YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ s(?x_1) = s(+(?x_1,0)),
     ?x = +(0,?x),
     s(+(?x_1,?y_1)) = +(?y_1,s(?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(+(?x_1,0)) = s(?x_1),
     +(0,?x) = ?x,
     s(?x) = s(+(?x,0)),
     +(?y,s(?x)) = s(+(?x,?y)),
     ?x = +(0,?x),
     s(+(?x_2,?y)) = +(?y,s(?x_2)) ]
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(+(?x_1,0)), s(?x_1)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(0,?x_2), ?x_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <+(?y_2,s(?x_1)), s(+(?x_1,?y_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(+(?x_1,0)) and s(?x_1)
strenghten s(+(?x_1,?x_9)) and +(?x_9,s(?x_1))
strenghten s(+(?x_1,?x_10)) and +(s(?x_1),?x_10)
strenghten s(+(?x_1,s(0))) and s(s(?x_1))
obtain 16 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,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(2)*x1*x1+(1)*x2+(2)*x2*x2
  0:= 0
  s:= (2)+(1)*x1
retract +(s(?x),?y) -> s(+(?x,?y))
Polynomial Interpretation:
  +:= (2)*x1*x1+(2)*x1*x2+(2)*x2*x2
  0:= (3)
  s:= (2)+(1)*x1*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(?x_1,0)), s(?x_1)> --> <s(?x_1), s(?x_1)> => yes
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 1)
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
Added P-Rules:
   [ ]
replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(+(?y,?x))
STEP: 2 (linear)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(?x_1,0)), s(?x_1)> --> <s(?x_1), s(?x_1)> => yes
CP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 2)
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
Added P-Rules:
   [ ]
replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(+(?y,?x))
STEP: 3 (relative)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(2)*x1*x1+(1)*x2+(2)*x2*x2
  0:= 0
  s:= (2)+(1)*x1
retract +(s(?x),?y) -> s(+(?x,?y))
Polynomial Interpretation:
  +:= (2)*x1*x1+(2)*x1*x2+(2)*x2*x2
  0:= (3)
  s:= (2)+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(?x_1,0)), s(?x_1)> --> <s(?x_1), s(?x_1)> => yes
<0, 0> --> <0, 0> => yes
<s(+(?x_3,s(?x_1))), s(+(?x_1,s(?x_3)))> --> <s(s(+(?x_1,?x_3))), s(s(+(?x_3,?x_1)))> => yes
<s(+(?x_3,0)), s(?x_3)> --> <s(?x_3), s(?x_3)> => yes
PCP_in(symP,S):
CP(S,symP):
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<s(+(?x,?y)), +(s(?x),?y)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
186.trs: Success(CR)
(290 msec.)