(ignored inputs)COMMENT Example 10 from [KH12] doi: http://dx.doi.org/10.1007/978-3-642-28717-6_21
Rewrite Rules:
   [ eq(?n,?xs,?xs) -> T,
     eq(s(?n),c(?x,?xs),c(?x,?ys)) -> eq(?n,?xs,?ys),
     nats -> c(0,inc(nats)),
     inc(c(?x,?xs)) -> c(s(?x),inc(?xs)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ T = eq(?n_1,?xs_1,?xs_1) ]
Overlay, check Innermost Termination...
unknown Innermost 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...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ eq(?n,?xs,?xs) -> T,
     eq(s(?n),c(?x,?xs),c(?x,?ys)) -> eq(?n,?xs,?ys),
     nats -> c(0,inc(nats)),
     inc(c(?x,?xs)) -> c(s(?x),inc(?xs)) ]
Sort Assignment:
 0 : =>11
 T : =>25
 c : 11*23=>23
 s : 11=>11
 eq : 11*23*23=>25
 inc : 23=>23
 nats : =>23
maximal types: {11,23,25}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ eq(?n,?xs,?xs) -> T,
     eq(s(?n),c(?x,?xs),c(?x,?ys)) -> eq(?n,?xs,?ys),
     nats -> c(0,inc(nats)),
     inc(c(?x,?xs)) -> c(s(?x),inc(?xs)) ]
Layer Preserving Decomposition failed: Can't judge

Try Commutative Decomposition for...
   [ eq(?n,?xs,?xs) -> T,
     eq(s(?n),c(?x,?xs),c(?x,?ys)) -> eq(?n,?xs,?ys),
     nats -> c(0,inc(nats)),
     inc(c(?x,?xs)) -> c(s(?x),inc(?xs)) ]
Commutative Decomposition failed (not left-linear): Can't judge
No further decomposition possible


Final result: Can't judge
/local-scratch/hzankl/2012/cops2012/118.trs: Failure(unknown)
(6 msec.)