NO
Input:
   [ a -> b,
     a -> c,
     c -> c,
     d -> c,
     d -> e ]

Make it flat:
   [ a -> b,
     a -> c,
     c -> c,
     d -> c,
     d -> e ]
Time: 0.000 [s]

Make it Complete (R^):
   [ b = c,
     b = e,
     b = d,
     a = b,
     a = d,
     a = e,
     a = c,
     c = e,
     d = c,
     d = e ]
Time: 0.001 [s]

CPNF:
   [ a … b,
     a … e ]
Time: 0.000 [s]

The TRS doesn't have Uniqueness of Normal Forms.
Counter Example: 
     b
<->* e

proof:
     b
->R^ a

     e
->R^ d
->R^ c
->R^ a
Total Time: 0.001 [s]

problems/r1.trs: Success(not UNC)
real 0.04
user 0.01
sys 0.01