(ignored inputs)COMMENT Example 1 of cite{AYT09}
Rewrite Rules:
   [ f(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)),
     g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
Apply Direct Methods...
Inner CPs:
   [ f(b(?x_4),?x_4) = f(?x_4,a(?x_4)) ]
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...
STEP: 1 (parallel)
S:
   [ g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
P:
   [ f(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)) ]
P: not reversible
failure(Step 1)
STEP: 2 (linear)
S:
   [ g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
P:
   [ f(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)) ]
P: not reversible
failure(Step 2)
STEP: 3 (relative)
S:
   [ g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
P:
   [ f(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)) ]
P: not reversible
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(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)),
     g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
Sort Assignment:
 a : 20=>20
 b : 20=>20
 c : 12=>20
 f : 20*20=>23
 g : 20*21=>21
maximal types: {12,20,21}{12,20,23}
{12,20,21}
(ps)Rewrite Rules:
   [ g(b(?x),?y) -> g(a(a(?x)),?y),
     g(c(?x),?y) -> ?y,
     a(?x) -> b(?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Linear
Development Closed
Direct Methods: CR
{12,20,23}
(ps)Rewrite Rules:
   [ f(a(?x),?x) -> f(?x,a(?x)),
     f(b(?x),?x) -> f(?x,b(?x)),
     a(?x) -> b(?x) ]
Apply Direct Methods...
Inner CPs:
   [ f(b(?x_2),?x_2) = f(?x_2,a(?x_2)) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
Terminating, WCR
Knuth&Bendix
Direct Methods: CR
Result by Persistent Decomposition: CR

Final result: CR
/local-scratch/hzankl/2012/cops2012/3.trs: Success(CR)
(45 msec.)