MAYBE
(ignored inputs)COMMENT from the collection of \cite{AT2012}
Rewrite Rules:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ +(?x,?z_1) = +(0,+(?x,?z_1)),
     +(0,?z_1) = +(1,+(-(1),?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),
     0 = +(-(1),1),
     +(?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),
     +(-(1),1) = 0,
     +(1,+(-(1),?z_1)) = +(0,?z_1),
     +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)),
     +(?z,?y) = +(0,+(?y,?z)),
     +(?z,0) = +(1,+(-(1),?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)),
     +(0,?z) = +(1,+(-(1),?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),
     +(0,+(?z,?z_1)) = +(+(1,+(-(1),?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),
     0 = +(-(1),1),
     +(?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, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <+(0,?z_1), +(1,+(-(1),?z_1))> by Rules <1, 2> preceded by [(+,1)]
 joinable by a reduction of rules <[([],3)], [([(+,2)],3),([],3),([],2),([(+,2)],3),([(+,2)],1)]>
 joinable by a reduction of rules <[([],3)], [([],3),([(+,1)],3),([],2),([(+,2)],3),([(+,2)],1)]>
 joinable by a reduction of rules <[([],3)], [([],3),([],2),([],3),([],2),([(+,2)],1)]>
Critical Pair <+(+(?y_2,?x_2),?z_1), +(?x_2,+(?y_2,?z_1))> 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 <+(?y_2,0), ?y_2> by Rules <3, 0> preceded by []
 joinable by a reduction of rules <[([],3),([],0)], []>
Critical Pair <+(-(1),1), 0> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[([],3),([],1)], []>
Critical Pair <+(?y_2,+(?x_1,?y_1)), +(?x_1,+(?y_1,?y_2))> 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_5,0) and ?x_5
strenghten +(0,?x_3) and ?x_3
strenghten +(-(1),1) and 0
strenghten +(1,-(1)) and 0
strenghten +(?x_8,?y_8) and +(?y_8,?x_8)
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 +(-(1),+(1,?z_1)) and +(0,?z_1)
strenghten +(1,+(-(1),?z_1)) and +(0,?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,?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,
     +(1,-(1)) -> 0 ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (13)+(2)*x1*x1
  0:= 0
  1:= 0
retract +(1,-(1)) -> 0
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (8)+(2)*x1*x1
  0:= (8)
  1:= 0
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0 ]
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
<+(0,?z), +(1,+(-(1),?z))> --> <?z, +(1,+(-(1),?z))> => no
<+(?x,0), +(+(?x,1),-(1))> --> <+(?x,0), +(+(?x,1),-(1))> => no
<0, +(-(1),1)> --> <0, +(-(1),1)> => 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
check modulo reachablity from ?z to +(1,+(-(1),?z)): maybe not reachable
check modulo reachablity from +(?x,0) to +(+(?x,1),-(1)): maybe not reachable
check modulo reachablity from 0 to +(-(1),1): maybe not reachable
failed
failure(Step 1)
   [ +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
Added S-Rules:
   [ +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0 ]
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
<+(0,?z), +(1,+(-(1),?z))> --> <?z, +(1,+(-(1),?z))> => no
<+(?x,0), +(+(?x,1),-(1))> --> <+(?x,0), +(+(?x,1),-(1))> => no
<0, +(-(1),1)> --> <0, +(-(1),1)> => 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
check modulo reachablity from ?z to +(1,+(-(1),?z)): maybe not reachable
check modulo reachablity from +(?x,0) to +(+(?x,1),-(1)): maybe not reachable
check modulo reachablity from 0 to +(-(1),1): maybe not reachable
failed
failure(Step 2)
   [ +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
Added S-Rules:
   [ +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0 ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0 ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (13)+(2)*x1*x1
  0:= 0
  1:= 0
retract +(1,-(1)) -> 0
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (8)+(2)*x1*x1
  0:= (8)
  1:= 0
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
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
<+(0,?z), +(1,+(-(1),?z))> --> <?z, +(1,+(-(1),?z))> => no
<+(?x,0), +(+(?x,1),-(1))> --> <?x, +(+(?x,1),-(1))> => no
<0, +(-(1),1)> --> <0, 0> => 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
<+(0,?z), +(-(1),+(1,?z))> --> <?z, +(-(1),+(1,?z))> => no
<+(?x,0), +(+(?x,-(1)),1)> --> <?x, +(+(?x,-(1)),1)> => no
<0, +(1,-(1))> --> <0, 0> => yes
check joinability condition:
check modulo reachablity from ?z to +(1,+(-(1),?z)): maybe not reachable
check modulo reachablity from ?x to +(+(?x,1),-(1)): maybe not reachable
check modulo reachablity from ?z to +(-(1),+(1,?z)): maybe not reachable
check modulo reachablity from ?x to +(+(?x,-(1)),1): maybe not reachable
failed
failure(Step 4)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
STEP: 5 (linear)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
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,?x)> => yes
<?x, +(?x,0)> --> <?x, ?x> => yes
<+(0,?z), +(1,+(-(1),?z))> --> <?z, +(1,+(-(1),?z))> => no
<+(?x,0), +(+(?x,1),-(1))> --> <?x, +(+(?x,1),-(1))> => no
<0, +(-(1),1)> --> <0, 0> => 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
<+(0,?z), +(-(1),+(1,?z))> --> <?z, +(-(1),+(1,?z))> => no
<+(?x,0), +(+(?x,-(1)),1)> --> <?x, +(+(?x,-(1)),1)> => no
<0, +(1,-(1))> --> <0, 0> => yes
check joinability condition:
check modulo reachablity from ?z to +(1,+(-(1),?z)): maybe not reachable
check modulo reachablity from ?x to +(+(?x,1),-(1)): maybe not reachable
check modulo reachablity from ?z to +(-(1),+(1,?z)): maybe not reachable
check modulo reachablity from ?x to +(+(?x,-(1)),1): maybe not reachable
failed
failure(Step 5)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
STEP: 6 (relative)
S:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Check relative termination:
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(?x,0) -> ?x,
     +(-(1),1) -> 0 ]
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (13)+(2)*x1*x1
  0:= 0
  1:= 0
retract +(1,-(1)) -> 0
retract +(-(1),1) -> 0
Polynomial Interpretation:
  +:= (1)*x1+(1)*x2
  -:= (12)+(2)*x1*x1
  0:= (8)
  1:= 0
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 6)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Sort Assignment:
 + : 14*14=>14
 - : 14=>14
 0 : =>14
 1 : =>14
maximal types: {14}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Layer Preserving Decomposition failed: Can't judge

Try Commutative Decomposition for...
   [ +(0,?x) -> ?x,
     +(1,-(1)) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
Outside Critical Pair: <+(?y_2,0), ?y_2> by Rules <3, 0>
develop reducts from lhs term...
<{3}, +(0,?y_2)>
<{}, +(?y_2,0)>
develop reducts from rhs term...
<{}, ?y_2>
Outside Critical Pair: <+(-(1),1), 0> by Rules <3, 1>
develop reducts from lhs term...
<{3}, +(1,-(1))>
<{}, +(-(1),1)>
develop reducts from rhs term...
<{}, 0>
Outside Critical Pair: <+(?y_2,+(?x_1,?y_1)), +(?x_1,+(?y_1,?y_2))> by Rules <3, 2>
develop reducts from lhs term...
<{3}, +(+(?y_1,?x_1),?y_2)>
<{3}, +(+(?x_1,?y_1),?y_2)>
<{3}, +(?y_2,+(?y_1,?x_1))>
<{}, +(?y_2,+(?x_1,?y_1))>
develop reducts from rhs term...
<{3}, +(+(?y_2,?y_1),?x_1)>
<{3}, +(+(?y_1,?y_2),?x_1)>
<{3}, +(?x_1,+(?y_2,?y_1))>
<{}, +(?x_1,+(?y_1,?y_2))>
Inside Critical Pair: <+(?x,?z_1), +(0,+(?x,?z_1))> by Rules <0, 2>
develop reducts from lhs term...
<{3}, +(?z_1,?x)>
<{}, +(?x,?z_1)>
develop reducts from rhs term...
<{3}, +(+(?z_1,?x),0)>
<{3}, +(+(?x,?z_1),0)>
<{0,3}, +(?z_1,?x)>
<{0}, +(?x,?z_1)>
<{3}, +(0,+(?z_1,?x))>
<{}, +(0,+(?x,?z_1))>
Inside Critical Pair: <+(0,?z_1), +(1,+(-(1),?z_1))> by Rules <1, 2>
develop reducts from lhs term...
<{3}, +(?z_1,0)>
<{0}, ?z_1>
<{}, +(0,?z_1)>
develop reducts from rhs term...
<{3}, +(+(?z_1,-(1)),1)>
<{3}, +(+(-(1),?z_1),1)>
<{3}, +(1,+(?z_1,-(1)))>
<{}, +(1,+(-(1),?z_1))>
Inside Critical Pair: <+(+(?y_2,?x_2),?z_1), +(?x_2,+(?y_2,?z_1))> by Rules <3, 2>
develop reducts from lhs term...
<{3}, +(?z_1,+(?x_2,?y_2))>
<{3}, +(?z_1,+(?y_2,?x_2))>
<{2}, +(?y_2,+(?x_2,?z_1))>
<{3}, +(+(?x_2,?y_2),?z_1)>
<{}, +(+(?y_2,?x_2),?z_1)>
develop reducts from rhs term...
<{3}, +(+(?z_1,?y_2),?x_2)>
<{3}, +(+(?y_2,?z_1),?x_2)>
<{3}, +(?x_2,+(?z_1,?y_2))>
<{}, +(?x_2,+(?y_2,?z_1))>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Final result: Can't judge
206.trs: Failure(unknown)
(3119 msec.)