NO
Input:
   [ f(?x) -> g(a,?x),
     f(?x) -> g(?x,a),
     g(?x,?x) -> f(?x) ]

Make it flat:
   [ f(?x) -> g(a,?x),
     f(?x) -> g(?x,a),
     g(?x,?x) -> f(?x) ]
Time: 0.000 [s]

Make it Complete (R^):
   [ g(a,?x) = g(?x,?x),
     g(a,?x) = g(?x,a),
     f(?x) = g(a,?x),
     g(a,a) = f(a),
     g(?x,a) = g(?x,?x),
     f(?x) = g(?x,a),
     g(?x,?x) = f(?x) ]
Time: 0.003 [s]

The number of normal forms that must be checked: 9
Time: 0.000 [s]

Now checking all the pairs...

Time to check pairs: 0.000 [s]

The TRS doesn't have Uniqueness of Normal Forms.
Counter Example: 
     g(a,?!cx)
<->* g(?!cx,a)

proof:
     g(a,?!cx)
->R^ f(?!cx)

     g(?!cx,a)
->R^ f(?!cx)
Total Time: 0.003 [s]

problems/r11.trs: Success(not UNC)
real 0.32
user 0.02
sys 0.00