YES
(ignored inputs)COMMENT the following rules are removed from the original TRS * ( x , 0 ( ) ) -> 0 ( ) * ( x , s ( y ) ) -> + ( * ( x , y ) , x )
Rewrite Rules:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ +(0,?x) = ?x,
     +(s(?y_2),?x) = s(+(?x,?y_2)) ]
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:
   [ ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     +(0,?x) = ?x,
     +(s(?y),?x) = s(+(?x,?y)) ]
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_1, +(0,?x_1)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],1)]>
Critical Pair <s(+(?x_2,?y_2)), +(s(?y_2),?x_2)> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],2)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 6 rules by 3 steps unfolding
obtain 14 candidates for checking non-joinability
check by TCAP-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)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(2)*x1*x2+(1)*x2
  0:= 0
  s:= (4)+(8)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)*x1*x1+(1)*x2*x2
  0:= 0
  s:= (5)+(1)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
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(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 1)
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
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(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), +(s(?y),?x)> => no
check joinability condition:
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(s(?y),?x): maybe not reachable
failed
failure(Step 2)
   [ +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(2)*x1*x2+(1)*x2
  0:= 0
  s:= (4)+(8)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)*x1*x1+(1)*x2*x2
  0:= 0
  s:= (5)+(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(+(?x,?y)),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(0,?y_3)), s(?y_3)> --> <s(?y_3), s(?y_3)> => yes
<s(?y_1), s(+(0,?y_1))> --> <s(?y_1), s(?y_1)> => 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
<0, 0> --> <0, 0> => yes
<s(+(0,?y_1)), s(?y_1)> --> <s(?y_1), s(?y_1)> => yes
<s(?y_3), s(+(0,?y_3))> --> <s(?y_3), s(?y_3)> => yes
<s(+(s(?y_3),?y_1)), s(+(s(?y_1),?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(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<s(+(?x,?y)), +(?x,s(?y))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?x) -> ?x,
     +(s(?y),?x) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Combined result: CR
/tmp/file21lWNy.trs: Success(CR)
(182 msec.)