YES
(ignored inputs)COMMENT the following rules are removed from the original TRS + ( x , s ( y ) ) -> + ( s ( y ) , x )
Rewrite Rules:
   [ +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x_4,+(?y,?x)) = +(+(?x_4,?x),?y),
     +(?x_4,?y_1) = +(+(?x_4,0),?y_1),
     +(?x_4,?x_2) = +(+(?x_4,?x_2),0),
     +(?x_4,s(+(?x_3,?y_3))) = +(+(?x_4,s(?x_3)),?y_3),
     +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ]
Outer CPs:
   [ +(?y,0) = ?y,
     +(0,?x) = ?x,
     +(?y,s(?x_3)) = s(+(?x_3,?y)),
     +(+(?y_4,?z_4),?x) = +(+(?x,?y_4),?z_4),
     0 = 0,
     +(?y_4,?z_4) = +(+(0,?y_4),?z_4),
     s(?x_3) = s(+(?x_3,0)),
     s(+(?x_3,+(?y_4,?z_4))) = +(+(s(?x_3),?y_4),?z_4) ]
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 = +(?y,0),
     ?x = +(0,?x),
     s(+(?x_3,?y)) = +(?y,s(?x_3)),
     +(+(?x,?y_4),?z_4) = +(+(?y_4,?z_4),?x),
     +(+(?x_5,?x),?y) = +(?x_5,+(?y,?x)),
     +(?y,0) = ?y,
     0 = 0,
     +(+(0,?y_4),?z_4) = +(?y_4,?z_4),
     +(+(?x_5,0),?y) = +(?x_5,?y),
     +(0,?x) = ?x,
     s(+(?x_3,0)) = s(?x_3),
     +(+(?x_5,?x),0) = +(?x_5,?x),
     +(?y,s(?x)) = s(+(?x,?y)),
     s(?x) = s(+(?x,0)),
     +(+(s(?x),?y_4),?z_4) = s(+(?x,+(?y_4,?z_4))),
     +(+(?x_5,s(?x)),?y) = +(?x_5,s(+(?x,?y))),
     +(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)),
     +(+(?z,?y),?x) = +(+(?x,?y),?z),
     +(?z,?x) = +(+(?x,0),?z),
     +(?y,?x) = +(+(?x,?y),0),
     +(s(+(?x_5,?z)),?x) = +(+(?x,s(?x_5)),?z),
     +(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)),
     +(?z,?y) = +(+(0,?y),?z),
     ?z = +(+(0,0),?z),
     ?y = +(+(0,?y),0),
     s(+(?x_5,?z)) = +(+(0,s(?x_5)),?z),
     s(+(?x_4,+(+(?y,?y_5),?z_5))) = +(+(s(?x_4),?y),+(?y_5,?z_5)),
     s(+(?x_4,+(?z,?y))) = +(+(s(?x_4),?y),?z),
     s(+(?x_4,?z)) = +(+(s(?x_4),0),?z),
     s(+(?x_4,?y)) = +(+(s(?x_4),?y),0),
     s(+(?x_4,s(+(?x_9,?z)))) = +(+(s(?x_4),s(?x_9)),?z),
     +(+(?y,?z),?x) = +(+(?x,?y),?z),
     +(?y,?z) = +(+(0,?y),?z),
     s(+(?x_4,+(?y,?z))) = +(+(s(?x_4),?y),?z),
     +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
     +(?x,+(?z,?y)) = +(+(?x,?y),?z),
     +(?x,?z) = +(+(?x,0),?z),
     +(?x,?y) = +(+(?x,?y),0),
     +(?x,s(+(?x_5,?z))) = +(+(?x,s(?x_5)),?z),
     +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
     +(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)),
     +(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)),
     +(+(?x_1,?x),?y) = +(?x_1,+(+(?x,?y),0)),
     +(+(?x_1,?x),s(+(?x_6,?z))) = +(?x_1,+(+(?x,s(?x_6)),?z)),
     +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)) ]
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_4,+(?y,?x)), +(+(?x_4,?x),?y)> by Rules <0, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],0),([],4)], []>
 joinable by a reduction of rules <[([],4),([(+,1)],0)], [([],0),([],4)]>
Critical Pair <+(?x_4,?y_1), +(+(?x_4,0),?y_1)> by Rules <1, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([(+,1)],2)]>
Critical Pair <+(?x_4,?x_2), +(+(?x_4,?x_2),0)> by Rules <2, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(?x_4,s(+(?x_3,?y_3))), +(+(?x_4,s(?x_3)),?y_3)> by Rules <3, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([],0),([],3),([(s,1)],0),([(s,1)],4)], [([(+,1)],0),([(+,1)],3),([(+,1),(s,1)],0),([],3)]>
 joinable by a reduction of rules <[([],0),([],3),([(s,1)],0),([(s,1)],4)], [([(+,1)],0),([(+,1)],3),([],3),([(s,1),(+,1)],0)]>
Critical Pair <+(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?y,?z))> by Rules <4, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([],4),([(+,1)],4)], [([],4)]>
Critical Pair <?y_1, +(?y_1,0)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <?x_2, +(0,?x_2)> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], [([],1)]>
Critical Pair <s(+(?x_3,?y_3)), +(?y_3,s(?x_3))> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],3)]>
Critical Pair <+(+(?x_4,?y_4),?z_4), +(+(?y_4,?z_4),?x_4)> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[], [([],0),([],4)]>
Critical Pair <0, 0> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <+(+(0,?y_4),?z_4), +(?y_4,?z_4)> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([(+,1)],1)], []>
Critical Pair <s(+(?x_3,0)), s(?x_3)> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([(s,1)],2)], []>
Critical Pair <+(+(s(?x_3),?y_4),?z_4), s(+(?x_3,+(?y_4,?z_4)))> by Rules <4, 3> preceded by []
 joinable by a reduction of rules <[([(+,1)],3),([],3)], [([(s,1)],4)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 7 rules by 3 steps unfolding
obtain 100 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,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (12)
  s:= (1)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (2)+(1)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<0, 0> --> <0, 0> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
<s(?x_2), s(+(?x_2,0))> --> <s(?x_2), s(?x_2)> => yes
PCP_in(symP,S):
CP(S,symP):
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<?y, +(?y,0)> --> <?y, ?y> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, ?x> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
<s(+(?x,+(?y_1,?z_1))), +(+(s(?x),?y_1),?z_1)> --> <s(+(?x,+(?y_1,?z_1))), s(+(+(?x,?y_1),?z_1))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
check joinability condition:
check modulo reachablity from +(?x_1,s(+(?x,?y))) to +(+(?x_1,s(?x)),?y): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 1)
   [ +(?y,s(?x)) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(?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:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<0, 0> --> <0, 0> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
<s(?x_2), s(+(?x_2,0))> --> <s(?x_2), s(?x_2)> => yes
CP_in(symP,S):
CP(S,symP):
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<?y, +(?y,0)> --> <?y, ?y> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, ?x> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
<s(+(?x,+(?y_1,?z_1))), +(+(s(?x),?y_1),?z_1)> --> <s(+(?x,+(?y_1,?z_1))), s(+(+(?x,?y_1),?z_1))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
check joinability condition:
check modulo reachablity from +(?x_1,s(+(?x,?y))) to +(+(?x_1,s(?x)),?y): maybe not reachable
check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 2)
   [ +(?y,s(?x)) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(?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:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (12)
  s:= (1)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (2)+(1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_3,0)), s(?x_3)> --> <s(?x_3), s(?x_3)> => yes
<0, 0> --> <0, 0> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
<s(?x_2), s(+(?x_2,0))> --> <s(?x_2), s(?x_2)> => yes
<s(+(?x_3,s(?x_2))), s(+(?x_2,s(?x_3)))> --> <s(s(+(?x_2,?x_3))), s(s(+(?x_3,?x_2)))> => yes
<s(?x_3), s(+(?x_3,0))> --> <s(?x_3), s(?x_3)> => yes
<s(+(?x_2,s(?x_3))), s(+(?x_3,s(?x_2)))> --> <s(s(+(?x_3,?x_2))), s(s(+(?x_2,?x_3)))> => yes
PCP_in(symP,S):
CP(S,symP):
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<?y, +(?y,0)> --> <?y, ?y> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, ?x> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
<s(+(?x,+(?y_1,?z_1))), +(+(s(?x),?y_1),?z_1)> --> <s(+(?x,+(?y_1,?z_1))), s(+(+(?x,?y_1),?z_1))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <s(+(+(?x,?y),?x_1)), s(+(+(?x,?x_1),?y))> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?y),s(?x))> --> <s(+(+(?x,?y),?x_1)), s(+(?x,+(?x_1,?y)))> => no
<s(+(?x,?y)), +(s(?x),?y)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<s(+(?x,+(?x_1,?y_1))), +(?x_1,+(?y_1,s(?x)))> --> <s(+(?x,+(?x_1,?y_1))), s(+(+(?x,?y_1),?x_1))> => no
<+(s(+(?x,?y)),?z_1), +(?y,+(s(?x),?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(+(?x,?z_1),?y))> => no
check joinability condition:
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(+(?x,?x_1),?y)): joinable by {2}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?x,+(?x_1,?y))): joinable by {2}
check modulo joinability of s(+(?x,+(?x_1,?y_1))) and s(+(+(?x,?y_1),?x_1)): joinable by {2}
check modulo joinability of s(+(+(?x,?y),?z_1)) and s(+(+(?x,?z_1),?y)): joinable by {2}
success
P':
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= (2)
  s:= (11)+(12)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (9)
  s:= (4)+(4)*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,?y) -> +(?y,?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Combined result: CR
/tmp/fileHVYORf.trs: Success(CR)
(1494 msec.)