3.76/1.75	YES
3.76/1.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.76/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.76/1.76	
3.76/1.76	
3.76/1.76	Termination w.r.t. Q of the given QTRS could be proven:
3.76/1.76	
3.76/1.76	(0) QTRS
3.76/1.76	(1) QTRSRRRProof [EQUIVALENT, 41 ms]
3.76/1.76	(2) QTRS
3.76/1.76	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.76/1.76	(4) YES
3.76/1.76	
3.76/1.76	
3.76/1.76	----------------------------------------
3.76/1.76	
3.76/1.76	(0)
3.76/1.76	Obligation:
3.76/1.76	Q restricted rewrite system:
3.76/1.76	The TRS R consists of the following rules:
3.76/1.76	
3.76/1.76	   a(a(x1)) -> b(b(b(x1)))
3.76/1.76	   b(b(b(b(b(x1))))) -> a(a(a(x1)))
3.76/1.76	
3.76/1.76	Q is empty.
3.76/1.76	
3.76/1.76	----------------------------------------
3.76/1.76	
3.76/1.76	(1) QTRSRRRProof (EQUIVALENT)
3.76/1.76	Used ordering:
3.76/1.76	Polynomial interpretation [POLO]:
3.76/1.76	
3.76/1.76	   POL(a(x_1)) = 8 + x_1
3.76/1.76	   POL(b(x_1)) = 5 + x_1
3.76/1.76	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.76/1.76	
3.76/1.76	   a(a(x1)) -> b(b(b(x1)))
3.76/1.76	   b(b(b(b(b(x1))))) -> a(a(a(x1)))
3.76/1.76	
3.76/1.76	
3.76/1.76	
3.76/1.76	
3.76/1.76	----------------------------------------
3.76/1.76	
3.76/1.76	(2)
3.76/1.76	Obligation:
3.76/1.76	Q restricted rewrite system:
3.76/1.76	R is empty.
3.76/1.76	Q is empty.
3.76/1.76	
3.76/1.76	----------------------------------------
3.76/1.76	
3.76/1.76	(3) RisEmptyProof (EQUIVALENT)
3.76/1.76	The TRS R is empty. Hence, termination is trivially proven.
3.76/1.76	----------------------------------------
3.76/1.76	
3.76/1.76	(4)
3.76/1.76	YES
3.76/1.78	EOF