YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ s(?y_1) = s(+(0,?y_1)),
     ?y = +(?y,0),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1) ]
Overlay, check Innermost Termination...
unknown Innermost 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:
   [ s(+(0,?y_1)) = s(?y_1),
     +(?y,0) = ?y,
     s(?y) = s(+(0,?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     ?y = +(?y,0),
     s(+(?x,?y_2)) = +(s(?y_2),?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 <s(+(0,?y_1)), s(?y_1)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(?y_2,0), ?y_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <+(s(?y_1),?x_2), s(+(?x_2,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
strenghten +(?x_5,0) and ?x_5
strenghten +(0,?x_3) and ?x_3
strenghten +(?x_8,?y_8) and +(?y_8,?x_8)
strenghten +(?x_11,s(0)) and s(?x_11)
strenghten +(s(0),?x_11) and s(?x_11)
strenghten s(+(?x_11,0)) and s(?x_11)
strenghten s(+(0,?y_1)) and s(?y_1)
strenghten s(+(?x_2,?y_1)) and +(s(?y_1),?x_2)
strenghten s(+(?x_7,?y_1)) and +(?x_7,s(?y_1))
obtain 12 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:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x1*x2+(1)*x2
  0:= 0
  s:= (9)+(2)*x1
retract +(0,?y) -> ?y
Polynomial Interpretation:
  +:= (1)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= 0
  s:= (6)+(3)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?y_1)), s(?y_1)> --> <s(?y_1), s(?y_1)> => yes
PCP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, +(?y,0)> => no
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?y to +(?y,0): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 1)
   [ +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added P-Rules:
   [ ]
replace: +(?x,s(?y)) -> s(+(?x,?y)) => +(?x,s(?y)) -> s(+(?y,?x))
STEP: 2 (linear)
S:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?y_1)), s(?y_1)> --> <s(?y_1), s(?y_1)> => yes
CP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, +(?y,0)> => no
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?y to +(?y,0): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 2)
   [ +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added P-Rules:
   [ ]
replace: +(?x,s(?y)) -> s(+(?x,?y)) => +(?x,s(?y)) -> s(+(?y,?x))
STEP: 3 (relative)
S:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x1*x2+(1)*x2
  0:= 0
  s:= (9)+(2)*x1
retract +(0,?y) -> ?y
Polynomial Interpretation:
  +:= (1)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= 0
  s:= (6)+(3)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?y_1)), s(?y_1)> --> <s(?y_1), s(?y_1)> => yes
<0, 0> --> <0, 0> => yes
<s(+(s(?y_1),?y_3)), s(+(s(?y_3),?y_1))> --> <s(s(+(?y_3,?y_1))), s(s(+(?y_1,?y_3)))> => yes
<s(+(0,?y_3)), s(?y_3)> --> <s(?y_3), s(?y_3)> => yes
PCP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, ?y> => yes
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<?y, +(0,?y)> --> <?y, ?y> => yes
<s(+(?x,?y)), +(?x,s(?y))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
S:
   [ +(0,?y) -> ?y,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?y,0) -> ?y,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
128.trs: Success(CR)
(308 msec.)