4.03/1.78	YES
4.03/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.03/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.03/1.79	
4.03/1.79	
4.03/1.79	Termination w.r.t. Q of the given QTRS could be proven:
4.03/1.79	
4.03/1.79	(0) QTRS
4.03/1.79	(1) QTRSRRRProof [EQUIVALENT, 28 ms]
4.03/1.79	(2) QTRS
4.03/1.79	(3) RisEmptyProof [EQUIVALENT, 0 ms]
4.03/1.79	(4) YES
4.03/1.79	
4.03/1.79	
4.03/1.79	----------------------------------------
4.03/1.79	
4.03/1.79	(0)
4.03/1.79	Obligation:
4.03/1.79	Q restricted rewrite system:
4.03/1.79	The TRS R consists of the following rules:
4.03/1.79	
4.03/1.79	   a(a(a(x1))) -> b(b(x1))
4.03/1.79	   b(b(b(x1))) -> c(x1)
4.03/1.79	   c(x1) -> d(d(x1))
4.03/1.79	   d(x1) -> a(a(x1))
4.03/1.79	
4.03/1.79	Q is empty.
4.03/1.79	
4.03/1.79	----------------------------------------
4.03/1.79	
4.03/1.79	(1) QTRSRRRProof (EQUIVALENT)
4.03/1.79	Used ordering:
4.03/1.79	Polynomial interpretation [POLO]:
4.03/1.79	
4.03/1.79	   POL(a(x_1)) = 11 + x_1
4.03/1.79	   POL(b(x_1)) = 16 + x_1
4.03/1.79	   POL(c(x_1)) = 47 + x_1
4.03/1.79	   POL(d(x_1)) = 23 + x_1
4.03/1.79	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.03/1.79	
4.03/1.79	   a(a(a(x1))) -> b(b(x1))
4.03/1.79	   b(b(b(x1))) -> c(x1)
4.03/1.79	   c(x1) -> d(d(x1))
4.03/1.79	   d(x1) -> a(a(x1))
4.03/1.79	
4.03/1.79	
4.03/1.79	
4.03/1.79	
4.03/1.79	----------------------------------------
4.03/1.79	
4.03/1.79	(2)
4.03/1.79	Obligation:
4.03/1.79	Q restricted rewrite system:
4.03/1.79	R is empty.
4.03/1.79	Q is empty.
4.03/1.79	
4.03/1.79	----------------------------------------
4.03/1.79	
4.03/1.79	(3) RisEmptyProof (EQUIVALENT)
4.03/1.79	The TRS R is empty. Hence, termination is trivially proven.
4.03/1.79	----------------------------------------
4.03/1.79	
4.03/1.79	(4)
4.03/1.79	YES
4.18/1.83	EOF