(ignored inputs)COMMENT Example 8 from [KH12] doi: http://dx.doi.org/10.1007/978-3-642-28717-6_21
Rewrite Rules:
   [ f(?x,?x) -> plus(plus(?x,?x),?x),
     plus(?x,?y) -> plus(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
not Left-Linear, not Right-Linear
unknown Strongly Depth-Preserving & Non-E-Overlapping
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ f(?x,?x) -> plus(plus(?x,?x),?x) ]
P:
   [ plus(?x,?y) -> plus(?y,?x) ]
S: not left-linear
failure(Step 1)
STEP: 2 (linear)
S:
   [ f(?x,?x) -> plus(plus(?x,?x),?x) ]
P:
   [ plus(?x,?y) -> plus(?y,?x) ]
S: not linear
failure(Step 2)
STEP: 3 (relative)
S:
   [ f(?x,?x) -> plus(plus(?x,?x),?x) ]
P:
   [ plus(?x,?y) -> plus(?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...
   [ f(?x,?x) -> plus(plus(?x,?x),?x),
     plus(?x,?y) -> plus(?y,?x) ]
Sort Assignment:
 f : 10*10=>10
 plus : 10*10=>10
maximal types: {10}
Persistent Decomposition failed: Can't judge

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

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


Final result: Can't judge
/local-scratch/hzankl/2012/cops2012/117.trs: Failure(unknown)
(1 msec.)