(ignored inputs)COMMENT R6 of \cite{AT2012}
Rewrite Rules:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x),
     dbl(?x) -> +(?x,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x_4,?y) = +(+(?x_4,0),?y),
     +(?x_4,?x_1) = +(+(?x_4,?x_1),0),
     +(?x_4,s(+(?x_2,?y_2))) = +(+(?x_4,s(?x_2)),?y_2),
     +(?x_4,+(s(?x_3),?y_3)) = +(+(?x_4,?x_3),s(?y_3)),
     +(?x_4,+(?y_5,?x_5)) = +(+(?x_4,?x_5),?y_5),
     +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ]
Outer CPs:
   [ 0 = 0,
     s(?y_3) = +(s(0),?y_3),
     +(?y_4,?z_4) = +(+(0,?y_4),?z_4),
     ?y = +(?y,0),
     s(?x_2) = s(+(?x_2,0)),
     ?x_1 = +(0,?x_1),
     s(+(?x_2,s(?y_3))) = +(s(s(?x_2)),?y_3),
     s(+(?x_2,+(?y_4,?z_4))) = +(+(s(?x_2),?y_4),?z_4),
     s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
     +(s(?x_3),?y_3) = +(s(?y_3),?x_3),
     +(+(?x_4,?y_4),?z_4) = +(+(?y_4,?z_4),?x_4) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ +(+(?x_6,?x),?y) = +(?x_6,+(?y,?x)),
     +(+(?x,?y_5),?z_5) = +(+(?y_5,?z_5),?x),
     s(+(?x_3,?y)) = +(?y,s(?x_3)),
     ?x = +(0,?x),
     ?y = +(?y,0),
     +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)),
     +(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)),
     +(+(?x_1,?x),+(s(?y),?y_6)) = +(?x_1,+(+(?x,?y),s(?y_6))),
     +(+(?x_1,?x),s(+(?x_5,?z))) = +(?x_1,+(+(?x,s(?x_5)),?z)),
     +(+(?x_1,?x),?y) = +(?x_1,+(+(?x,?y),0)),
     +(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)),
     +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
     +(?x,+(?z,?y)) = +(+(?x,?y),?z),
     +(?x,+(s(?y),?y_5)) = +(+(?x,?y),s(?y_5)),
     +(?x,s(+(?x_4,?z))) = +(+(?x,s(?x_4)),?z),
     +(?x,?y) = +(+(?x,?y),0),
     +(?x,?z) = +(+(?x,0),?z),
     +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
     +(+(?y,?z),?x) = +(+(?x,?y),?z),
     s(+(?x_3,+(?y,?z))) = +(+(s(?x_3),?y),?z),
     +(?y,?z) = +(+(0,?y),?z),
     +(+(?z,?y),?x) = +(+(?x,?y),?z),
     +(+(s(?y),?y_5),?x) = +(+(?x,?y),s(?y_5)),
     +(s(+(?x_4,?z)),?x) = +(+(?x,s(?x_4)),?z),
     +(?y,?x) = +(+(?x,?y),0),
     +(?z,?x) = +(+(?x,0),?z),
     +(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)),
     s(+(?x_3,+(?z,?y))) = +(+(s(?x_3),?y),?z),
     s(+(?x_3,+(s(?y),?y_8))) = +(+(s(?x_3),?y),s(?y_8)),
     s(+(?x_3,s(+(?x_7,?z)))) = +(+(s(?x_3),s(?x_7)),?z),
     s(+(?x_3,?y)) = +(+(s(?x_3),?y),0),
     s(+(?x_3,?z)) = +(+(s(?x_3),0),?z),
     s(+(?x_3,+(+(?y,?y_4),?z_4))) = +(+(s(?x_3),?y),+(?y_4,?z_4)),
     +(?z,?y) = +(+(0,?y),?z),
     +(s(?y),?y_5) = +(+(0,?y),s(?y_5)),
     s(+(?x_4,?z)) = +(+(0,s(?x_4)),?z),
     ?y = +(+(0,?y),0),
     ?z = +(+(0,0),?z),
     +(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)),
     +(+(?x_5,?x),s(?y)) = +(?x_5,+(s(?x),?y)),
     +(s(?y),?x) = +(s(?x),?y),
     s(+(?x_3,s(?y))) = +(s(s(?x_3)),?y),
     s(?y) = +(s(0),?y),
     +(+(?x_5,s(?x)),?y) = +(?x_5,s(+(?x,?y))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(+(s(?x),?y_4),?z_4) = s(+(?x,+(?y_4,?z_4))),
     +(s(s(?x)),?y_3) = s(+(?x,s(?y_3))),
     s(?x) = s(+(?x,0)),
     +(+(?x_5,?x),0) = +(?x_5,?x),
     +(0,?x) = ?x,
     s(+(?x_2,0)) = s(?x_2),
     +(+(?x_5,0),?y) = +(?x_5,?y),
     +(?y,0) = ?y,
     +(+(0,?y_4),?z_4) = +(?y_4,?z_4),
     +(s(0),?y_3) = s(?y_3),
     0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <+(?x_4,?y), +(+(?x_4,0),?y)> by Rules <0, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([(+,1)],1)]>
Critical Pair <+(?x_4,?x_1), +(+(?x_4,?x_1),0)> by Rules <1, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[], [([],1)]>
Critical Pair <+(?x_4,s(+(?x_2,?y_2))), +(+(?x_4,s(?x_2)),?y_2)> by Rules <2, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([],3),([],4)], [([(+,1)],3)]>
Critical Pair <+(?x_4,+(s(?x_3),?y_3)), +(+(?x_4,?x_3),s(?y_3))> by Rules <3, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([],4),([(+,1)],5),([(+,1)],2)], [([(+,1)],5),([],3)]>
 joinable by a reduction of rules <[([],4),([(+,1)],5),([(+,1)],2)], [([],3),([(+,1),(s,1)],5)]>
 joinable by a reduction of rules <[([],4),([(+,1)],3),([(+,1)],2)], [([],3)]>
 joinable by a reduction of rules <[([(+,2)],5),([(+,2)],3),([],5)], [([(+,1)],5),([],5),([],4)]>
 joinable by a reduction of rules <[([(+,2)],5),([(+,2)],3),([],5)], [([],5),([(+,2)],5),([],4)]>
 joinable by a reduction of rules <[([(+,2)],5),([(+,2)],3),([],4)], [([],5),([],4),([(+,1)],5)]>
 joinable by a reduction of rules <[([(+,2)],5),([],5),([(+,1)],3)], [([(+,1)],5),([],5),([],4)]>
 joinable by a reduction of rules <[([(+,2)],5),([],5),([(+,1)],3)], [([],5),([(+,2)],5),([],4)]>
 joinable by a reduction of rules <[([],5),([(+,1)],5),([(+,1)],3)], [([(+,1)],5),([],5),([],4)]>
 joinable by a reduction of rules <[([],5),([(+,1)],5),([(+,1)],3)], [([],5),([(+,2)],5),([],4)]>
 joinable by a reduction of rules <[([],4),([(+,1)],3),([(+,1)],2)], [([(+,1)],5),([],3),([(+,1),(s,1)],5)]>
Critical Pair <+(?x_4,+(?y_5,?x_5)), +(+(?x_4,?x_5),?y_5)> by Rules <5, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],5),([],4)], []>
 joinable by a reduction of rules <[([],4),([(+,1)],5)], [([],5),([],4)]>
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 <0, 0> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <+(s(0),?y_3), s(?y_3)> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],2),([(s,1)],0)], []>
Critical Pair <+(+(0,?y_4),?z_4), +(?y_4,?z_4)> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[([(+,1)],0)], []>
Critical Pair <+(?y_5,0), ?y_5> by Rules <5, 0> preceded by []
 joinable by a reduction of rules <[([],1)], []>
Critical Pair <s(+(?x_2,0)), s(?x_2)> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([(s,1)],1)], []>
Critical Pair <+(0,?x_5), ?x_5> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(s(s(?x_2)),?y_3), s(+(?x_2,s(?y_3)))> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([],2)], [([(s,1)],3)]>
Critical Pair <+(+(s(?x_2),?y_4),?z_4), s(+(?x_2,+(?y_4,?z_4)))> by Rules <4, 2> preceded by []
 joinable by a reduction of rules <[([(+,1)],2),([],2)], [([(s,1)],4)]>
Critical Pair <+(?y_5,s(?x_2)), s(+(?x_2,?y_5))> by Rules <5, 2> preceded by []
 joinable by a reduction of rules <[([],5),([],2)], []>
 joinable by a reduction of rules <[([],3),([],2)], [([(s,1)],5)]>
Critical Pair <+(s(?y_3),?x_5), +(s(?x_5),?y_3)> by Rules <5, 3> preceded by []
 joinable by a reduction of rules <[([],5),([],3)], []>
 joinable by a reduction of rules <[([],2),([(s,1)],5)], [([],2)]>
 joinable by a reduction of rules <[], [([],5),([],3)]>
 joinable by a reduction of rules <[([],2)], [([],2),([(s,1)],5)]>
Critical Pair <+(+(?y_4,?z_4),?x_5), +(+(?x_5,?y_4),?z_4)> by Rules <5, 4> preceded by []
 joinable by a reduction of rules <[([],5),([],4)], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
PCP_in(symP,S):
CP(S,symP):
<s(?y_1), +(s(0),?y_1)> --> <s(?y_1), s(?y_1)> => yes
<+(?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,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<?y, +(?y,0)> --> <?y, ?y> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<s(?x_1), +(?x_1,s(0))> --> <s(?x_1), +(?x_1,s(0))> => no
<+(?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
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(?x,s(?y_1))), +(s(s(?x)),?y_1)> --> <s(+(?x,s(?y_1))), s(s(+(?x,?y_1)))> => no
<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)), +(?x,s(?y))> --> <s(+(?x,?y)), +(?x,s(?y))> => no
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
check joinability condition:
check modulo reachablity from s(?x_1) to +(?x_1,s(0)): maybe not reachable
check modulo joinability of s(+(?x,s(?y_1))) and s(s(+(?x,?y_1))): joinable by {0}
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 +(?x,s(?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)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x_1,s(0)) -> s(?x_1) ]
Added S-Rules:
   [ +(?y,s(?x)) -> s(+(?x,?y)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x_1,s(0)) -> s(?x_1) ]
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)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (12)
  s:= (1)*x1
  dbl:= (9)+(11)*x1+(9)*x1*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
unknown relative termination
S/P: unknown relative termination
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,s(?x)) -> s(+(?x,?y)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x_1,s(0)) -> s(?x_1) ]
P:
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_4,0)), s(?x_4)> --> <s(?x_4), s(?x_4)> => yes
<s(+(0,?y_5)), s(?y_5)> --> <s(?y_5), s(?y_5)> => yes
<s(0), s(0)> --> <s(0), s(0)> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
<s(+(?x_4,s(?x_2))), s(+(?x_2,s(?x_4)))> --> <s(s(+(?x_2,?x_4))), s(s(+(?x_4,?x_2)))> => yes
<s(+(s(?x_2),?y_5)), s(+(?x_2,s(?y_5)))> --> <s(s(+(?x_2,?y_5))), s(s(+(?y_5,?x_2)))> => yes
<s(s(?x_2)), s(+(?x_2,s(0)))> --> <s(s(?x_2)), s(s(?x_2))> => yes
<s(+(?x_5,?y_5)), s(+(?y_5,?x_5))> --> <s(+(?x_5,?y_5)), s(+(?y_5,?x_5))> => yes
<s(?x_6), s(+(0,?x_6))> --> <s(?x_6), s(?x_6)> => yes
<s(?x_6), s(+(?x_6,0))> --> <s(?x_6), s(?x_6)> => yes
PCP_in(symP,S):
CP(S,symP):
<s(?y_1), +(s(0),?y_1)> --> <s(?y_1), s(?y_1)> => yes
<+(?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,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<?y, +(?y,0)> --> <?y, ?y> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<s(?x_1), +(?x_1,s(0))> --> <s(?x_1), s(?x_1)> => 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
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(?x,s(?y_1))), +(s(s(?x)),?y_1)> --> <s(s(+(?y_1,?x))), s(s(+(?x,?y_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)), +(?x,s(?y))> --> <s(+(?x,?y)), s(+(?y,?x))> => yes
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?y,?x))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?y),s(?x))> --> <s(+(+(?x,?y),?x_1)), s(+(?x,+(?x_1,?y)))> => no
<s(+(?x,s(?x_1))), +(?x_1,s(s(?x)))> --> <s(s(+(?x_1,?x))), s(s(+(?x,?x_1)))> => 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
<s(+(?x,?y)), +(s(?x),?y)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<s(+(?x,?y)), +(s(?x),?y)> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?x),s(?y))> --> <s(+(+(?x,?y),?x_1)), s(+(?y,+(?x_1,?x)))> => no
<s(+(s(?x_1),?y)), +(?x_1,s(s(?y)))> --> <s(s(+(?x_1,?y))), s(s(+(?y,?x_1)))> => yes
<s(+(+(?x_1,?y_1),?y)), +(?x_1,+(?y_1,s(?y)))> --> <s(+(+(?x_1,?y_1),?y)), s(+(+(?y,?y_1),?x_1))> => no
<+(s(+(?x,?y)),?z_1), +(?x,+(s(?y),?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(+(?y,?z_1),?x))> => no
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?y,?x))> => yes
<s(?x), +(s(?x),0)> --> <s(?x), s(?x)> => yes
<+(?x_1,s(?x)), +(+(?x_1,?x),s(0))> --> <s(+(?x,?x_1)), s(+(?x_1,?x))> => yes
<s(s(?x_1)), +(?x_1,s(s(0)))> --> <s(s(?x_1)), s(s(?x_1))> => yes
<s(+(?x_1,?y_1)), +(?x_1,+(?y_1,s(0)))> --> <s(+(?x_1,?y_1)), s(+(?y_1,?x_1))> => yes
<+(s(?x),?z_1), +(?x,+(s(0),?z_1))> --> <s(+(?x,?z_1)), s(+(?z_1,?x))> => yes
<s(?x), +(s(0),?x)> --> <s(?x), s(?x)> => yes
check joinability condition:
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(+(?x,?x_1),?y)): joinable by {1}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?x,+(?x_1,?y))): joinable by {1}
check modulo joinability of s(+(?x,+(?x_1,?y_1))) and s(+(+(?x,?y_1),?x_1)): joinable by {1}
check modulo joinability of s(+(+(?x,?y),?z_1)) and s(+(+(?x,?z_1),?y)): joinable by {1}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?y,+(?x_1,?x))): joinable by {0}
check modulo joinability of s(+(+(?x_1,?y_1),?y)) and s(+(+(?y,?y_1),?x_1)): joinable by {0,1}
check modulo joinability of s(+(+(?x,?y),?z_1)) and s(+(+(?y,?z_1),?x)): joinable by {0}
success
P':
   [ +(?y,?x) -> +(?x,?y),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,s(?x)) -> s(+(?x,?y)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x_1,s(0)) -> s(?x_1) ]
   [ +(?y,?x) -> +(?x,?y),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= (2)
  s:= (7)+(2)*x1
  dbl:= (9)+(8)*x1+(1)*x1*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
retract +(?x_1,s(0)) -> s(?x_1)
Polynomial Interpretation:
  +:= (2)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
  dbl:= (13)+(10)*x1+(1)*x1*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,s(?x)) -> s(+(?x,?y)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x_1,s(0)) -> s(?x_1) ]
P:
   [ +(?x,s(?y)) -> +(s(?x),?y),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/122.trs: Success(CR)
(817 msec.)