YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(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:
   [ +(s(?x_4),?y_1) = s(+(s(?x_4),?y_1)),
     +(s(s(?x_5)),?y_1) = s(+(?x_5,?y_1)),
     +(?y,?z_2) = +(0,+(?y,?z_2)),
     +(s(+(?x_1,?y_1)),?z_2) = +(s(?x_1),+(?y_1,?z_2)),
     +(+(?y_3,?x_3),?z_2) = +(?x_3,+(?y_3,?z_2)),
     s(s(s(?x_5))) = s(?x_5),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)),
     s(s(?x)) = s(s(?x)) ]
Outer CPs:
   [ ?y = +(?y,0),
     s(+(?x_1,?y_1)) = +(?y_1,s(?x_1)),
     +(?x_2,+(?y_2,?z_2)) = +(?z_2,+(?x_2,?y_2)),
     s(?x_4) = s(s(s(?x_4))) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly 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_3)) = +(?y,?z_3),
     +(?y,s(?x_5)) = s(+(s(?x_5),?y)),
     +(?y,s(s(?x))) = s(+(?x,?y)),
     +(?y,s(?x)) = s(+(?x,?y)),
     +(s(?x_5),?y) = s(+(s(?x_5),?y)),
     +(s(s(?x)),?y) = s(+(?x,?y)),
     +(s(?x_8),+(?y,?z_3)) = +(s(+(s(?x_8),?y)),?z_3),
     +(s(s(?x)),+(?y,?z_3)) = +(s(+(?x,?y)),?z_3),
     +(s(?x),+(?y,?z_3)) = +(s(+(?x,?y)),?z_3),
     +(?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_3,+(?y_3,?y)) = +(?y,+(?x_3,?y_3)),
     +(?x,+(?y,?z_4)) = +(+(?y,?x),?z_4),
     s(s(s(?x_1))) = s(s(?x_1)),
     s(s(s(s(?x)))) = s(?x),
     s(s(s(?x))) = s(?x),
     s(s(?x_1)) = s(s(?x_1)),
     s(s(?x_1)) = s(s(s(?x_1))),
     s(+(s(?x_4),?y_3)) = +(s(s(?x_4)),?y_3),
     s(+(s(s(?x)),?y_3)) = +(s(?x),?y_3),
     s(+(s(?x),?y_3)) = +(s(?x),?y_3),
     s(?x_5) = s(s(s(?x_5))),
     s(+(?x,?y_3)) = +(s(s(?x)),?y_3) ]
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 <+(s(?x_4),?y_1), s(+(s(?x_4),?y_1))> by Rules <4, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],5),([],1)], []>
 joinable by a reduction of rules <[([],1),([],5)], [([(s,1)],1)]>
 joinable by a reduction of rules <[([],1)], [([(s,1)],1),([],4)]>
Critical Pair <+(s(s(?x_5)),?y_1), s(+(?x_5,?y_1))> by Rules <5, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],4),([],1)], []>
 joinable by a reduction of rules <[([],1),([(s,1)],1)], [([],5)]>
Critical Pair <+(?y,?z_2), +(0,+(?y,?z_2))> by Rules <0, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <+(s(+(?x_1,?y_1)),?z_2), +(s(?x_1),+(?y_1,?z_2))> by Rules <1, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[([],1),([(s,1)],2)], [([],1)]>
Critical Pair <+(+(?y_3,?x_3),?z_2), +(?x_3,+(?y_3,?z_2))> by Rules <3, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],3),([],2)], []>
 joinable by a reduction of rules <[([],2),([(+,2)],3)], [([],3),([],2)]>
Critical Pair <s(s(s(?x_5))), s(?x_5)> by Rules <5, 4> preceded by [(s,1)]
 joinable by a reduction of rules <[([(s,1)],4)], [([],5)]>
 joinable by a reduction of rules <[([],4)], [([],5)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <2, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[([],2),([(+,2)],2)], [([],2)]>
Critical Pair <s(s(?x)), s(s(?x))> by Rules <4, 4> preceded by [(s,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <+(?y_3,0), ?y_3> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],3),([],0)], []>
Critical Pair <+(?y_3,s(?x_1)), s(+(?x_1,?y_3))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([],3),([],1)], []>
Critical Pair <+(?y_3,+(?x_2,?y_2)), +(?x_2,+(?y_2,?y_3))> by Rules <3, 2> preceded by []
 joinable by a reduction of rules <[([],3),([],2)], []>
Critical Pair <s(s(s(?x_4))), s(?x_4)> by Rules <5, 4> preceded by []
 joinable by a reduction of rules <[([(s,1)],4)], [([],5)]>
 joinable by a reduction of rules <[([],4)], [([],5)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 16 rules by 3 steps unfolding
strenghten +(?x_10,0) and ?x_10
strenghten +(0,?x_8) and ?x_8
strenghten s(s(?x_4)) and s(?x_4)
strenghten s(+(?x_1,0)) and s(?x_1)
strenghten s(s(s(?x_4))) and s(?x_4)
strenghten s(+(?x_1,?y_3)) and +(?y_3,s(?x_1))
strenghten s(+(?x_6,?y_1)) and +(s(?x_6),?y_1)
strenghten +(?x_2,+(?y_2,0)) and +(?x_2,?y_2)
strenghten +(?x_8,+(0,?z_2)) and +(?x_8,?z_2)
strenghten +(0,+(?x_10,?z_2)) and +(?x_10,?z_2)
strenghten +(s(s(?x_5)),?y_1) and s(+(?x_5,?y_1))
strenghten s(+(s(?x_4),?y_1)) and +(s(?x_4),?y_1)
strenghten +(?x_2,+(?y_2,?y_3)) and +(?y_3,+(?x_2,?y_2))
strenghten +(?x_3,+(?y_3,?z_2)) and +(+(?y_3,?x_3),?z_2)
strenghten +(s(?x_1),+(?y_1,?z_2)) and +(s(+(?x_1,?y_1)),?z_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,
     +(s(?x),?y) -> s(+(?x,?y)),
     s(s(?x)) -> s(?x),
     s(?x) -> s(s(?x)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
not relatively terminatiing
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(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):
PCP_in(symP,S):
<+(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):
<+(?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 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}
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))
replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(s(+(?x,?y)))
STEP: 2 (linear)
S:
   [ +(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):
CP_in(symP,S):
<+(s(s(?x)),?y_1), s(+(?x,?y_1))> --> <s(s(+(?x,?y_1))), s(+(?x,?y_1))> => yes
<+(s(?x),?y_1), s(+(s(?x),?y_1))> --> <s(+(?x,?y_1)), s(s(+(?x,?y_1)))> => yes
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 2)
   [ +(?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))
replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(s(+(?x,?y)))
STEP: 3 (relative)
S:
   [ +(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)) ]
Check relative termination:
   [ +(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)) ]
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (12)
  s:= (1)*x1
retract +(0,?y) -> ?y
Polynomial Interpretation:
  +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2
  0:= (10)
  s:= (1)+(1)*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)),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> 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(?x_1), s(+(?x_1,0))> --> <s(?x_1), s(?x_1)> => yes
<s(+(?x_3,s(?x_1))), s(+(?x_1,s(?x_3)))> --> <s(s(+(?x_1,?x_3))), s(s(+(?x_3,?x_1)))> => yes
PCP_in(symP,S):
<+(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
<+(?y,s(?x_5)), s(+(s(?x_5),?y))> --> <s(+(?x_5,?y)), s(s(+(?x_5,?y)))> => no
<+(?y,s(s(?x_4))), s(+(?x_4,?y))> --> <s(s(+(?x_4,?y))), s(+(?x_4,?y))> => no
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_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}
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}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(+(?x,?x_1),?y)): joinable by {3}
check modulo joinability of s(+(?x,+(?x_1,?y_1))) and s(+(+(?x,?y_1),?x_1)): joinable by {3}
check modulo joinability of s(+(+(?x,?y),?z_1)) and s(+(+(?x,?z_1),?y)): joinable by {3}
check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?x,+(?x_1,?y))): joinable by {3}
success
P':
   [ s(s(?x)) -> s(?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Check relative termination:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> s(+(?x,?y)) ]
   [ s(s(?x)) -> s(?x),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (8)
  s:= (1)*x1
retract +(0,?y) -> ?y
retract +(?y,0) -> ?y
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= (15)
  s:= (4)+(1)*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?y,0) -> ?y,
     +(?y,s(?x)) -> 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
170.trs: Success(CR)
(3017 msec.)