(?y_5,?y_6)) and +(?y_6,+(?x_5,?y_5))
strenghten +(?x_6,+(?y_6,?z_5)) and +(+(?y_6,?x_6),?z_5)
strenghten s(+(s(?x_3),?y_1)) and s(+(?x_3,s(?y_1)))
strenghten +(?x_1,+(s(?y_1),?z_5)) and +(s(+(?x_1,?y_1)),?z_5)
strenghten +(s(?x_3),+(?y_3,?z_5)) and +(s(+(?x_3,?y_3)),?z_5)
strenghten s(+(+(?x_5,?y_5),?y_1)) and +(?x_5,+(?y_5,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)),
     sum(0) -> 0,
     sum(s(?x)) -> +(s(?x),sum(?x)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= 0
  s:= (1)+(1)*x1
  sum:= (6)*x1*x1
retract sum(s(?x)) -> +(s(?x),sum(?x))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (1)*x1
  sum:= (6)+(1)*x1*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract sum(0) -> 0
retract sum(s(?x)) -> +(s(?x),sum(?x))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
  sum:= (1)*x1
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Final result: CR
198.trs: Success(CR)
(8253 msec.)