0.00/0.21	YES
0.00/0.21	(ignored inputs)COMMENT Cops #647 - #721: generated ground TRSs; evenly distributed in the UNR/UNC/NFP/CR hierarchy submitted by: Bertram Felgenhauer
0.00/0.21	input TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	unknown Strongly Non-Overlapping
0.00/0.21	unknown Right-Reducible
0.00/0.21	Call a UNC decision procedure for shallow TRSs...
0.00/0.21	Input:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	
0.00/0.21	Make it flat:
0.00/0.21	   [ f(!cl_3) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(!cl_2,a) -> c,
0.00/0.21	     !cl_3 -> !cl_3,
0.00/0.21	     h(b,!cl_1) -> !cl_3,
0.00/0.21	     f(a) -> !cl_1,
0.00/0.21	     h(a,a) -> a,
0.00/0.21	     f(b) -> !cl_0,
0.00/0.21	     !cl_0 -> !cl_0,
0.00/0.21	     b -> a,
0.00/0.21	     f(!cl_0) -> !cl_2,
0.00/0.21	     !cl_2 -> !cl_2 ]
0.00/0.21	Time: 0.000 [s]
0.00/0.21	
0.00/0.21	Make it Complete (R^):
0.00/0.21	   [ f(!cl_3) = f(c),
0.00/0.21	     h(c,c) = h(!cl_2,a),
0.00/0.21	     h(c,c) = h(!cl_2,b),
0.00/0.21	     c = h(c,c),
0.00/0.21	     h(!cl_2,b) = h(!cl_2,a),
0.00/0.21	     h(!cl_2,b) = c,
0.00/0.21	     h(!cl_2,a) = c,
0.00/0.21	     h(a,!cl_1) = !cl_3,
0.00/0.21	     h(a,!cl_0) = !cl_3,
0.00/0.21	     h(a,!cl_1) = h(b,!cl_0),
0.00/0.21	     h(a,!cl_1) = h(a,!cl_0),
0.00/0.21	     h(b,!cl_0) = h(a,!cl_0),
0.00/0.21	     h(a,!cl_1) = h(b,!cl_1),
0.00/0.21	     h(a,!cl_0) = h(b,!cl_1),
0.00/0.21	     h(b,!cl_0) = h(b,!cl_1),
0.00/0.21	     h(b,!cl_0) = !cl_3,
0.00/0.21	     h(b,!cl_1) = !cl_3,
0.00/0.21	     f(b) = !cl_1,
0.00/0.21	     f(!cl_1) = f(!cl_0),
0.00/0.21	     f(!cl_1) = !cl_2,
0.00/0.21	     !cl_1 = !cl_0,
0.00/0.21	     f(a) = !cl_1,
0.00/0.21	     b = h(a,a),
0.00/0.21	     h(a,b) = a,
0.00/0.21	     b = h(b,a),
0.00/0.21	     h(a,b) = b,
0.00/0.21	     b = h(b,b),
0.00/0.21	     h(b,b) = a,
0.00/0.21	     h(a,b) = h(b,a),
0.00/0.21	     h(a,b) = h(b,b),
0.00/0.21	     h(b,a) = h(b,b),
0.00/0.21	     h(a,b) = h(a,a),
0.00/0.21	     h(b,b) = h(a,a),
0.00/0.21	     h(b,a) = h(a,a),
0.00/0.21	     h(b,a) = a,
0.00/0.21	     h(a,a) = a,
0.00/0.21	     f(a) = f(b),
0.00/0.21	     f(a) = !cl_0,
0.00/0.21	     f(b) = !cl_0,
0.00/0.21	     b = a,
0.00/0.21	     f(!cl_0) = !cl_2 ]
0.00/0.21	Time: 0.010 [s]
0.00/0.21	
0.00/0.21	CPNF:
0.00/0.21	   [ c … h(h(f(!cl_1),a),h(f(!cl_1),a)),
0.00/0.21	     c … h(f(!cl_1),a),
0.00/0.21	     !cl_3 … h(a,!cl_1),
0.00/0.21	     !cl_2 … f(!cl_1),
0.00/0.21	     !cl_0 … !cl_1,
0.00/0.21	     a … a ]
0.00/0.21	Time: 0.000 [s]
0.00/0.21	
0.00/0.21	The TRS doesn't have Uniqueness of Normal Forms.
0.00/0.21	Counter Example: 
0.00/0.21	     h(h(f(!cl_1),a),h(f(!cl_1),a))
0.00/0.21	<->* h(f(!cl_1),a)
0.00/0.21	
0.00/0.21	proof:
0.00/0.21	     h(h(f(!cl_1),a),h(f(!cl_1),a))
0.00/0.21	->R^ h(h(f(!cl_0),a),h(f(!cl_1),a))
0.00/0.21	->R^ h(h(f(!cl_0),a),h(f(!cl_0),a))
0.00/0.21	->R^ h(h(!cl_2,a),h(f(!cl_0),a))
0.00/0.21	->R^ h(h(!cl_2,a),h(!cl_2,a))
0.00/0.21	->R^ h(c,h(!cl_2,a))
0.00/0.21	->R^ h(c,c)
0.00/0.21	->R^ c
0.00/0.21	
0.00/0.21	     h(f(!cl_1),a)
0.00/0.21	->R^ h(f(!cl_0),a)
0.00/0.21	->R^ h(!cl_2,a)
0.00/0.21	->R^ c
0.00/0.21	Total Time: 0.012 [s]
0.00/0.21	
0.00/0.21	...Not UNC found by the decision procedure.
0.00/0.21	Check distinct normal forms in critical pair closure...failed
0.00/0.21	UNR Proof by the Knuth-Bendix criterion
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	not Overlay, check Termination...
0.00/0.21	unknown/not Terminating...failed
0.00/0.21	
0.00/0.21	UNR proof by Transformation (Strongly Closed)
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	with auxiliary rules:
0.00/0.21	   [ ]
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	with auxiliary rules:
0.00/0.21	   [ c -> h(f(f(a)),a),
0.00/0.21	     f(c) -> f(h(a,f(a))) ]
0.00/0.21	...failed
0.00/0.21	UNR proof by Transformation (Parallel Closed)
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	with auxiliary rules:
0.00/0.21	   [ ]
0.00/0.21	CPin(R):
0.00/0.21	   [ f(h(a,f(a))) = f(c),
0.00/0.21	     h(f(f(a)),a) = c ]
0.00/0.21	CPout(R):
0.00/0.21	   [ ]
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(c),
0.00/0.21	     c -> h(c,c),
0.00/0.21	     h(f(f(b)),a) -> c,
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	with auxiliary rules:
0.00/0.21	   [ c -> h(f(f(a)),a),
0.00/0.21	     f(c) -> f(h(a,f(a))) ]
0.00/0.21	CPin(R):
0.00/0.21	   [ f(h(a,f(a))) = f(c),
0.00/0.21	     h(f(f(a)),a) = c ]
0.00/0.21	CPout(R):
0.00/0.21	   [ ]
0.00/0.21	...failed
0.00/0.21	UNR proof by Collapse Mapping
0.00/0.21	Modify rules by collapse mapping
0.00/0.21	checking TRS:
0.00/0.21	   [ b -> a,
0.00/0.21	     f(h(b,f(a))) -> f(h(b,f(a))),
0.00/0.21	     h(f(f(b)),a) -> h(f(f(b)),a),
0.00/0.21	     h(a,a) -> a ]
0.00/0.21	with auxiliary rules:
0.00/0.21	   [ ]
0.00/0.21	CP:
0.00/0.21	   [ f(h(a,f(a))) = f(h(b,f(a))),
0.00/0.21	     h(f(f(a)),a) = h(f(f(b)),a) ]
0.00/0.21	
0.00/0.21	(approximated) TRS is linear, strongly closed; hence confluent
0.00/0.21	/export/starexec/sandbox/benchmark/theBenchmark.trs: Success(UNR)
0.00/0.21	(174 msec.)
0.00/0.21	EOF