5.48/2.29	YES
5.79/2.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.79/2.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.79/2.31	
5.79/2.31	
5.79/2.31	Termination w.r.t. Q of the given QTRS could be proven:
5.79/2.31	
5.79/2.31	(0) QTRS
5.79/2.31	(1) QTRSRRRProof [EQUIVALENT, 45 ms]
5.79/2.31	(2) QTRS
5.79/2.31	(3) QTRSRRRProof [EQUIVALENT, 2 ms]
5.79/2.31	(4) QTRS
5.79/2.31	(5) RisEmptyProof [EQUIVALENT, 0 ms]
5.79/2.31	(6) YES
5.79/2.31	
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(0)
5.79/2.31	Obligation:
5.79/2.31	Q restricted rewrite system:
5.79/2.31	The TRS R consists of the following rules:
5.79/2.31	
5.79/2.31	   a(a(a(a(x1)))) -> b(b(b(b(b(b(x1))))))
5.79/2.31	   b(b(b(b(x1)))) -> c(c(c(c(c(c(x1))))))
5.79/2.31	   c(c(c(c(x1)))) -> d(d(d(d(d(d(x1))))))
5.79/2.31	   b(b(x1)) -> d(d(d(d(x1))))
5.79/2.31	   c(c(d(d(d(d(x1)))))) -> a(a(x1))
5.79/2.31	
5.79/2.31	Q is empty.
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(1) QTRSRRRProof (EQUIVALENT)
5.79/2.31	Used ordering:
5.79/2.31	Polynomial interpretation [POLO]:
5.79/2.31	
5.79/2.31	   POL(a(x_1)) = 66 + x_1
5.79/2.31	   POL(b(x_1)) = 44 + x_1
5.79/2.31	   POL(c(x_1)) = 29 + x_1
5.79/2.31	   POL(d(x_1)) = 19 + x_1
5.79/2.31	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.79/2.31	
5.79/2.31	   b(b(b(b(x1)))) -> c(c(c(c(c(c(x1))))))
5.79/2.31	   c(c(c(c(x1)))) -> d(d(d(d(d(d(x1))))))
5.79/2.31	   b(b(x1)) -> d(d(d(d(x1))))
5.79/2.31	   c(c(d(d(d(d(x1)))))) -> a(a(x1))
5.79/2.31	
5.79/2.31	
5.79/2.31	
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(2)
5.79/2.31	Obligation:
5.79/2.31	Q restricted rewrite system:
5.79/2.31	The TRS R consists of the following rules:
5.79/2.31	
5.79/2.31	   a(a(a(a(x1)))) -> b(b(b(b(b(b(x1))))))
5.79/2.31	
5.79/2.31	Q is empty.
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(3) QTRSRRRProof (EQUIVALENT)
5.79/2.31	Used ordering:
5.79/2.31	Polynomial interpretation [POLO]:
5.79/2.31	
5.79/2.31	   POL(a(x_1)) = 1 + x_1
5.79/2.31	   POL(b(x_1)) = x_1
5.79/2.31	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.79/2.31	
5.79/2.31	   a(a(a(a(x1)))) -> b(b(b(b(b(b(x1))))))
5.79/2.31	
5.79/2.31	
5.79/2.31	
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(4)
5.79/2.31	Obligation:
5.79/2.31	Q restricted rewrite system:
5.79/2.31	R is empty.
5.79/2.31	Q is empty.
5.79/2.31	
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(5) RisEmptyProof (EQUIVALENT)
5.79/2.31	The TRS R is empty. Hence, termination is trivially proven.
5.79/2.31	----------------------------------------
5.79/2.31	
5.79/2.31	(6)
5.79/2.31	YES
5.93/2.39	EOF