YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,s(?y)) -> +(s(?y),?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
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(?y_3),?x_3)) = +(+(?x_4,?x_3),s(?y_3)),
     +(?x_4,+(?y_5,?x_5)) = +(+(?x_4,?x_5),?y_5),
     dbl(?y) = +(dbl(0),dbl(?y)),
     dbl(?x_1) = +(dbl(?x_1),dbl(0)),
     dbl(s(+(?x_2,?y_2))) = +(dbl(s(?x_2)),dbl(?y_2)),
     dbl(+(s(?y_3),?x_3)) = +(dbl(?x_3),dbl(s(?y_3))),
     dbl(+(+(?x_4,?y_4),?z_4)) = +(dbl(?x_4),dbl(+(?y_4,?z_4))),
     dbl(+(?y_5,?x_5)) = +(dbl(?x_5),dbl(?y_5)),
     +(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ]
Outer CPs:
   [ 0 = 0,
     s(?y_3) = +(s(?y_3),0),
     +(?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(?y_3),s(?x_2)),
     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(?y_3),?x_3) = +(s(?y_3),?x_3),
     +(+(?x_4,?y_4),?z_4) = +(+(?y_4,?z_4),?x_4),
     +(+(?x_7,?y_7),+(?x_7,?y_7)) = +(dbl(?x_7),dbl(?y_7)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Left-Linear, not Right-Linear
unknown Development 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:
   [ 0 = 0,
     +(s(?y_3),0) = s(?y_3),
     +(+(0,?y_4),?z_4) = +(?y_4,?z_4),
     +(?y,0) = ?y,
     +(+(?x_5,0),?y) = +(?x_5,?y),
     +(dbl(0),dbl(?y)) = dbl(?y),
     s(+(?x_2,0)) = s(?x_2),
     +(0,?x) = ?x,
     +(+(?x_5,?x),0) = +(?x_5,?x),
     +(dbl(?x),dbl(0)) = dbl(?x),
     s(?x) = s(+(?x,0)),
     +(s(?y_3),s(?x)) = s(+(?x,s(?y_3))),
     +(+(s(?x),?y_4),?z_4) = s(+(?x,+(?y_4,?z_4))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(+(?x_5,s(?x)),?y) = +(?x_5,s(+(?x,?y))),
     +(dbl(s(?x)),dbl(?y)) = dbl(s(+(?x,?y))),
     s(?y) = +(s(?y),0),
     s(+(?x_3,s(?y))) = +(s(?y),s(?x_3)),
     +(s(?y),?x) = +(s(?y),?x),
     +(+(?x_5,?x),s(?y)) = +(?x_5,+(s(?y),?x)),
     +(dbl(?x),dbl(s(?y))) = dbl(+(s(?y),?x)),
     +(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)),
     ?z = +(+(0,0),?z),
     ?y = +(+(0,?y),0),
     s(+(?x_4,?z)) = +(+(0,s(?x_4)),?z),
     +(s(?y_5),?y) = +(+(0,?y),s(?y_5)),
     +(?z,?y) = +(+(0,?y),?z),
     s(+(?x_3,+(+(?y,?y_4),?z_4))) = +(+(s(?x_3),?y),+(?y_4,?z_4)),
     s(+(?x_3,?z)) = +(+(s(?x_3),0),?z),
     s(+(?x_3,?y)) = +(+(s(?x_3),?y),0),
     s(+(?x_3,s(+(?x_7,?z)))) = +(+(s(?x_3),s(?x_7)),?z),
     s(+(?x_3,+(s(?y_8),?y))) = +(+(s(?x_3),?y),s(?y_8)),
     s(+(?x_3,+(?z,?y))) = +(+(s(?x_3),?y),?z),
     +(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)),
     +(?z,?x) = +(+(?x,0),?z),
     +(?y,?x) = +(+(?x,?y),0),
     +(s(+(?x_4,?z)),?x) = +(+(?x,s(?x_4)),?z),
     +(+(s(?y_5),?y),?x) = +(+(?x,?y),s(?y_5)),
     +(+(?z,?y),?x) = +(+(?x,?y),?z),
     +(?y,?z) = +(+(0,?y),?z),
     s(+(?x_3,+(?y,?z))) = +(+(s(?x_3),?y),?z),
     +(+(?y,?z),?x) = +(+(?x,?y),?z),
     +(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
     +(?x,?z) = +(+(?x,0),?z),
     +(?x,?y) = +(+(?x,?y),0),
     +(?x,s(+(?x_4,?z))) = +(+(?x,s(?x_4)),?z),
     +(?x,+(s(?y_5),?y)) = +(+(?x,?y),s(?y_5)),
     +(?x,+(?z,?y)) = +(+(?x,?y),?z),
     +(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
     +(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)),
     +(+(?x_1,?x),?y) = +(?x_1,+(+(?x,?y),0)),
     +(+(?x_1,?x),s(+(?x_5,?z))) = +(?x_1,+(+(?x,s(?x_5)),?z)),
     +(+(?x_1,?x),+(s(?y_6),?y)) = +(?x_1,+(+(?x,?y),s(?y_6))),
     +(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)),
     +(dbl(?x),dbl(+(+(?y,?y_1),?z_1))) = dbl(+(+(?x,?y),+(?y_1,?z_1))),
     +(dbl(?x),dbl(?z)) = dbl(+(+(?x,0),?z)),
     +(dbl(?x),dbl(?y)) = dbl(+(+(?x,?y),0)),
     +(dbl(?x),dbl(s(+(?x_4,?z)))) = dbl(+(+(?x,s(?x_4)),?z)),
     +(dbl(?x),dbl(+(s(?y_5),?y))) = dbl(+(+(?x,?y),s(?y_5))),
     +(dbl(?x),dbl(+(?z,?y))) = dbl(+(+(?x,?y),?z)),
     +(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)),
     +(dbl(?x),dbl(+(?y,?z))) = dbl(+(+(?x,?y),?z)),
     ?y = +(?y,0),
     ?x = +(0,?x),
     s(+(?x_3,?y)) = +(?y,s(?x_3)),
     +(+(?x,?y_5),?z_5) = +(+(?y_5,?z_5),?x),
     +(+(?x_6,?x),?y) = +(?x_6,+(?y,?x)),
     +(dbl(?x),dbl(?y)) = dbl(+(?y,?x)),
     +(dbl(?x_7),dbl(?y_7)) = +(+(?x_7,?y_7),+(?x_7,?y_7)),
     +(?y,?y) = +(dbl(0),dbl(?y)),
     +(?x,?x) = +(dbl(?x),dbl(0)),
     +(s(+(?x_4,?y)),s(+(?x_4,?y))) = +(dbl(s(?x_4)),dbl(?y)),
     +(+(s(?y_5),?x),+(s(?y_5),?x)) = +(dbl(?x),dbl(s(?y_5))),
     +(+(+(?x,?y_6),?z_6),+(+(?x,?y_6),?z_6)) = +(dbl(?x),dbl(+(?y_6,?z_6))),
     +(+(?y,?x),+(?y,?x)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),+(?x,?y)) = +(dbl(?x),dbl(?y)),
     dbl(?y) = +(dbl(0),dbl(?y)),
     dbl(?x) = +(dbl(?x),dbl(0)),
     dbl(s(+(?x_4,?y))) = +(dbl(s(?x_4)),dbl(?y)),
     dbl(+(s(?y_5),?x)) = +(dbl(?x),dbl(s(?y_5))),
     dbl(+(+(?x,?y_6),?z_6)) = +(dbl(?x),dbl(+(?y_6,?z_6))),
     dbl(+(?y,?x)) = +(dbl(?x),dbl(?y)) ]
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_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 <[([],5),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],5),([(+,1)],2),([(+,1),(s,1)],5),([],2)]>
 joinable by a reduction of rules <[([],5),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],5),([(+,1)],2),([],2),([(s,1),(+,1)],5)]>
 joinable by a reduction of rules <[([],5),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],3),([(+,1)],2),([(+,1),(s,1)],5),([],2)]>
 joinable by a reduction of rules <[([],5),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],3),([(+,1)],2),([],2),([(s,1),(+,1)],5)]>
 joinable by a reduction of rules <[([],3),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],5),([(+,1)],2),([(+,1),(s,1)],5),([],2)]>
 joinable by a reduction of rules <[([],3),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],5),([(+,1)],2),([],2),([(s,1),(+,1)],5)]>
 joinable by a reduction of rules <[([],3),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],3),([(+,1)],2),([(+,1),(s,1)],5),([],2)]>
 joinable by a reduction of rules <[([],3),([],2),([(s,1)],5),([(s,1)],4)], [([(+,1)],3),([(+,1)],2),([],2),([(s,1),(+,1)],5)]>
Critical Pair <+(?x_4,+(s(?y_3),?x_3)), +(+(?x_4,?x_3),s(?y_3))> by Rules <3, 4> preceded by [(+,2)]
 joinable by a reduction of rules <[([(+,2)],5),([],4)], []>
 joinable by a reduction of rules <[([],4),([(+,1)],5)], [([],5),([],4)]>
 joinable by a reduction of rules <[([],4),([(+,1)],5)], [([],3),([],4)]>
 joinable by a reduction of rules <[([],4),([(+,1)],3)], [([],5),([],4)]>
 joinable by a reduction of rules <[([],4),([(+,1)],3)], [([],3),([],4)]>
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 <dbl(?y), +(dbl(0),dbl(?y))> by Rules <0, 7> preceded by [(dbl,1)]
 joinable by a reduction of rules <[], [([(+,1)],6),([(+,1)],1),([],0)]>
 joinable by a reduction of rules <[], [([(+,1)],6),([(+,1)],0),([],0)]>
Critical Pair <dbl(?x_1), +(dbl(?x_1),dbl(0))> by Rules <1, 7> preceded by [(dbl,1)]
 joinable by a reduction of rules <[], [([(+,2)],6),([(+,2)],1),([],1)]>
 joinable by a reduction of rules <[], [([(+,2)],6),([(+,2)],0),([],1)]>
Critical Pair <dbl(s(+(?x_2,?y_2))), +(dbl(s(?x_2)),dbl(?y_2))> by Rules <2, 7> preceded by [(dbl,1)]
unknown Diagram Decreasing
   [ +(0,?y) -> ?y,
     +(?x_1,0) -> ?x_1,
     +(s(?x_2),?y_2) -> s(+(?x_2,?y_2)),
     +(?x_3,s(?y_3)) -> +(s(?y_3),?x_3),
     +(?x_4,+(?y_4,?z_4)) -> +(+(?x_4,?y_4),?z_4),
     +(?x_5,?y_5) -> +(?y_5,?x_5),
     dbl(?x_6) -> +(?x_6,?x_6),
     dbl(+(?x_7,?y_7)) -> +(dbl(?x_7),dbl(?y_7)) ]
Sort Assignment:
 + : 23*23=>23
 0 : =>23
 s : 23=>23
 dbl : 23=>23
non-linear variables: {?x_6}
non-linear types: {23}
types leq non-linear types: {23}
rules applicable to terms of non-linear types:
   [ +(0,?y) -> ?y,
     +(?x_1,0) -> ?x_1,
     +(s(?x_2),?y_2) -> s(+(?x_2,?y_2)),
     +(?x_3,s(?y_3)) -> +(s(?y_3),?x_3),
     +(?x_4,+(?y_4,?z_4)) -> +(+(?x_4,?y_4),?z_4),
     +(?x_5,?y_5) -> +(?y_5,?x_5),
     dbl(?x_6) -> +(?x_6,?x_6),
     dbl(+(?x_7,?y_7)) -> +(dbl(?x_7),dbl(?y_7)) ]
NLR:
0: {}
1: {}
2: {}
3: {}
4: {}
5: {}
6: {0,1,2,3,4,5,6,7}
7: {}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(0,?y) -> ?y,
     +(?x_1,0) -> ?x_1,
     +(s(?x_2),?y_2) -> s(+(?x_2,?y_2)),
     +(?x_3,s(?y_3)) -> +(s(?y_3),?x_3),
     +(?x_4,+(?y_4,?z_4)) -> +(+(?x_4,?y_4),?z_4),
     +(?x_5,?y_5) -> +(?y_5,?x_5),
     dbl(?x_6) -> +(?x_6,?x_6),
     dbl(+(?x_7,?y_7)) -> +(dbl(?x_7),dbl(?y_7)) ]
Sort Assignment:
 + : 23*23=>23
 0 : =>23
 s : 23=>23
 dbl : 23=>23
non-linear variables: {?x_6}
non-linear types: {23}
types leq non-linear types: {23}
rules applicable to terms of non-linear types:
   [ +(0,?y) -> ?y,
     +(?x_1,0) -> ?x_1,
     +(s(?x_2),?y_2) -> s(+(?x_2,?y_2)),
     +(?x_3,s(?y_3)) -> +(s(?y_3),?x_3),
     +(?x_4,+(?y_4,?z_4)) -> +(+(?x_4,?y_4),?z_4),
     +(?x_5,?y_5) -> +(?y_5,?x_5),
     dbl(?x_6) -> +(?x_6,?x_6),
     dbl(+(?x_7,?y_7)) -> +(dbl(?x_7),dbl(?y_7)) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 18 rules by 3 steps unfolding
strenghten dbl(0) and 0
strenghten +(?x_10,0) and ?x_10
strenghten +(0,?x_5) and ?x_5
strenghten +(0,0) and 0
strenghten +(dbl(0),?x_14) and ?x_14
strenghten dbl(+(0,0)) and 0
strenghten +(s(?y_3),0) and s(?y_3)
strenghten s(+(?x_2,0)) and s(?x_2)
strenghten dbl(+(0,0)) and dbl(0)
strenghten +(?x_14,+(0,0)) and ?x_14
strenghten +(dbl(0),dbl(0)) and 0
strenghten +(dbl(?x_1),dbl(0)) and dbl(?x_1)
strenghten +(dbl(0),dbl(?x_10)) and dbl(?x_10)
strenghten +(dbl(0),dbl(0)) and dbl(0)
strenghten s(+(?x_2,dbl(0))) and s(?x_2)
strenghten s(+(?x_2,?y_5)) and +(?y_5,s(?x_2))
strenghten +(+(?x_4,?x_1),0) and +(?x_4,?x_1)
strenghten +(+(?x_4,0),?x_10) and +(?x_4,?x_10)
strenghten +(+(0,?y_4),?z_4) and +(?y_4,?z_4)
strenghten +(dbl(?x_14),dbl(dbl(0))) and dbl(?x_14)
strenghten +(dbl(?x_5),dbl(?y_5)) and dbl(+(?y_5,?x_5))
strenghten +(+(?x_4,?x_14),dbl(0)) and +(?x_4,?x_14)
strenghten +(+(?x_4,?x_5),?y_5) and +(?x_4,+(?y_5,?x_5))
strenghten +(+(?x_5,?y_4),?z_4) and +(+(?y_4,?z_4),?x_5)
strenghten s(+(?x_2,s(?y_3))) and +(s(?y_3),s(?x_2))
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-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,s(?y)) -> +(s(?y),?x),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x2
  0:= (4)
  s:= (1)*x1
  dbl:= (1)+(8)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (4)
  s:= (1)*x1
  dbl:= (4)+(2)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
  dbl:= (1)+(2)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract +(s(?x),?y) -> s(+(?x,?y))
retract dbl(?x) -> +(?x,?x)
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract +(s(?x),?y) -> s(+(?x,?y))
retract dbl(?x) -> +(?x,?x)
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
unknown relative termination
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)),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
P:
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?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
<+(dbl(?x_4),dbl(?y_4)), +(+(?x_4,?y_4),+(?x_4,?y_4))> --> <+(+(?x_4,?x_4),+(?y_4,?y_4)), +(+(?x_4,?y_4),+(?x_4,?y_4))> => no
<dbl(?y), +(dbl(0),dbl(?y))> --> <+(?y,?y), +(?y,?y)> => yes
<dbl(?x_1), +(dbl(?x_1),dbl(0))> --> <+(?x_1,?x_1), +(?x_1,?x_1)> => yes
<dbl(s(+(?x_2,?y_2))), +(dbl(s(?x_2)),dbl(?y_2))> --> <s(+(+(?x_2,?y_2),s(+(?x_2,?y_2)))), s(+(+(?x_2,s(?x_2)),+(?y_2,?y_2)))> => no
PCP_in(symP,S):
<dbl(+(?x_5,?y_5)), +(dbl(?y_5),dbl(?x_5))> --> <+(+(?x_5,?y_5),+(?x_5,?y_5)), +(+(?y_5,?y_5),+(?x_5,?x_5))> => no
<dbl(+(?x_4,+(?y_4,?z_4))), +(dbl(+(?x_4,?y_4)),dbl(?z_4))> --> <+(+(?x_4,+(?y_4,?z_4)),+(?x_4,+(?y_4,?z_4))), +(+(+(?x_4,?y_4),+(?x_4,?y_4)),+(?z_4,?z_4))> => no
<dbl(+(?x_3,s(?y_3))), +(dbl(s(?y_3)),dbl(?x_3))> --> <+(+(?x_3,s(?y_3)),+(?x_3,s(?y_3))), s(+(+(?y_3,s(?y_3)),+(?x_3,?x_3)))> => no
<dbl(+(+(?x_2,?y_2),?z_2)), +(dbl(?x_2),dbl(+(?y_2,?z_2)))> --> <+(+(+(?x_2,?y_2),?z_2),+(+(?x_2,?y_2),?z_2)), +(+(?x_2,?x_2),+(+(?y_2,?z_2),+(?y_2,?z_2)))> => no
<dbl(+(s(?y_1),?x_1)), +(dbl(?x_1),dbl(s(?y_1)))> --> <s(+(+(?y_1,?x_1),s(+(?y_1,?x_1)))), +(+(?x_1,?x_1),s(+(?y_1,s(?y_1))))> => no
CP(S,symP):
<s(?y_1), +(s(?y_1),0)> --> <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(?y_1), +(0,s(?y_1))> --> <s(?y_1), s(?y_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(?y_1),s(?x))> --> <s(+(?x,s(?y_1))), s(+(?y_1,s(?x)))> => 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)), +(?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
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), +(?y,s(?x))> => no
check joinability condition:
check modulo joinability of +(+(?x_4,?x_4),+(?y_4,?y_4)) and +(+(?x_4,?y_4),+(?x_4,?y_4)): joinable by {0,1}
check modulo joinability of s(+(+(?x_2,?y_2),s(+(?x_2,?y_2)))) and s(+(+(?x_2,s(?x_2)),+(?y_2,?y_2))): joinable by {0,1}
check modulo joinability of +(+(?x_5,?y_5),+(?x_5,?y_5)) and +(+(?y_5,?y_5),+(?x_5,?x_5)): joinable by {0,1}
check modulo joinability of +(+(?x_4,+(?y_4,?z_4)),+(?x_4,+(?y_4,?z_4))) and +(+(+(?x_4,?y_4),+(?x_4,?y_4)),+(?z_4,?z_4)): joinable by {1,3}
check modulo joinability of +(+(?x_3,s(?y_3)),+(?x_3,s(?y_3))) and s(+(+(?y_3,s(?y_3)),+(?x_3,?x_3))): joinable by {0,1}
check modulo joinability of +(+(+(?x_2,?y_2),?z_2),+(+(?x_2,?y_2),?z_2)) and +(+(?x_2,?x_2),+(+(?y_2,?z_2),+(?y_2,?z_2))): joinable by {0,1,3}
check modulo joinability of s(+(+(?y_1,?x_1),s(+(?y_1,?x_1)))) and +(+(?x_1,?x_1),s(+(?y_1,s(?y_1)))): joinable by {0,1}
check modulo joinability of s(+(?x,s(?y_1))) and s(+(?y_1,s(?x))): 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 +(?y,s(?x)): 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)),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
P:
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?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),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
P:
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?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),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= 0
  s:= (1)*x1
  dbl:= (4)+(8)*x1+(13)*x1*x1
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (2)
  s:= (1)*x1
  dbl:= (8)*x1*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (15)
  s:= (3)+(1)*x1
  dbl:= (1)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract +(s(?x),?y) -> s(+(?x,?y))
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (2)+(1)*x1+(1)*x2
  0:= (14)
  s:= (3)+(2)*x1
  dbl:= (10)*x1+(8)*x1*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)),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
<s(+(?x_5,0)), s(?x_5)> --> <s(?x_5), s(?x_5)> => yes
<s(+(?x_2,0)), s(?x_2)> --> <s(?x_2), s(?x_2)> => yes
<s(+(?x_5,s(?x_2))), s(+(?x_2,s(?x_5)))> --> <s(s(+(?x_2,?x_5))), s(s(+(?x_5,?x_2)))> => yes
<+(dbl(?x_4),dbl(?y_4)), +(+(?x_4,?y_4),+(?x_4,?y_4))> --> <+(+(?x_4,?x_4),+(?y_4,?y_4)), +(+(?x_4,?y_4),+(?x_4,?y_4))> => no
<dbl(?y), +(dbl(0),dbl(?y))> --> <+(?y,?y), +(?y,?y)> => yes
<dbl(?x_1), +(dbl(?x_1),dbl(0))> --> <+(?x_1,?x_1), +(?x_1,?x_1)> => yes
<dbl(s(+(?x_2,?y_2))), +(dbl(s(?x_2)),dbl(?y_2))> --> <s(s(+(+(?x_2,?y_2),+(?x_2,?y_2)))), s(s(+(+(?x_2,?x_2),+(?y_2,?y_2))))> => no
<dbl(s(+(?x_5,?y_5))), +(dbl(?y_5),dbl(s(?x_5)))> --> <s(s(+(+(?x_5,?y_5),+(?x_5,?y_5)))), s(s(+(+(?x_5,?x_5),+(?y_5,?y_5))))> => no
PCP_in(symP,S):
<dbl(+(?x_5,?y_5)), +(dbl(?y_5),dbl(?x_5))> --> <+(+(?x_5,?y_5),+(?x_5,?y_5)), +(+(?y_5,?y_5),+(?x_5,?x_5))> => no
<dbl(+(?x_4,+(?y_4,?z_4))), +(dbl(+(?x_4,?y_4)),dbl(?z_4))> --> <+(+(?x_4,+(?y_4,?z_4)),+(?x_4,+(?y_4,?z_4))), +(+(+(?x_4,?y_4),+(?x_4,?y_4)),+(?z_4,?z_4))> => no
<dbl(+(?x_3,s(?y_3))), +(dbl(s(?y_3)),dbl(?x_3))> --> <s(s(+(+(?y_3,?x_3),+(?y_3,?x_3)))), s(s(+(+(?y_3,?y_3),+(?x_3,?x_3))))> => no
<dbl(+(+(?x_2,?y_2),?z_2)), +(dbl(?x_2),dbl(+(?y_2,?z_2)))> --> <+(+(+(?x_2,?y_2),?z_2),+(+(?x_2,?y_2),?z_2)), +(+(?x_2,?x_2),+(+(?y_2,?z_2),+(?y_2,?z_2)))> => no
<dbl(+(s(?y_1),?x_1)), +(dbl(?x_1),dbl(s(?y_1)))> --> <s(s(+(+(?y_1,?x_1),+(?y_1,?x_1)))), s(s(+(+(?y_1,?y_1),+(?x_1,?x_1))))> => no
CP(S,symP):
<s(?y_1), +(s(?y_1),0)> --> <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(?y_1), +(0,s(?y_1))> --> <s(?y_1), s(?y_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(?y_1),s(?x))> --> <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)), +(?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
<s(+(?x,?y)), +(?y,s(?x))> --> <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,?y),s(?x))> --> <s(+(+(?x,?y),?x_1)), s(+(?x,+(?x_1,?y)))> => no
<s(+(?x,s(?y_1))), +(s(?x),s(?y_1))> --> <s(s(+(?y_1,?x))), s(s(+(?y_1,?x)))> => 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
check joinability condition:
check modulo joinability of +(+(?x_4,?x_4),+(?y_4,?y_4)) and +(+(?x_4,?y_4),+(?x_4,?y_4)): joinable by {0,1}
check modulo joinability of s(s(+(+(?x_2,?y_2),+(?x_2,?y_2)))) and s(s(+(+(?x_2,?x_2),+(?y_2,?y_2)))): joinable by {0,1}
check modulo joinability of s(s(+(+(?x_5,?y_5),+(?x_5,?y_5)))) and s(s(+(+(?x_5,?x_5),+(?y_5,?y_5)))): joinable by {0,1}
check modulo joinability of +(+(?x_5,?y_5),+(?x_5,?y_5)) and +(+(?y_5,?y_5),+(?x_5,?x_5)): joinable by {0,1}
check modulo joinability of +(+(?x_4,+(?y_4,?z_4)),+(?x_4,+(?y_4,?z_4))) and +(+(+(?x_4,?y_4),+(?x_4,?y_4)),+(?z_4,?z_4)): joinable by {1,3}
check modulo joinability of s(s(+(+(?y_3,?x_3),+(?y_3,?x_3)))) and s(s(+(+(?y_3,?y_3),+(?x_3,?x_3)))): joinable by {0,1}
check modulo joinability of +(+(+(?x_2,?y_2),?z_2),+(+(?x_2,?y_2),?z_2)) and +(+(?x_2,?x_2),+(+(?y_2,?z_2),+(?y_2,?z_2))): joinable by {0,1,3}
check modulo joinability of s(s(+(+(?y_1,?x_1),+(?y_1,?x_1)))) and s(s(+(+(?y_1,?y_1),+(?x_1,?x_1)))): joinable by {0,1}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(+(?x,?x_1),?y)): joinable by {1}
check modulo joinability of s(+(+(?x,?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}
success
P':
   [ +(?y,?x) -> +(?x,?y),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?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),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
   [ +(?y,?x) -> +(?x,?y),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (8)
  s:= (1)*x1
  dbl:= (7)+(2)*x1+(1)*x1*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (2)+(1)*x1+(1)*x2
  0:= (12)
  s:= (1)*x1
  dbl:= (1)+(10)*x1
retract +(0,?y) -> ?y
retract +(?x,0) -> ?x
retract dbl(?x) -> +(?x,?x)
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (3)
  s:= (6)+(1)*x1
  dbl:= (1)*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)),
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(?x,s(?y)) -> +(s(?y),?x),
     +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
     +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
171.trs: Success(CR)
(6074 msec.)