YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?y,?z_3) = +(0,+(?y,?z_3)),
     +(s(+(?x_1,?y_1)),?z_3) = +(s(?x_1),+(?y_1,?z_3)),
     +(+(?y_4,?x_4),?z_3) = +(?x_4,+(?y_4,?z_3)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ ?y = +(?y,0),
     s(+(?x_1,?y_1)) = +(?y_1,s(?x_1)),
     +(?x_3,+(?y_3,?z_3)) = +(?z_3,+(?x_3,?y_3)) ]
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:
   [ +(?y,0) = ?y,
     +(0,+(?y,?z_4)) = +(?y,?z_4),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(s(?x),+(?y,?z_4)) = +(s(+(?x,?y)),?z_4),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,s(+(?x_3,?y))) = +(s(?x_3),+(?y,?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?y,?z) = +(0,+(?y,?z)),
     +(s(+(?x_3,?y)),?z) = +(s(?x_3),+(?y,?z)),
     +(+(?y,?x),?z) = +(?x,+(?y,?z)),
     +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1),
     +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1),
     +(s(+(?x_4,?y)),+(?z,?z_1)) = +(+(s(?x_4),+(?y,?z)),?z_1),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     ?y = +(?y,0),
     s(+(?x_2,?y)) = +(?y,s(?x_2)),
     +(?x_4,+(?y_4,?y)) = +(?y,+(?x_4,?y_4)),
     +(?x,+(?y,?z_5)) = +(+(?y,?x),?z_5) ]
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 <+(?y,?z_3), +(0,+(?y,?z_3))> by Rules <0, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_3), +(s(?x_1),+(?y_1,?z_3))> by Rules <1, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[([],1),([(s,1)],3)], [([],1)]>
Critical Pair <+(+(?y_4,?x_4),?z_3), +(?x_4,+(?y_4,?z_3))> by Rules <4, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],4),([],3)], []>
 joinable by a reduction of rules <[([],3),([(+,2)],4)], [([],4),([],3)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <3, 3> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3),([(+,2)],3)], [([],3)]>
Critical Pair <+(?y_4,0), ?y_4> by Rules <4, 0> preceded by []
 joinable by a reduction of rules <[([],4),([],0)], []>
Critical Pair <+(?y_4,s(?x_1)), s(+(?x_1,?y_4))> by Rules <4, 1> preceded by []
 joinable by a reduction of rules <[([],4),([],1)], []>
Critical Pair <+(?y_4,+(?x_3,?y_3)), +(?x_3,+(?y_3,?y_4))> by Rules <4, 3> preceded by []
 joinable by a reduction of rules <[([],4),([],3)], []>
unknown Diagram Decreasing
   [ +(0,?y) -> ?y,
     +(s(?x_1),?y_1) -> s(+(?x_1,?y_1)),
     dbl(?x_2) -> +(?x_2,?x_2),
     +(+(?x_3,?y_3),?z_3) -> +(?x_3,+(?y_3,?z_3)),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
Sort Assignment:
 + : 18*18=>18
 0 : =>18
 s : 18=>18
 dbl : 18=>18
non-linear variables: {?x_2}
non-linear types: {18}
types leq non-linear types: {18}
rules applicable to terms of non-linear types:
   [ +(0,?y) -> ?y,
     +(s(?x_1),?y_1) -> s(+(?x_1,?y_1)),
     dbl(?x_2) -> +(?x_2,?x_2),
     +(+(?x_3,?y_3),?z_3) -> +(?x_3,+(?y_3,?z_3)),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
NLR:
0: {}
1: {}
2: {0,1,2,3,4}
3: {}
4: {}
unknown innermost-termination for terms of non-linear types
unknown Quasi-Linear & Linearized-Decreasing
   [ +(0,?y) -> ?y,
     +(s(?x_1),?y_1) -> s(+(?x_1,?y_1)),
     dbl(?x_2) -> +(?x_2,?x_2),
     +(+(?x_3,?y_3),?z_3) -> +(?x_3,+(?y_3,?z_3)),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
Sort Assignment:
 + : 18*18=>18
 0 : =>18
 s : 18=>18
 dbl : 18=>18
non-linear variables: {?x_2}
non-linear types: {18}
types leq non-linear types: {18}
rules applicable to terms of non-linear types:
   [ +(0,?y) -> ?y,
     +(s(?x_1),?y_1) -> s(+(?x_1,?y_1)),
     dbl(?x_2) -> +(?x_2,?x_2),
     +(+(?x_3,?y_3),?z_3) -> +(?x_3,+(?y_3,?z_3)),
     +(?x_4,?y_4) -> +(?y_4,?x_4) ]
unknown innermost-termination for terms of non-linear types
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 15 rules by 3 steps unfolding
strenghten dbl(0) and 0
strenghten +(?x_7,0) and ?x_7
strenghten +(0,?x_5) and ?x_5
strenghten +(0,0) and 0
strenghten +(dbl(0),?x_11) and ?x_11
strenghten dbl(+(0,0)) and 0
strenghten s(+(?x_1,0)) and s(?x_1)
strenghten dbl(+(0,0)) and dbl(0)
strenghten +(?x_11,+(0,0)) and ?x_11
strenghten +(dbl(0),dbl(0)) and 0
strenghten +(dbl(0),dbl(0)) and dbl(0)
strenghten s(+(?x_1,dbl(0))) and s(?x_1)
strenghten s(+(?x_1,?y_4)) and +(?y_4,s(?x_1))
strenghten +(?x_3,+(?y_3,0)) and +(?x_3,?y_3)
strenghten +(?x_5,+(0,?z_3)) and +(?x_5,?z_3)
strenghten +(0,+(?x_7,?z_3)) and +(?x_7,?z_3)
strenghten +(?x_3,+(?y_3,dbl(0))) and +(?x_3,?y_3)
strenghten +(?x_11,+(dbl(0),?z_3)) and +(?x_11,?z_3)
strenghten +(?x_3,+(?y_3,?y_4)) and +(?y_4,+(?x_3,?y_3))
strenghten +(?x_4,+(?y_4,?z_3)) and +(+(?y_4,?x_4),?z_3)
strenghten +(s(?x_1),+(?y_1,?z_3)) and +(s(+(?x_1,?y_1)),?z_3)
strenghten +(+(?x,?y),+(?z,?z_1)) and +(+(?x,+(?y,?z)),?z_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:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (15)
  s:= (1)*x1
  dbl:= (13)+(4)*x1+(6)*x1*x1
retract +(0,?y) -> ?y
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (10)
  s:= (6)+(3)*x1
  dbl:= (4)+(3)*x1+(15)*x1*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
<+(?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_1,0),?y)> => no
<?y, +(?y,0)> --> <?y, +(?y,0)> => 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_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
check joinability condition:
check modulo reachablity from +(?x_1,?y) to +(+(?x_1,0),?y): maybe not reachable
check modulo reachablity from ?y to +(?y,0): maybe not reachable
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,0) -> ?y,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
Added S-Rules:
   [ +(?y,0) -> ?y,
     +(?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,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (15)
  s:= (1)*x1
  dbl:= (13)+(4)*x1+(6)*x1*x1
retract +(0,?y) -> ?y
retract dbl(?x) -> +(?x,?x)
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (10)
  s:= (6)+(3)*x1
  dbl:= (4)+(3)*x1+(15)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(+(?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(?x_1), s(+(?x_1,0))> --> <s(?x_1), s(?x_1)> => yes
<s(+(?x_4,s(?x_1))), s(+(?x_1,s(?x_4)))> --> <s(s(+(?x_1,?x_4))), s(s(+(?x_4,?x_1)))> => yes
PCP_in(symP,S):
CP(S,symP):
<+(?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,?y),?x_1)), s(+(+(?x,?x_1),?y))> => no
<s(+(?x,?y)), +(?y,s(?x))> --> <s(+(?x,?y)), s(+(?x,?y))> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?y,?z_1), +(?y,+(0,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,?y),0)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<?y, +(0,?y)> --> <?y, ?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
<+(?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
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,+(?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}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?x,+(?x_1,?y))): joinable by {2}
success
P':
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x1*x2+(1)*x2
  0:= (14)
  s:= (8)+(8)*x1
  dbl:= (6)+(8)*x1+(4)*x1*x1
retract +(0,?y) -> ?y
retract dbl(?x) -> +(?x,?x)
retract +(?y,0) -> ?y
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (15)
  s:= (4)+(1)*x1
  dbl:= (2)*x1+(4)*x1*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     dbl(?x) -> +(?x,?x),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
192.trs: Success(CR)
(2531 msec.)