(ignored inputs)COMMENT Example 1 of \cite{Toy98}
Rewrite Rules:
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Apply Direct Methods...
Inner CPs:
   [ f(?x,g(?x)) = g(f(?x,?x)) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth&Bendix
not Left-Linear, not 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...
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Sort Assignment:
 a : 12=>12
 f : 12*12=>12
 g : 12=>12
 h : 12=>12
maximal types: {12}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
(lp)   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x ]
Rewrite Rules:
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [ f(?x,g(?x)) = g(f(?x,?x)) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
Terminating, WCR
Knuth&Bendix
Direct Methods: CR
(lp)   [ a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Rewrite Rules:
   [ a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Linear
Development Closed
Direct Methods: CR
Result by Layer Preserving Decomposition: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/89.trs: Success(CR)
(7 msec.)