YES
af/ONE-SORTED/506.trs
Input rules:
   [ f(f(f(?x:o))) -> a,
     f(f(a)) -> a,
     f(a) -> a,
     f(f(g(g(?x:o)))) -> f(a),
     g(f(a)) -> a,
     g(a) -> a ]
Sorts having no ground terms: 
Rules applicable to ground terms:
   [ f(f(f(?x:o))) -> a,
     f(f(a)) -> a,
     f(a) -> a,
     f(f(g(g(?x:o)))) -> f(a),
     g(f(a)) -> a,
     g(a) -> a ]
Constructor pattern: [a]
Defined pattern: [g(?x_1),f(?x_1)]
Constructor subsystem:
   [ ]
Modified Constructor subsystem:
   [ ]
candidate for g(?x_1):
   [ g(a) -> a ]
candidate for f(?x_1):
   [ f(a) -> a ]
Find a quasi-ordering ...
order successfully found
Precedence:
 a : Mul;
 g : Mul,  f : Mul;
Rules:
   [ g(a) -> a,
     f(a) -> a ]
Conjectures:
   [ f(f(f(?x))) = a,
     f(f(a)) = a,
     f(f(g(g(?x)))) = f(a),
     g(f(a)) = a ]
STEP 0
ES:
   [ f(f(f(?x))) = a,
     f(f(a)) = a,
     f(f(g(g(?x)))) = f(a),
     g(f(a)) = a ]
HS:
   [ ]
ES0:
   [ f(f(f(?x))) = a,
     a = a,
     f(f(g(g(?x)))) = a,
     a = a ]
HS0:
   [ ]
ES1:
   [ f(f(f(?x))) = a,
     f(f(g(g(?x)))) = a ]
HS1:
   [ ]
Expand f(f(f(?x))) = a
   [ f(f(a)) = a ]
ES2:
   [ a = a,
     f(f(g(g(?x)))) = a ]
HS2:
   [ f(f(f(?x))) -> a ]
STEP 1
ES:
   [ a = a,
     f(f(g(g(?x)))) = a ]
HS:
   [ f(f(f(?x))) -> a ]
ES0:
   [ a = a,
     f(f(g(g(?x)))) = a ]
HS0:
   [ f(f(f(?x))) -> a ]
ES1:
   [ f(f(g(g(?x)))) = a ]
HS1:
   [ f(f(f(?x))) -> a ]
Expand f(f(g(g(?x)))) = a
   [ f(f(g(a))) = a ]
ES2:
   [ a = a ]
HS2:
   [ f(f(g(g(?x)))) -> a,
     f(f(f(?x))) -> a ]
STEP 2
ES:
   [ a = a ]
HS:
   [ f(f(g(g(?x)))) -> a,
     f(f(f(?x))) -> a ]
ES0:
   [ a = a ]
HS0:
   [ f(f(g(g(?x)))) -> a,
     f(f(f(?x))) -> a ]
ES1:
   [ ]
HS1:
   [ f(f(g(g(?x)))) -> a,
     f(f(f(?x))) -> a ]
: Success(GCR)
(3 msec.)
0.01