YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?x = +(0,?x),
     s(+(?y_1,?x_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:
   [ +(0,?x) = ?x,
     +(s(?y),?x) = s(+(?y,?x)),
     ?x = +(0,?x),
     s(+(?y_2,?x)) = +(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 <+(0,?x_2), ?x_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <+(s(?y_1),?x_2), s(+(?y_1,?x_2))> 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_2) and ?x_2
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(+(?y_1,0)) and s(?y_1)
strenghten s(+(0,?x_11)) and s(?x_11)
strenghten s(+(?y_1,?x_2)) and +(s(?y_1),?x_2)
strenghten s(+(?y_1,?x_7)) 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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (1)
  s:= (1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (2)+(2)*x1+(2)*x2
  0:= (15)
  s:= (3)+(1)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<s(+(?y,?x)), +(s(?y),?x)> --> <s(+(?y,?x)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?y,?x)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 1)
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
Added P-Rules:
   [ ]
replace: +(?x,s(?y)) -> s(+(?y,?x)) => +(?x,s(?y)) -> s(+(?x,?y))
STEP: 2 (linear)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<s(+(?y,?x)), +(s(?y),?x)> --> <s(+(?y,?x)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?y,?x)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 2)
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
Added P-Rules:
   [ ]
replace: +(?x,s(?y)) -> s(+(?y,?x)) => +(?x,s(?y)) -> s(+(?x,?y))
STEP: 3 (relative)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (1)
  s:= (1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (2)+(2)*x1+(2)*x2
  0:= (15)
  s:= (3)+(1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?y_3,0)), s(?y_3)> --> <s(?y_3), s(?y_3)> => yes
<s(?y_1), s(+(?y_1,0))> --> <s(?y_1), s(?y_1)> => yes
<s(+(?y_3,s(?y_1))), s(+(?y_1,s(?y_3)))> --> <s(s(+(?y_1,?y_3))), s(s(+(?y_3,?y_1)))> => yes
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(?y,?x)), +(s(?y),?x)> --> <s(+(?y,?x)), s(+(?y,?x))> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<s(+(?y,?x)), +(?x,s(?y))> --> <s(+(?y,?x)), s(+(?y,?x))> => yes
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?y,?x)),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?y,?x)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
200.trs: Success(CR)
(301 msec.)