(ignored inputs)COMMENT R5 of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     s(s(?x)) -> s(?x),
     s(?x) -> s(s(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x_1,s(?x_6)) = s(+(?x_1,s(?x_6))),
     +(?x_1,s(s(?x_7))) = s(+(?x_1,?x_7)),
     +(s(?x_6),?y_3) = s(+(s(?x_6),?y_3)),
     +(s(s(?x_7)),?y_3) = s(+(?x_7,?y_3)),
     +(?x,?z_4) = +(?x,+(0,?z_4)),
     +(s(+(?x_1,?y_1)),?z_4) = +(?x_1,+(s(?y_1),?z_4)),
     +(?y_2,?z_4) = +(0,+(?y_2,?z_4)),
     +(s(+(?x_3,?y_3)),?z_4) = +(s(?x_3),+(?y_3,?z_4)),
     +(+(?y_5,?x_5),?z_4) = +(?x_5,+(?y_5,?z_4)),
     s(s(s(?x_7))) = s(?x_7),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)),
     s(s(?x)) = s(s(?x)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     +(?x_4,?y_4) = +(?x_4,+(?y_4,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(+(?x_4,?y_4),?y_1)) = +(?x_4,+(?y_4,s(?y_1))),
     s(+(?x_1,?y_1)) = +(s(?y_1),?x_1),
     ?y_2 = +(?y_2,0),
     s(+(?x_3,?y_3)) = +(?y_3,s(?x_3)),
     +(?x_4,+(?y_4,?z_4)) = +(?z_4,+(?x_4,?y_4)),
     s(?x_6) = s(s(s(?x_6))) ]
not Overlay, check Termination...
unknown/not 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_5)) = +(s(s(?x)),?y_5),
     s(+(?x_3,?x)) = +(?x_3,s(s(?x))),
     s(?x_7) = s(s(s(?x_7))),
     s(+(s(?x),?y_5)) = +(s(?x),?y_5),
     s(+(?x_3,s(?x))) = +(?x_3,s(?x)),
     s(+(s(s(?x)),?y_5)) = +(s(?x),?y_5),
     s(+(s(?x_6),?y_5)) = +(s(s(?x_6)),?y_5),
     s(+(?x_3,s(s(?x)))) = +(?x_3,s(?x)),
     s(+(?x_3,s(?x_4))) = +(?x_3,s(s(?x_4))),
     s(s(?x_1)) = s(s(s(?x_1))),
     s(s(?x_1)) = s(s(?x_1)),
     s(s(s(?x))) = s(?x),
     s(s(s(s(?x)))) = s(?x),
     s(s(s(?x_1))) = s(s(?x_1)),
     +(?x,+(?y,?z_6)) = +(+(?y,?x),?z_6),
     +(?x_5,+(?y_5,?y)) = +(?y,+(?x_5,?y_5)),
     s(+(?x_4,?y)) = +(?y,s(?x_4)),
     ?y = +(?y,0),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?x = +(0,?x),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(s(+(?x_6,?y)),+(?z,?z_1)) = +(+(s(?x_6),+(?y,?z)),?z_1),
     +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1),
     +(s(+(?x,?y_4)),+(?z,?z_1)) = +(+(?x,+(s(?y_4),?z)),?z_1),
     +(?x,+(?z,?z_1)) = +(+(?x,+(0,?z)),?z_1),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(s(+(?x_5,?y)),?z) = +(s(?x_5),+(?y,?z)),
     +(?y,?z) = +(0,+(?y,?z)),
     +(s(+(?x,?y_3)),?z) = +(?x,+(s(?y_3),?z)),
     +(?x,?z) = +(?x,+(0,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     s(+(+(?x,?y),?y_2)) = +(?x,+(?y,s(?y_2))),
     +(?x,?y) = +(?x,+(?y,0)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?z,s(+(?x_5,?y))) = +(s(?x_5),+(?y,?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,s(+(?x,?y_3))) = +(?x,+(s(?y_3),?z)),
     +(?z,?x) = +(?x,+(0,?z)),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     s(+(+(?y,?x),?y_2)) = +(?x,+(?y,s(?y_2))),
     s(+(s(+(?x_7,?y)),?y_2)) = +(s(?x_7),+(?y,s(?y_2))),
     s(+(?y,?y_2)) = +(0,+(?y,s(?y_2))),
     s(+(s(+(?x,?y_5)),?y_2)) = +(?x,+(s(?y_5),s(?y_2))),
     s(+(?x,?y_2)) = +(?x,+(0,s(?y_2))),
     s(+(+(?x_3,+(?y_3,?y)),?y_2)) = +(+(?x_3,?y_3),+(?y,s(?y_2))),
     +(?y,?x) = +(?x,+(?y,0)),
     s(+(?x_5,?y)) = +(s(?x_5),+(?y,0)),
     ?y = +(0,+(?y,0)),
     s(+(?x,?y_3)) = +(?x,+(s(?y_3),0)),
     ?x = +(?x,+(0,0)),
     +(?x_1,+(?y_1,?y)) = +(+(?x_1,?y_1),+(?y,0)),
     +(s(?x),+(?y,?z_5)) = +(s(+(?x,?y)),?z_5),
     +(s(s(?x)),+(?y,?z_5)) = +(s(+(?x,?y)),?z_5),
     +(s(?x_12),+(?y,?z_5)) = +(s(+(s(?x_12),?y)),?z_5),
     +(s(s(?x)),?y) = s(+(?x,?y)),
     +(s(?x_7),?y) = s(+(s(?x_7),?y)),
     +(?y,s(?x)) = s(+(?x,?y)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))),
     s(?x) = s(+(?x,0)),
     +(?y,s(s(?x))) = s(+(?x,?y)),
     +(?y,s(?x_7)) = s(+(s(?x_7),?y)),
     s(+(s(s(?x)),?y_2)) = s(+(?x,s(?y_2))),
     s(+(s(?x_9),?y_2)) = s(+(s(?x_9),s(?y_2))),
     s(s(?x)) = s(+(?x,0)),
     s(?x_7) = s(+(s(?x_7),0)),
     +(0,+(?y,?z_5)) = +(?y,?z_5),
     +(?y,0) = ?y,
     s(+(0,?y_2)) = s(?y_2),
     +(?x,+(s(?y),?z_5)) = +(s(+(?x,?y)),?z_5),
     +(?x,+(s(s(?y)),?z_5)) = +(s(+(?x,?y)),?z_5),
     +(?x,+(s(?x_12),?z_5)) = +(s(+(?x,s(?x_12))),?z_5),
     +(?x,s(s(?y))) = s(+(?x,?y)),
     +(?x,s(?x_7)) = s(+(?x,s(?x_7))),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x_4,+(?y_4,s(?y))) = s(+(+(?x_4,?y_4),?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     s(?y) = s(+(0,?y)),
     +(s(s(?y)),?x) = s(+(?x,?y)),
     +(s(?x_7),?x) = s(+(?x,s(?x_7))),
     +(?x_4,+(?y_4,s(s(?y)))) = s(+(+(?x_4,?y_4),?y)),
     +(?x_4,+(?y_4,s(?x_11))) = s(+(+(?x_4,?y_4),s(?x_11))),
     s(+(?x_3,s(s(?y)))) = s(+(s(?x_3),?y)),
     s(+(?x_3,s(?x_10))) = s(+(s(?x_3),s(?x_10))),
     s(s(?y)) = s(+(0,?y)),
     s(?x_7) = s(+(0,s(?x_7))),
     +(?x,+(0,?z_5)) = +(?x,?z_5),
     +(0,?x) = ?x,
     +(?x_4,+(?y_4,0)) = +(?x_4,?y_4),
     s(+(?x_3,0)) = s(?x_3),
     0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <+(?x_1,s(?x_6)), s(+(?x_1,s(?x_6)))> by Rules <6, 1> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],7),([],1)], []>
 joinable by a reduction of rules <[([],1),([],7)], [([(s,1)],1)]>
 joinable by a reduction of rules <[([],1)], [([(s,1)],1),([],6)]>
Critical Pair <+(?x_1,s(s(?x_7))), s(+(?x_1,?x_7))> by Rules <7, 1> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],6),([],1)], []>
 joinable by a reduction of rules <[([],1),([(s,1)],1)], [([],7)]>
Critical Pair <+(s(?x_6),?y_3), s(+(s(?x_6),?y_3))> by Rules <6, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],7),([],3)], []>
 joinable by a reduction of rules <[([],3),([],7)], [([(s,1)],3)]>
 joinable by a reduction of rules <[([],3)], [([(s,1)],3),([],6)]>
Critical Pair <+(s(s(?x_7)),?y_3), s(+(?x_7,?y_3))> by Rules <7, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],6),([],3)], []>
 joinable by a reduction of rules <[([],3),([(s,1)],3)], [([],7)]>
Critical Pair <+(?x,?z_4), +(?x,+(0,?z_4))> by Rules <0, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],2)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_4), +(?x_1,+(s(?y_1),?z_4))> by Rules <1, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],4)], [([(+,2)],3),([],1)]>
Critical Pair <+(?y_2,?z_4), +(0,+(?y_2,?z_4))> by Rules <2, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(s(+(?x_3,?y_3)),?z_4), +(s(?x_3),+(?y_3,?z_4))> by Rules <3, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],4)], [([],3)]>
Critical Pair <+(+(?y_5,?x_5),?z_4), +(?x_5,+(?y_5,?z_4))> by Rules <5, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],5),([],4)], []>
 joinable by a reduction of rules <[([],4),([(+,2)],5)], [([],5),([],4)]>
Critical Pair <s(s(s(?x_7))), s(?x_7)> by Rules <7, 6> preceded by [(s,1)]
 joinable by a reduction of rules <[([(s,1)],6)], [([],7)]>
 joinable by a reduction of rules <[([],6)], [([],7)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <4, 4> preceded by [(+,1)]
 joinable by a reduction of rules <[([],4),([(+,2)],4)], [([],4)]>
Critical Pair <s(s(?x)), s(s(?x))> by Rules <6, 6> preceded by [(s,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <0, 0> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[], []>
Critical Pair <s(+(?x_3,0)), s(?x_3)> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(?x_4,+(?y_4,0)), +(?x_4,?y_4)> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
Critical Pair <+(0,?x_5), ?x_5> by Rules <5, 0> preceded by []
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <s(?y_1), s(+(0,?y_1))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], [([(s,1)],2)]>
Critical Pair <s(+(?x_3,s(?y_1))), s(+(s(?x_3),?y_1))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([(s,1)],1)], [([(s,1)],3)]>
Critical Pair <+(?x_4,+(?y_4,s(?y_1))), s(+(+(?x_4,?y_4),?y_1))> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([(+,2)],1),([],1)], [([(s,1)],4)]>
Critical Pair <+(s(?y_1),?x_5), s(+(?x_5,?y_1))> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([],3)], [([(s,1)],5)]>
Critical Pair <+(?y_5,0), ?y_5> by Rules <5, 2> preceded by []
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <+(?y_5,s(?x_3)), s(+(?x_3,?y_5))> by Rules <5, 3> preceded by []
 joinable by a reduction of rules <[([],1)], [([(s,1)],5)]>
Critical Pair <+(?y_5,+(?x_4,?y_4)), +(?x_4,+(?y_4,?y_5))> by Rules <5, 4> preceded by []
 joinable by a reduction of rules <[([],5),([],4)], []>
Critical Pair <s(s(s(?x_6))), s(?x_6)> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[([(s,1)],6)], [([],7)]>
 joinable by a reduction of rules <[([],6)], [([],7)]>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     s(s(?x)) -> s(?x),
     s(?x) -> s(s(?x)) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_3,0)), s(?x_3)> --> <s(?x_3), s(?x_3)> => yes
<s(?y_1), s(+(0,?y_1))> --> <s(?y_1), s(?y_1)> => yes
<s(+(?x_3,s(?y_1))), s(+(s(?x_3),?y_1))> --> <s(s(+(?x_3,?y_1))), s(s(+(?x_3,?y_1)))> => yes
PCP_in(symP,S):
<+(?x,s(?x_5)), s(+(?x,s(?x_5)))> --> <s(+(?x,?x_5)), s(s(+(?x,?x_5)))> => no
<+(?x,s(s(?x_4))), s(+(?x,?x_4))> --> <s(s(+(?x,?x_4))), s(+(?x,?x_4))> => no
<+(s(?x_5),?y), s(+(s(?x_5),?y))> --> <s(+(?x_5,?y)), s(s(+(?x_5,?y)))> => no
<+(s(s(?x_4)),?y), s(+(?x_4,?y))> --> <s(s(+(?x_4,?y))), s(+(?x_4,?y))> => no
CP(S,symP):
<+(?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_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, ?x> => yes
<s(+(+(?x_1,?y_1),?y)), +(?x_1,+(?y_1,s(?y)))> --> <s(+(+(?x_1,?y_1),?y)), s(+(?x_1,+(?y_1,?y)))> => yes
<+(s(+(?x,?y)),?z_1), +(?x,+(s(?y),?z_1))> --> <s(+(+(?x,?y),?z_1)), s(+(?x,+(?y,?z_1)))> => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?x),s(?y))> --> <s(+(?x_1,+(?x,?y))), s(+(+(?x_1,?x),?y))> => yes
<s(+(?x,?y)), +(s(?y),?x)> --> <s(+(?x,?y)), s(+(?y,?x))> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_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, +(?y,0)> --> <?y, ?y> => 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_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_1,+(?x,?y))), s(+(+(?x_1,?x),?y))> => yes
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), s(+(?y,?x))> => yes
check joinability condition:
check modulo joinability of s(+(?x,?x_5)) and s(s(+(?x,?x_5))): joinable by {0}
check modulo joinability of s(s(+(?x,?x_4))) and s(+(?x,?x_4)): joinable by {0}
check modulo joinability of s(+(?x_5,?y)) and s(s(+(?x_5,?y))): joinable by {0}
check modulo joinability of s(s(+(?x_4,?y))) and s(+(?x_4,?y)): joinable by {0}
success
P':
   [ s(s(?x)) -> s(?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ s(s(?x)) -> s(?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= (8)
  s:= (8)+(8)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract s(s(?x)) -> s(?x)
Polynomial Interpretation:
  +:= (1)*x1*x1+(1)*x2*x2
  0:= (6)
  s:= (2)+(1)*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     s(s(?x)) -> s(?x),
     s(?x) -> s(s(?x)) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/124.trs: Success(CR)
(262 msec.)