YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_5) = +(?x,+(0,?z_5)),
     +(s(+(?x_1,?y_1)),?z_5) = +(?x_1,+(s(?y_1),?z_5)),
     +(?y_2,?z_5) = +(0,+(?y_2,?z_5)),
     +(s(+(?x_3,?y_3)),?z_5) = +(s(?x_3),+(?y_3,?z_5)),
     +(+(?y_6,?x_6),?z_5) = +(?x_6,+(?y_6,?z_5)),
     dbl(?x) = +(dbl(?x),dbl(0)),
     dbl(s(+(?x_1,?y_1))) = +(dbl(?x_1),dbl(s(?y_1))),
     dbl(?y_2) = +(dbl(0),dbl(?y_2)),
     dbl(s(+(?x_3,?y_3))) = +(dbl(s(?x_3)),dbl(?y_3)),
     dbl(+(?x_5,+(?y_5,?z_5))) = +(dbl(+(?x_5,?y_5)),dbl(?z_5)),
     dbl(+(?y_6,?x_6)) = +(dbl(?x_6),dbl(?y_6)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ 0 = 0,
     s(?x_3) = s(+(?x_3,0)),
     +(?x_5,?y_5) = +(?x_5,+(?y_5,0)),
     ?x = +(0,?x),
     s(+(0,?y_1)) = s(?y_1),
     s(+(s(?x_3),?y_1)) = s(+(?x_3,s(?y_1))),
     s(+(+(?x_5,?y_5),?y_1)) = +(?x_5,+(?y_5,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_7,?y_7),+(?x_7,?y_7)) = +(dbl(?x_7),dbl(?y_7)),
     +(?x_5,+(?y_5,?z_5)) = +(?z_5,+(?x_5,?y_5)) ]
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(+(?x_3,0)) = s(?x_3),
     +(?x_5,+(?y_5,0)) = +(?x_5,?y_5),
     +(0,?x) = ?x,
     +(?x,+(0,?z_6)) = +(?x,?z_6),
     +(dbl(?x),dbl(0)) = dbl(?x),
     s(?y) = s(+(0,?y)),
     s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
     +(?x_5,+(?y_5,s(?y))) = s(+(+(?x_5,?y_5),?y)),
     +(s(?y),?x) = s(+(?x,?y)),
     +(?x,+(s(?y),?z_6)) = +(s(+(?x,?y)),?z_6),
     +(dbl(?x),dbl(s(?y))) = dbl(s(+(?x,?y))),
     s(+(0,?y_2)) = s(?y_2),
     +(?y,0) = ?y,
     +(0,+(?y,?z_6)) = +(?y,?z_6),
     +(dbl(0),dbl(?y)) = dbl(?y),
     s(?x) = s(+(?x,0)),
     s(+(s(?x),?y_2)) = s(+(?x,s(?y_2))),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(s(?x),+(?y,?z_6)) = +(s(+(?x,?y)),?z_6),
     +(dbl(s(?x)),dbl(?y)) = dbl(s(+(?x,?y))),
     +(dbl(?x_7),dbl(?y_7)) = +(+(?x_7,?y_7),+(?x_7,?y_7)),
     +(?x_1,+(?y_1,?y)) = +(+(?x_1,?y_1),+(?y,0)),
     ?x = +(?x,+(0,0)),
     s(+(?x,?y_3)) = +(?x,+(s(?y_3),0)),
     ?y = +(0,+(?y,0)),
     s(+(?x_5,?y)) = +(s(?x_5),+(?y,0)),
     +(?y,?x) = +(?x,+(?y,0)),
     s(+(+(?x_3,+(?y_3,?y)),?y_2)) = +(+(?x_3,?y_3),+(?y,s(?y_2))),
     s(+(?x,?y_2)) = +(?x,+(0,s(?y_2))),
     s(+(s(+(?x,?y_5)),?y_2)) = +(?x,+(s(?y_5),s(?y_2))),
     s(+(?y,?y_2)) = +(0,+(?y,s(?y_2))),
     s(+(s(+(?x_7,?y)),?y_2)) = +(s(?x_7),+(?y,s(?y_2))),
     s(+(+(?y,?x),?y_2)) = +(?x,+(?y,s(?y_2))),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?x) = +(?x,+(0,?z)),
     +(?z,s(+(?x,?y_3))) = +(?x,+(s(?y_3),?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,s(+(?x_5,?y))) = +(s(?x_5),+(?y,?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?x,+(?y,0)),
     s(+(+(?x,?y),?y_2)) = +(?x,+(?y,s(?y_2))),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?x,?z) = +(?x,+(0,?z)),
     +(s(+(?x,?y_3)),?z) = +(?x,+(s(?y_3),?z)),
     +(?y,?z) = +(0,+(?y,?z)),
     +(s(+(?x_5,?y)),?z) = +(s(?x_5),+(?y,?z)),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(?x,+(?z,?z_1)) = +(+(?x,+(0,?z)),?z_1),
     +(s(+(?x,?y_4)),+(?z,?z_1)) = +(+(?x,+(s(?y_4),?z)),?z_1),
     +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1),
     +(s(+(?x_6,?y)),+(?z,?z_1)) = +(+(s(?x_6),+(?y,?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(dbl(+(?x_1,+(?y_1,?y))),dbl(?z)) = dbl(+(+(?x_1,?y_1),+(?y,?z))),
     +(dbl(?x),dbl(?z)) = dbl(+(?x,+(0,?z))),
     +(dbl(s(+(?x,?y_3))),dbl(?z)) = dbl(+(?x,+(s(?y_3),?z))),
     +(dbl(?y),dbl(?z)) = dbl(+(0,+(?y,?z))),
     +(dbl(s(+(?x_5,?y))),dbl(?z)) = dbl(+(s(?x_5),+(?y,?z))),
     +(dbl(+(?y,?x)),dbl(?z)) = dbl(+(?x,+(?y,?z))),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(dbl(+(?x,?y)),dbl(?z)) = dbl(+(?x,+(?y,?z))),
     ?x = +(0,?x),
     s(+(?x,?y_2)) = +(s(?y_2),?x),
     ?y = +(?y,0),
     s(+(?x_4,?y)) = +(?y,s(?x_4)),
     +(?x_6,+(?y_6,?y)) = +(?y,+(?x_6,?y_6)),
     +(?x,+(?y,?z_7)) = +(+(?y,?x),?z_7),
     +(dbl(?x),dbl(?y)) = dbl(+(?y,?x)),
     +(?x,?x) = +(dbl(?x),dbl(0)),
     +(s(+(?x,?y_3)),s(+(?x,?y_3))) = +(dbl(?x),dbl(s(?y_3))),
     +(?y,?y) = +(dbl(0),dbl(?y)),
     +(s(+(?x_5,?y)),s(+(?x_5,?y))) = +(dbl(s(?x_5)),dbl(?y)),
     +(+(?x_7,+(?y_7,?y)),+(?x_7,+(?y_7,?y))) = +(dbl(+(?x_7,?y_7)),dbl(?y)),
     +(+(?y,?x),+(?y,?x)) = +(dbl(?x),dbl(?y)),
     +(+(?x,?y),+(?x,?y)) = +(dbl(?x),dbl(?y)),
     dbl(?x) = +(dbl(?x),dbl(0)),
     dbl(s(+(?x,?y_3))) = +(dbl(?x),dbl(s(?y_3))),
     dbl(?y) = +(dbl(0),dbl(?y)),
     dbl(s(+(?x_5,?y))) = +(dbl(s(?x_5)),dbl(?y)),
     dbl(+(?x_7,+(?y_7,?y))) = +(dbl(+(?x_7,?y_7)),dbl(?y)),
     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,?z_5), +(?x,+(0,?z_5))> by Rules <0, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],2)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_5), +(?x_1,+(s(?y_1),?z_5))> by Rules <1, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],5)], [([(+,2)],3),([],1)]>
Critical Pair <+(?y_2,?z_5), +(0,+(?y_2,?z_5))> by Rules <2, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],2)]>
Critical Pair <+(s(+(?x_3,?y_3)),?z_5), +(s(?x_3),+(?y_3,?z_5))> by Rules <3, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(s,1)],5)], [([],3)]>
Critical Pair <+(+(?y_6,?x_6),?z_5), +(?x_6,+(?y_6,?z_5))> by Rules <6, 5> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],6),([],5)], []>
 joinable by a reduction of rules <[([],5),([(+,2)],6)], [([],6),([],5)]>
Critical Pair <dbl(?x), +(dbl(?x),dbl(0))> by Rules <0, 7> preceded by [(dbl,1)]
 joinable by a reduction of rules <[], [([(+,2)],4),([(+,2)],2),([],0)]>
 joinable by a reduction of rules <[], [([(+,2)],4),([(+,2)],0),([],0)]>
Critical Pair <dbl(s(+(?x_1,?y_1))), +(dbl(?x_1),dbl(s(?y_1)))> by Rules <1, 7> preceded by [(dbl,1)]
unknown Diagram Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     dbl(?x_4) -> +(?x_4,?x_4),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_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_4}
non-linear types: {23}
types leq non-linear types: {23}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     dbl(?x_4) -> +(?x_4,?x_4),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_6,?x_6),
     dbl(+(?x_7,?y_7)) -> +(dbl(?x_7),dbl(?y_7)) ]
NLR:
0: {}
1: {}
2: {}
3: {}
4: {0,1,2,3,4,5,6,7}
5: {}
6: {}
7: {}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     dbl(?x_4) -> +(?x_4,?x_4),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_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_4}
non-linear types: {23}
types leq non-linear types: {23}
rules applicable to terms of non-linear types:
   [ +(?x,0) -> ?x,
     +(?x_1,s(?y_1)) -> s(+(?x_1,?y_1)),
     +(0,?y_2) -> ?y_2,
     +(s(?x_3),?y_3) -> s(+(?x_3,?y_3)),
     dbl(?x_4) -> +(?x_4,?x_4),
     +(+(?x_5,?y_5),?z_5) -> +(?x_5,+(?y_5,?z_5)),
     +(?x_6,?y_6) -> +(?y_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_6) and ?x_6
strenghten +(0,0) and 0
strenghten +(dbl(0),?x_14) and ?x_14
strenghten dbl(+(0,0)) and 0
strenghten s(+(?x_3,0)) and s(?x_3)
strenghten s(+(0,?y_1)) and s(?y_1)
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),dbl(0)) and dbl(?x)
strenghten +(dbl(0),dbl(?x_10)) and dbl(?x_10)
strenghten +(dbl(0),dbl(0)) and dbl(0)
strenghten s(+(?x_3,dbl(0))) and s(?x_3)
strenghten s(+(?x_3,?y_6)) and +(?y_6,s(?x_3))
strenghten s(+(?x_6,?y_1)) and +(s(?y_1),?x_6)
strenghten +(?x,+(0,?z_5)) and +(?x,?z_5)
strenghten +(?x_5,+(?y_5,0)) and +(?x_5,?y_5)
strenghten +(0,+(?x_10,?z_5)) and +(?x_10,?z_5)
strenghten +(dbl(?x_14),dbl(dbl(0))) and dbl(?x_14)
strenghten +(dbl(?x_6),dbl(?y_6)) and dbl(+(?y_6,?x_6))
strenghten +(?x_5,+(?y_5,dbl(0))) and +(?x_5,?y_5)
strenghten +(?x_14,+(dbl(0),?z_5)) and +(?x_14,?z_5)
strenghten +(?x_5,+(?y_5,?y_6)) and +(?y_6,+(?x_5,?y_5))
strenghten +(?x_6,+(?y_6,?z_5)) and +(+(?y_6,?x_6),?z_5)
strenghten s(+(s(?x_3),?y_1)) and s(+(?x_3,s(?y_1)))
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:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (12)
  s:= (1)*x1
  dbl:= (7)+(2)*x1+(15)*x1*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x2
  0:= (8)
  s:= (3)+(1)*x1
  dbl:= (3)*x1
retract +(?x,0) -> ?x
retract +(0,?y) -> ?y
retract dbl(?x) -> +(?x,?x)
retract dbl(+(?x,?y)) -> +(dbl(?x),dbl(?y))
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (12)
  s:= (2)+(2)*x1
  dbl:= (1)*x1
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Final result: CR
155.trs: Success(CR)
(4619 msec.)