(ignored inputs)COMMENT addition + commutative law
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?x = +(0,?x),
     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 Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?x = +(0,?x),
     +(s(?y),?x) = s(+(?x,?y)),
     +(0,?x) = ?x ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & 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(+(?x_2,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
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)
   [ +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
Added S-Rules:
   [ +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
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)
   [ +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
Added S-Rules:
   [ +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
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)+(2)*x1*x1+(1)*x1*x2+(2)*x2*x2
  0:= (8)
  s:= (1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (2)*x1*x1+(1)*x1*x2+(2)*x2*x2
  0:= 0
  s:= (4)+(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)),
     +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?y_2)), s(?y_2)> --> <s(?y_2), s(?y_2)> => yes
<0, 0> --> <0, 0> => yes
<s(+(s(?y_1),?y_2)), s(+(s(?y_2),?y_1))> --> <s(s(+(?y_2,?y_1))), s(s(+(?y_1,?y_2)))> => yes
<s(?y_1), s(+(0,?y_1))> --> <s(?y_1), s(?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
<s(+(?x,?y)), +(?x,s(?y))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(s(?y),?x) -> s(+(?x,?y)),
     +(0,?x) -> ?x ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/106.trs: Success(CR)
(118 msec.)