MAYBE
af/ONE-SORTED/213.trs
Input rules:
   [ a -> b,
     a -> f(a),
     b -> f(f(b)),
     f(f(f(b))) -> b ]
Sorts having no ground terms: 
Rules applicable to ground terms:
   [ a -> b,
     a -> f(a),
     b -> f(f(b)),
     f(f(f(b))) -> b ]
Constructor pattern: []
Defined pattern: [f(?x_1),b,a]
Constructor subsystem:
   [ ]
Modified Constructor subsystem:
   [ ]
No orientable rules for f(?x_1). Add rules, and retry...
failed to construct defining rules
Retry with a different D/C-partition.
Constructor pattern: [f(?x_1)]
Defined pattern: [b,a]
Constructor subsystem:
   [ ]
Modified Constructor subsystem:
   [ ]
No orientable rules for b. Add rules, and retry...
failed to construct defining rules
Retry with a different D/C-partition.
Constructor pattern: [f(?x_1),b]
Defined pattern: [a]
Constructor subsystem:
   [ b -> f(f(b)),
     f(f(f(b))) -> b ]
Modified Constructor subsystem:
   [ b -> f(f(b)),
     f(f(f(b))) -> b ]
candidate for a:
   [ a -> b ]
Find a quasi-ordering ...
order successfully found
Precedence:
 a : Mul;
 b : Mul;
 f : Mul;
Rules:
   [ a -> b,
     f(f(f(b))) -> b ]
Check confluence of constructor subsystem...
Check Termination...
Terminating, WCR: CR
Conjectures:
   [ a = f(a),
     b = f(f(b)) ]
STEP 0
ES:
   [ a = f(a),
     b = f(f(b)) ]
HS:
   [ ]
ES0:
   [ b = f(b),
     b = f(f(b)) ]
HS0:
   [ ]
ES1:
   [ b = f(b),
     b = f(f(b)) ]
HS1:
   [ ]
No equation to expand
check Non-Ground-Confluence...
ground constructor terms for instantiation: {}
ground terms for instantiation: {a:o}
obtain 11 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly) (failure)
: Failure(unknown)
(1385 msec.)
5.83