YES
(ignored inputs)COMMENT experiments for [36] submitted by: Takahito Aoto
Rewrite Rules:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     +(?x,?y) -> +(?y,?x),
     s(s(?x)) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [ s(?x) = s(?x) ]
Outer CPs:
   [ ?y = +(?y,0),
     s(+(0,?y_1)) = +(?y_1,s(0)) ]
not Overlay, check Termination...
unknown/not 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:
   [ +(?y,0) = ?y,
     +(?y,s(0)) = s(+(0,?y)),
     ?y = +(?y,0),
     s(+(0,?y)) = +(?y,s(0)),
     s(?x_1) = s(?x_1),
     ?x_1 = s(s(?x_1)) ]
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), s(?x)> by Rules <3, 3> preceded by [(s,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <+(?y_2,0), ?y_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <+(?y_2,s(0)), s(+(0,?y_2))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 7 rules by 3 steps unfolding
obtain 24 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:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (2)*x1+(1)*x1*x1+(2)*x2+(1)*x2*x2
  0:= 0
  s:= (4)+(1)*x1
retract +(s(0),?y) -> s(+(0,?y))
retract s(s(?x)) -> ?x
Polynomial Interpretation:
  +:= (2)+(1)*x1+(1)*x2
  0:= 0
  s:= (4)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(?x), s(?x)> --> <s(?x), s(?x)> => yes
PCP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, +(?y,0)> => no
<s(+(0,?y)), +(?y,s(0))> --> <s(?y), +(?y,s(0))> => no
check joinability condition:
check modulo reachablity from ?y to +(?y,0): maybe not reachable
check modulo reachablity from s(?y) to +(?y,s(0)): maybe not reachable
failed
failure(Step 1)
   [ +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
Added S-Rules:
   [ +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
Added P-Rules:
   [ ]
replace: +(s(0),?y) -> s(+(0,?y)) => +(s(0),?y) -> s(+(?y,0))
STEP: 2 (linear)
S:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(?x), s(?x)> --> <s(?x), s(?x)> => yes
CP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, +(?y,0)> => no
<s(+(0,?y)), +(?y,s(0))> --> <s(?y), +(?y,s(0))> => no
check joinability condition:
check modulo reachablity from ?y to +(?y,0): maybe not reachable
check modulo reachablity from s(?y) to +(?y,s(0)): maybe not reachable
failed
failure(Step 2)
   [ +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
Added S-Rules:
   [ +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
Added P-Rules:
   [ ]
replace: +(s(0),?y) -> s(+(0,?y)) => +(s(0),?y) -> s(+(?y,0))
STEP: 3 (relative)
S:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x ]
   [ +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (2)*x1+(1)*x1*x1+(2)*x2+(1)*x2*x2
  0:= 0
  s:= (4)+(1)*x1
retract +(s(0),?y) -> s(+(0,?y))
retract s(s(?x)) -> ?x
Polynomial Interpretation:
  +:= (2)+(1)*x1+(1)*x2
  0:= 0
  s:= (4)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x,
     +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(0), s(0)> --> <s(0), s(0)> => yes
<s(0), s(+(0,0))> --> <s(0), s(0)> => yes
<s(s(0)), s(+(0,s(0)))> --> <0, 0> => yes
<s(?x), s(?x)> --> <s(?x), s(?x)> => yes
<0, 0> --> <0, 0> => yes
<s(+(0,0)), s(0)> --> <s(0), s(0)> => yes
<s(0), s(0)> --> <s(0), s(0)> => yes
<s(+(0,s(0))), s(s(0))> --> <0, 0> => yes
PCP_in(symP,S):
CP(S,symP):
<?y, +(?y,0)> --> <?y, ?y> => yes
<s(+(0,?y)), +(?y,s(0))> --> <s(?y), s(?y)> => yes
<?y, +(0,?y)> --> <?y, ?y> => yes
<s(?y), +(s(0),?y)> --> <s(?y), s(?y)> => yes
S:
   [ +(0,?y) -> ?y,
     +(s(0),?y) -> s(+(0,?y)),
     s(s(?x)) -> ?x,
     +(?y,0) -> ?y,
     +(?y,s(0)) -> s(?y) ]
P:
   [ +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Combined result: CR
581.trs: Success(CR)
(310 msec.)