YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)),
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_2) = +(?x,+(0,?z_2)),
     +(s(+(?y_1,?x_1)),?z_2) = +(s(?x_1),+(?y_1,?z_2)),
     +(+(?y_3,?x_3),?z_2) = +(?x_3,+(?y_3,?z_2)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ s(?x_1) = s(+(0,?x_1)),
     +(?x_2,?y_2) = +(?x_2,+(?y_2,0)),
     ?x = +(0,?x),
     s(+(?y_1,?x_1)) = +(?y_1,s(?x_1)),
     +(?x_2,+(?y_2,?z_2)) = +(?z_2,+(?x_2,?y_2)) ]
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:
   [ s(+(0,?x_1)) = s(?x_1),
     +(?x_2,+(?y_2,0)) = +(?x_2,?y_2),
     +(0,?x) = ?x,
     +(?x,+(0,?z_3)) = +(?x,?z_3),
     s(?x) = s(+(0,?x)),
     +(?y,s(?x)) = s(+(?y,?x)),
     +(s(?x),+(?y,?z_3)) = +(s(+(?y,?x)),?z_3),
     +(?x_1,+(?y_1,?y)) = +(+(?x_1,?y_1),+(?y,0)),
     ?x = +(?x,+(0,0)),
     s(+(?y,?x_3)) = +(s(?x_3),+(?y,0)),
     +(?y,?x) = +(?x,+(?y,0)),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?x) = +(?x,+(0,?z)),
     +(?z,s(+(?y,?x_3))) = +(s(?x_3),+(?y,?z)),
     +(?z,+(?y,?x)) = +(?x,+(?y,?z)),
     +(?x,?y) = +(?x,+(?y,0)),
     +(?z,+(?x,?y)) = +(?x,+(?y,?z)),
     +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?x,?z) = +(?x,+(0,?z)),
     +(s(+(?y,?x_3)),?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),
     +(?x,+(?z,?z_1)) = +(+(?x,+(0,?z)),?z_1),
     +(s(+(?y,?x_4)),+(?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),
     ?x = +(0,?x),
     s(+(?y,?x_2)) = +(?y,s(?x_2)),
     +(?x_3,+(?y_3,?y)) = +(?y,+(?x_3,?y_3)),
     +(?x,+(?y,?z_4)) = +(+(?y,?x),?z_4) ]
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_2), +(?x,+(0,?z_2))> by Rules <0, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([(+,2)],3),([(+,2)],0)]>
Critical Pair <+(s(+(?y_1,?x_1)),?z_2), +(s(?x_1),+(?y_1,?z_2))> by Rules <1, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[([],1)], [([(+,2)],3),([],1),([(s,1)],2)]>
 joinable by a reduction of rules <[([],1)], [([],1),([(s,1),(+,1)],3),([(s,1)],2)]>
 joinable by a reduction of rules <[([(+,1),(s,1)],3),([],1),([(s,1),(+,2)],3)], [([(+,2)],3),([],1),([(s,1)],2)]>
 joinable by a reduction of rules <[([(+,1),(s,1)],3),([],1),([(s,1),(+,2)],3)], [([],1),([(s,1),(+,1)],3),([(s,1)],2)]>
 joinable by a reduction of rules <[([],1),([(s,1)],3),([(s,1)],2)], [([],1),([(s,1)],2),([(s,1),(+,2)],3)]>
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 <+(+(?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(+(0,?x_1)), s(?x_1)> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([(s,1)],3),([(s,1)],0)], []>
Critical Pair <+(?x_2,+(?y_2,0)), +(?x_2,?y_2)> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([(+,2)],0)], []>
Critical Pair <+(0,?x_3), ?x_3> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],3),([],0)], []>
Critical Pair <+(?y_3,s(?x_1)), s(+(?y_3,?x_1))> 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)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 14 rules by 3 steps unfolding
strenghten +(?x_6,0) and ?x_6
strenghten +(0,?x_3) and ?x_3
strenghten +(?x_9,?y_9) and +(?y_9,?x_9)
strenghten +(?x_12,s(0)) and s(?x_12)
strenghten +(s(0),?x_12) and s(?x_12)
strenghten s(+(0,?x_1)) and s(?x_1)
strenghten s(+(?x_10,?x_1)) and +(?x_10,s(?x_1))
strenghten s(+(?x_11,?x_1)) and +(s(?x_1),?x_11)
strenghten +(?x,+(0,?z_2)) and +(?x,?z_2)
strenghten +(?x_2,+(?y_2,0)) and +(?x_2,?y_2)
strenghten +(0,+(?x_6,?z_2)) and +(?x_6,?z_2)
strenghten s(+(s(0),?x_1)) and s(s(?x_1))
strenghten +(?x_2,+(?y_2,?x_10)) and +(?x_10,+(?x_2,?y_2))
strenghten +(?x_2,+(?y_2,?x_11)) and +(+(?x_2,?y_2),?x_11)
strenghten +(?x_3,+(?y_3,?z_2)) and +(+(?y_3,?x_3),?z_2)
strenghten +(?x_2,+(?y_2,s(0))) and s(+(?x_2,?y_2))
strenghten +(?x_12,+(s(0),?z_2)) and +(s(?x_12),?z_2)
strenghten +(s(?x_1),+(?y_1,?z_2)) and +(s(+(?y_1,?x_1)),?z_2)
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:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (5)
  s:= (1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?x_1)), s(?x_1)> --> <s(+(0,?x_1)), s(?x_1)> => no
PCP_in(symP,S):
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,+(0,?z_1))> => no
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<+(s(+(?y,?x)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(?z_1,+(?y,?x))), s(+(+(?y,?z_1),?x))> => no
<s(+(+(?y_1,?z_1),?x)), +(+(s(?x),?y_1),?z_1)> --> <s(+(+(?y_1,?z_1),?x)), s(+(?z_1,+(?y_1,?x)))> => no
<+(?x_1,s(+(?y,?x))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?y,?x))), +(+(?x_1,s(?x)),?y)> => no
<s(+(?y,?x)), +(?y,s(?x))> --> <s(+(?y,?x)), +(?y,s(?x))> => no
check joinability condition:
check modulo joinability of s(+(0,?x_1)) and s(?x_1): joinable by {0}
check modulo reachablity from +(?x,?z_1) to +(?x,+(0,?z_1)): maybe not reachable
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo joinability of s(+(?z_1,+(?y,?x))) and s(+(+(?y,?z_1),?x)): joinable by {1}
check modulo joinability of s(+(+(?y_1,?z_1),?x)) and s(+(?z_1,+(?y_1,?x))): joinable by {1}
check modulo reachablity from +(?x_1,s(+(?y,?x))) to +(+(?x_1,s(?x)),?y): maybe not reachable
check modulo reachablity from s(+(?y,?x)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 1)
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
Added P-Rules:
   [ ]
replace: +(s(?x),?y) -> s(+(?y,?x)) => +(s(?x),?y) -> s(+(?x,?y))
STEP: 2 (linear)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?x_1)), s(?x_1)> --> <s(+(0,?x_1)), s(?x_1)> => no
CP_in(symP,S):
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,+(0,?z_1))> => no
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(0,?x)> --> <?x, +(0,?x)> => no
<+(s(+(?y,?x)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(?z_1,+(?y,?x))), s(+(+(?y,?z_1),?x))> => no
<s(+(+(?y_1,?z_1),?x)), +(+(s(?x),?y_1),?z_1)> --> <s(+(+(?y_1,?z_1),?x)), s(+(?z_1,+(?y_1,?x)))> => no
<+(?x_1,s(+(?y,?x))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?y,?x))), +(+(?x_1,s(?x)),?y)> => no
<s(+(?y,?x)), +(?y,s(?x))> --> <s(+(?y,?x)), +(?y,s(?x))> => no
check joinability condition:
check modulo joinability of s(+(0,?x_1)) and s(?x_1): maybe not joinable
check modulo reachablity from +(?x,?z_1) to +(?x,+(0,?z_1)): maybe not reachable
check modulo reachablity from ?x to +(0,?x): maybe not reachable
check modulo joinability of s(+(?z_1,+(?y,?x))) and s(+(+(?y,?z_1),?x)): joinable by {1}
check modulo joinability of s(+(+(?y_1,?z_1),?x)) and s(+(?z_1,+(?y_1,?x))): joinable by {1}
check modulo reachablity from +(?x_1,s(+(?y,?x))) to +(+(?x_1,s(?x)),?y): maybe not reachable
check modulo reachablity from s(+(?y,?x)) to +(?y,s(?x)): maybe not reachable
failed
failure(Step 2)
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
Added S-Rules:
   [ +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
Added P-Rules:
   [ ]
replace: +(s(?x),?y) -> s(+(?y,?x)) => +(s(?x),?y) -> s(+(?x,?y))
STEP: 3 (relative)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)) ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  0:= (5)
  s:= (1)*x1
retract +(?x,0) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)),
     +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<s(+(0,?x_1)), s(?x_1)> --> <s(?x_1), s(?x_1)> => yes
<0, 0> --> <0, 0> => yes
<s(+(s(?x_1),?x_3)), s(+(s(?x_3),?x_1))> --> <s(s(+(?x_3,?x_1))), s(s(+(?x_1,?x_3)))> => yes
<s(+(0,?x_3)), s(?x_3)> --> <s(?x_3), s(?x_3)> => yes
PCP_in(symP,S):
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(+(?y,?x)),?z_1), +(s(?x),+(?y,?z_1))> --> <s(+(?z_1,+(?y,?x))), s(+(+(?y,?z_1),?x))> => no
<s(+(+(?y_1,?z_1),?x)), +(+(s(?x),?y_1),?z_1)> --> <s(+(+(?y_1,?z_1),?x)), s(+(?z_1,+(?y_1,?x)))> => no
<+(?x_1,s(+(?y,?x))), +(+(?x_1,s(?x)),?y)> --> <s(+(?x_1,+(?y,?x))), s(+(?y,+(?x_1,?x)))> => no
<s(+(?y,?x)), +(?y,s(?x))> --> <s(+(?y,?x)), s(+(?y,?x))> => yes
<+(?x,?z_1), +(0,+(?x,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?x), +(+(?x_1,0),?x)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<s(+(+(?x_1,?y_1),?x)), +(?x_1,+(?y_1,s(?x)))> --> <s(+(+(?x_1,?y_1),?x)), s(+(?x_1,+(?y_1,?x)))> => yes
<+(s(+(?y,?x)),?z_1), +(?y,+(s(?x),?z_1))> --> <s(+(?z_1,+(?y,?x))), s(+(?y,+(?z_1,?x)))> => no
<+(?x_1,s(+(?y,?x))), +(+(?x_1,?y),s(?x))> --> <s(+(?x_1,+(?y,?x))), s(+(+(?x_1,?y),?x))> => yes
<s(+(?y,?x)), +(s(?x),?y)> --> <s(+(?y,?x)), s(+(?y,?x))> => yes
check joinability condition:
check modulo joinability of s(+(?z_1,+(?y,?x))) and s(+(+(?y,?z_1),?x)): joinable by {1}
check modulo joinability of s(+(+(?y_1,?z_1),?x)) and s(+(?z_1,+(?y_1,?x))): joinable by {1}
check modulo joinability of s(+(?x_1,+(?y,?x))) and s(+(?y,+(?x_1,?x))): joinable by {1}
check modulo joinability of s(+(?z_1,+(?y,?x))) and s(+(?y,+(?z_1,?x))): joinable by {1}
success
P':
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Check relative termination:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)),
     +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
   [ +(?x,+(?y,?z)) -> +(+(?x,?y),?z) ]
Polynomial Interpretation:
  +:= (1)*x1+(2)*x1*x2+(1)*x2
  0:= (1)
  s:= (4)+(8)*x1
retract +(?x,0) -> ?x
retract +(0,?x) -> ?x
Polynomial Interpretation:
  +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2
  0:= 0
  s:= (4)+(1)*x1
relatively terminating
S/P': relatively terminating
S:
   [ +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?y,?x)),
     +(0,?x) -> ?x,
     +(?y,s(?x)) -> s(+(?y,?x)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
174.trs: Success(CR)
(3393 msec.)