5.93/2.34	YES
6.24/2.42	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.24/2.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.24/2.42	
6.24/2.42	
6.24/2.42	Termination w.r.t. Q of the given QTRS could be proven:
6.24/2.42	
6.24/2.42	(0) QTRS
6.24/2.42	(1) QTRSRRRProof [EQUIVALENT, 19 ms]
6.24/2.42	(2) QTRS
6.24/2.42	(3) RisEmptyProof [EQUIVALENT, 0 ms]
6.24/2.42	(4) YES
6.24/2.42	
6.24/2.42	
6.24/2.42	----------------------------------------
6.24/2.42	
6.24/2.42	(0)
6.24/2.42	Obligation:
6.24/2.42	Q restricted rewrite system:
6.24/2.42	The TRS R consists of the following rules:
6.24/2.42	
6.24/2.42	   a(a(x1)) -> b(c(x1))
6.24/2.42	   b(b(x1)) -> c(d(x1))
6.24/2.42	   c(c(x1)) -> d(d(d(x1)))
6.24/2.42	   d(d(d(x1))) -> a(c(x1))
6.24/2.42	
6.24/2.42	Q is empty.
6.24/2.42	
6.24/2.42	----------------------------------------
6.24/2.42	
6.24/2.42	(1) QTRSRRRProof (EQUIVALENT)
6.24/2.42	Used ordering:
6.24/2.42	Polynomial interpretation [POLO]:
6.24/2.42	
6.24/2.42	   POL(a(x_1)) = 30 + x_1
6.24/2.42	   POL(b(x_1)) = 27 + x_1
6.24/2.42	   POL(c(x_1)) = 32 + x_1
6.24/2.42	   POL(d(x_1)) = 21 + x_1
6.24/2.42	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.24/2.42	
6.24/2.42	   a(a(x1)) -> b(c(x1))
6.24/2.42	   b(b(x1)) -> c(d(x1))
6.24/2.42	   c(c(x1)) -> d(d(d(x1)))
6.24/2.42	   d(d(d(x1))) -> a(c(x1))
6.24/2.42	
6.24/2.42	
6.24/2.42	
6.24/2.42	
6.24/2.42	----------------------------------------
6.24/2.42	
6.24/2.42	(2)
6.24/2.42	Obligation:
6.24/2.42	Q restricted rewrite system:
6.24/2.42	R is empty.
6.24/2.42	Q is empty.
6.24/2.42	
6.24/2.42	----------------------------------------
6.24/2.42	
6.24/2.42	(3) RisEmptyProof (EQUIVALENT)
6.24/2.42	The TRS R is empty. Hence, termination is trivially proven.
6.24/2.42	----------------------------------------
6.24/2.42	
6.24/2.42	(4)
6.24/2.42	YES
6.44/2.51	EOF