YES
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(0,?x) -> ?x,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_1) = +(0,+(?x,?z_1)),
     +(+(?y_2,?x_2),?z_1) = +(?x_2,+(?y_2,?z_1)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)) ]
Outer CPs:
   [ ?x = +(?x,0),
     +(?x_1,+(?y_1,?z_1)) = +(?z_1,+(?x_1,?y_1)) ]
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:
   [ +(?x,0) = ?x,
     +(0,+(?x,?z_2)) = +(?x,?z_2),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?y) = +(0,+(?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)),
     +(+(?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),
     +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1),
     ?y = +(?y,0),
     +(?x_2,+(?y_2,?y)) = +(?y,+(?x_2,?y_2)),
     +(?x,+(?y,?z_3)) = +(+(?y,?x),?z_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 <+(?x,?z_1), +(0,+(?x,?z_1))> by Rules <0, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <+(+(?y_2,?x_2),?z_1), +(?x_2,+(?y_2,?z_1))> by Rules <2, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([(+,1)],2),([],1)], []>
 joinable by a reduction of rules <[([],1),([(+,2)],2)], [([],2),([],1)]>
Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <1, 1> preceded by [(+,1)]
 joinable by a reduction of rules <[([],1),([(+,2)],1)], [([],1)]>
Critical Pair <+(?y_2,0), ?y_2> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <+(?y_2,+(?x_1,?y_1)), +(?x_1,+(?y_1,?y_2))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
strenghten +(?x_5,0) and ?x_5
strenghten +(0,?x_3) and ?x_3
strenghten +(0,0) and 0
strenghten +(?x_8,?y_8) and +(?y_8,?x_8)
strenghten +(?x_11,+(0,0)) and ?x_11
strenghten +(+(0,0),?x_11) and ?x_11
strenghten +(?x_1,+(?y_1,0)) and +(?x_1,?y_1)
strenghten +(?x_3,+(0,?z_1)) and +(?x_3,?z_1)
strenghten +(0,+(?x,?z_1)) and +(?x,?z_1)
strenghten +(?x_1,+(?y_1,?x_9)) and +(?x_9,+(?x_1,?y_1))
strenghten +(?x_1,+(?y_1,?x_10)) and +(+(?x_1,?y_1),?x_10)
strenghten +(?x_2,+(?y_2,?z_1)) and +(+(?y_2,?x_2),?z_1)
strenghten +(?x_1,+(?y_1,+(0,0))) and +(?x_1,?y_1)
strenghten +(?x_11,+(+(0,0),?z_1)) and +(?x_11,?z_1)
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,?x) -> ?x ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x2
  0:= (7)
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?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):
<+(?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_1,0),?x)> => no
<?x, +(?x,0)> --> <?x, +(?x,0)> => no
check joinability condition:
check modulo reachablity from +(?x_1,?x) to +(+(?x_1,0),?x): maybe not reachable
check modulo reachablity from ?x to +(?x,0): maybe not reachable
failed
failure(Step 1)
   [ +(?x,0) -> ?x ]
Added S-Rules:
   [ +(?x,0) -> ?x ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ +(0,?x) -> ?x ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
CP(S,symP):
<+(?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_1,0),?x)> => no
<?x, +(?x,0)> --> <?x, +(?x,0)> => no
check joinability condition:
check modulo reachablity from +(?x_1,?x) to +(+(?x_1,0),?x): maybe not reachable
check modulo reachablity from ?x to +(?x,0): maybe not reachable
failed
failure(Step 2)
   [ +(?x,0) -> ?x ]
Added S-Rules:
   [ +(?x,0) -> ?x ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ +(0,?x) -> ?x ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?x) -> ?x ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)+(1)*x1+(1)*x2
  0:= (7)
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?x) -> ?x,
     +(?x,0) -> ?x ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
PCP_in(symP,S):
CP(S,symP):
<+(?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
<+(?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:
   [ +(0,?x) -> ?x,
     +(?x,0) -> ?x ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Final result: CR
199.trs: Success(CR)
(2594 msec.)