NO
(ignored inputs)COMMENT generated by FORT -f "a:0 b:0 f:2" -r 5 "UN & ~UNC & ~SN" submitted by: Franziska Rapp
Input:
   [ a -> a,
     a -> b,
     a -> f(a,a),
     f(b,b) -> f(a,a),
     f(?x0,a) -> f(a,b) ]

Make it flat:
   [ a -> a,
     a -> b,
     a -> f(a,a),
     f(b,b) -> f(a,a),
     f(?x0,a) -> f(a,b) ]
Time: 0.000 [s]

Make it Complete (R^):
   [ f(b,a) = f(a,a),
     f(b,a) = f(?x0_1,a),
     f(b,a) = f(a,b),
     f(b,a) = a,
     f(?x0,b) = f(a,b),
     f(?x0_1,b) = f(?x0,a),
     f(b,a) = f(b,b),
     f(?x0_1,b) = f(b,b),
     f(?x0_1,b) = f(a,a),
     b = f(a,a),
     f(b,a) = f(?x0_2,b),
     f(b,a) = b,
     b = f(b,b),
     f(?x0_1,b) = a,
     f(?x0_1,b) = f(?x0_3,b),
     f(?x0_1,b) = b,
     b = f(?x0_1,a),
     b = f(a,b),
     a = b,
     a = f(a,b),
     a = f(?x0_1,a),
     a = f(b,b),
     a = f(a,a),
     f(b,b) = f(a,b),
     f(a,a) = f(a,b),
     f(?x0_1,a) = f(a,a),
     f(b,b) = f(?x0_1,a),
     f(b,b) = f(a,a),
     f(?x0,a) = f(?x0_1,a),
     f(?x0,a) = f(a,b) ]
Time: 0.022 [s]

The number of normal forms that must be checked: 845
Time: 0.158 [s]

Now checking all the pairs...

Time to check pairs: 0.000 [s]

The TRS doesn't have Uniqueness of Normal Forms.
Counter Example: 
     f(f(?!cx_2,?!cx_3),b)
<->* f(f(?!cx_3,?!cx_3),b)

proof:
     f(f(?!cx_2,?!cx_3),b)
->R^ f(f(?!cx_2,?!cx_3),a)
->R^ f(a,b)
->R^ f(a,a)
->R^ f(?!,a)
->R^ a

     f(f(?!cx_3,?!cx_3),b)
->R^ f(f(?!cx_3,?!cx_3),a)
->R^ f(a,b)
->R^ f(a,a)
->R^ f(?!,a)
->R^ a
Total Time: 0.181 [s]

problems/732.trs: Success(not UNC)
real 0.20
user 0.20
sys 0.00