(ignored inputs)COMMENT Commutative Group Theory from \cite{PS81}
Rewrite Rules:
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
Apply Direct Methods...
Inner CPs:
   [ +(0,0) = 0,
     +(?x_4,-(?x_4)) = 0,
     +(+(-(?x_5),-(?y_5)),+(?x_5,?y_5)) = 0,
     +(?x,?z_2) = +(0,+(?x,?z_2)),
     +(0,?z_2) = +(-(?x_1),+(?x_1,?z_2)),
     +(+(?y_3,?x_3),?z_2) = +(?x_3,+(?y_3,?z_2)),
     -(0) = 0,
     -(+(-(?x_5),-(?y_5))) = +(?x_5,?y_5),
     -(?x) = +(-(0),-(?x)),
     -(0) = +(-(-(?x_1)),-(?x_1)),
     -(+(?x_2,+(?y_2,?z_2))) = +(-(+(?x_2,?y_2)),-(?z_2)),
     -(+(?y_3,?x_3)) = +(-(?x_3),-(?y_3)),
     +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)),
     -(?x) = -(?x) ]
Outer CPs:
   [ ?x = +(?x,0),
     0 = +(?x_1,-(?x_1)),
     +(?x_2,+(?y_2,?z_2)) = +(?z_2,+(?x_2,?y_2)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth&Bendix
not Left-Linear, Right-Linear
unknown Simple-Right-Linear
unknown Strongly Depth-Preserving & Non-E-Overlapping
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: not left-linear
failure(Step 1)
STEP: 2 (linear)
S:
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
P:
   [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x) ]
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
Sort Assignment:
 + : 17*17=>17
 - : 17=>17
 0 : =>17
maximal types: {17}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ +(0,?x) -> ?x,
     +(-(?x),?x) -> 0,
     +(+(?x,?y),?z) -> +(?x,+(?y,?z)),
     +(?x,?y) -> +(?y,?x),
     -(0) -> 0,
     -(-(?x)) -> ?x,
     -(+(?x,?y)) -> +(-(?x),-(?y)) ]
Commutative Decomposition failed (not left-linear): Can't judge
No further decomposition possible


Final result: Can't judge
/local-scratch/hzankl/2012/cops2012/76.trs: Failure(unknown)
(41 msec.)