YES
(ignored inputs)COMMENT cops number: 669 submitted by: Bertram Felgenhauer Cops #647 - #721: generated ground TRSs; evenly distributed in the UNR/UNC/NFP/CR hierarchy
input TRS:
   [ a : 0,
     b : 0,
     c : 0,
     f : 1,
     h : 2 ]
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
unknown Strongly Non-Overlapping
unknown Right-Reducible
Call a UNC decision procedure for shallow TRSs...
Input:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]

Make it flat:
   [ a -> c,
     f(a) -> b,
     b -> b,
     !cr_0 -> h(c,a),
     b -> h(b,!cr_0) ]
Time: 0.000 [s]

Make it Complete (R^):
   [ h(a,a) = !cr_0,
     h(a,c) = !cr_0,
     h(a,a) = h(c,c),
     h(a,a) = h(a,c),
     h(c,c) = h(a,c),
     h(a,a) = h(c,a),
     h(a,c) = h(c,a),
     h(c,c) = h(c,a),
     h(c,c) = !cr_0,
     f(c) = b,
     f(c) = f(a),
     f(c) = h(b,!cr_0),
     a = c,
     f(a) = h(b,!cr_0),
     f(a) = b,
     !cr_0 = h(c,a),
     b = h(b,!cr_0) ]
Time: 0.003 [s]

CPNF:
   [ b … h(f(c),h(c,c)),
     !cr_0 … h(c,c),
     b … f(c),
     a … c ]
Time: 0.000 [s]

The TRS doesn't have Uniqueness of Normal Forms.
Counter Example: 
     h(f(c),h(c,c))
<->* f(c)

proof:
     h(f(c),h(c,c))
->R^ h(f(a),h(c,c))
->R^ h(f(a),h(a,c))
->R^ h(f(a),h(a,a))
->R^ h(b,h(a,a))
->R^ h(b,!cr_0)
->R^ b

     f(c)
->R^ f(a)
->R^ b
Total Time: 0.003 [s]

...Not UNC found by the decision procedure.
Check distinct normal forms in critical pair closure...failed
UNR Proof by the Knuth-Bendix criterion
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
containing trivial rules..remove trivial rules and forget nonUNR
not Overlay, check Termination...
unknown/not Terminating...failed

UNR proof by Transformation (Strongly Closed)
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c),
     h(h(f(c),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,c)) -> f(c),
     h(h(b,h(c,c)),h(c,a)) -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c),
     h(h(f(c),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,c)) -> f(c),
     h(h(b,h(c,c)),h(c,a)) -> f(c),
     h(h(b,h(c,c)),h(c,c)) -> f(c),
     h(h(h(f(c),h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(h(h(b,h(c,a)),h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,c)) -> f(c),
     h(h(h(b,h(c,a)),h(c,c)),h(c,a)) -> f(c),
     h(h(h(b,h(c,c)),h(c,a)),h(c,a)) -> f(c),
     h(h(f(c),h(c,a)),h(c,c)) -> f(c),
     h(h(f(c),h(c,c)),h(c,a)) -> f(c) ]
Trivial rules are removed for checking CR
...failed
UNR proof by Transformation (Parallel Closed)
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     f(c) -> h(b,h(c,a)),
     f(c) -> b ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     f(c) -> h(b,h(c,a)),
     f(c) -> b,
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     f(c) -> h(b,h(c,a)),
     f(c) -> b,
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c),
     h(h(f(c),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,c)) -> f(c),
     h(h(b,h(c,c)),h(c,a)) -> f(c) ]
Trivial rules are removed for checking CR
checking TRS:
   [ a -> c,
     f(a) -> b,
     b -> b,
     b -> h(b,h(c,a)) ]
with auxiliary rules:
   [ b -> f(c),
     h(b,h(c,a)) -> f(c),
     f(c) -> h(b,h(c,a)),
     f(c) -> b,
     h(f(c),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,a)) -> f(c),
     h(b,h(c,c)) -> f(c),
     h(h(f(c),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(b,h(c,a)),h(c,c)) -> f(c),
     h(h(b,h(c,c)),h(c,a)) -> f(c),
     h(h(b,h(c,c)),h(c,c)) -> f(c),
     h(h(h(f(c),h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(h(h(b,h(c,a)),h(c,a)),h(c,a)),h(c,a)) -> f(c),
     h(h(h(b,h(c,a)),h(c,a)),h(c,c)) -> f(c),
     h(h(h(b,h(c,a)),h(c,c)),h(c,a)) -> f(c),
     h(h(h(b,h(c,c)),h(c,a)),h(c,a)) -> f(c),
     h(h(f(c),h(c,a)),h(c,c)) -> f(c),
     h(h(f(c),h(c,c)),h(c,a)) -> f(c) ]
Trivial rules are removed for checking CR
...failed
UNR proof by Collapse Mapping
Modify rules by collapse mapping
input TRS:
   [ a : 0,
     c : 0,
     f : 1 ]
   [ a -> c,
     f(a) -> f(a) ]
unknown Strongly Non-Overlapping
unknown Right-Reducible
Call a UNC decision procedure for shallow TRSs...
Input:
   [ a -> c,
     f(a) -> f(a) ]

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

Make it Complete (R^):
   [ a = c ]
Time: 0.000 [s]

CPNF:
   [ a … c ]
Time: 0.000 [s]

Now checking all the pairs in CW...

Time to check pairs: 0.000 [s]

The TRS has Uniqueness of Normal Forms.
Total Time: 0.000 [s]

...UNC found by the decision procedure (hence UNR).
cops-FXbH9eBu.trs: Success(UNR)
(69 msec.)