4.41/1.86	YES
4.65/1.90	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.65/1.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.65/1.90	
4.65/1.90	
4.65/1.90	Termination w.r.t. Q of the given QTRS could be proven:
4.65/1.90	
4.65/1.90	(0) QTRS
4.65/1.90	(1) QTRSRRRProof [EQUIVALENT, 43 ms]
4.65/1.90	(2) QTRS
4.65/1.90	(3) RisEmptyProof [EQUIVALENT, 4 ms]
4.65/1.90	(4) YES
4.65/1.90	
4.65/1.90	
4.65/1.90	----------------------------------------
4.65/1.90	
4.65/1.90	(0)
4.65/1.90	Obligation:
4.65/1.90	Q restricted rewrite system:
4.65/1.90	The TRS R consists of the following rules:
4.65/1.90	
4.65/1.90	   a(a(x1)) -> b(b(b(x1)))
4.65/1.90	   b(b(x1)) -> c(c(c(x1)))
4.65/1.90	   c(c(x1)) -> d(d(d(x1)))
4.65/1.90	   b(x1) -> d(d(x1))
4.65/1.90	   c(d(d(x1))) -> a(x1)
4.65/1.90	
4.65/1.90	Q is empty.
4.65/1.90	
4.65/1.90	----------------------------------------
4.65/1.90	
4.65/1.90	(1) QTRSRRRProof (EQUIVALENT)
4.65/1.90	Used ordering:
4.65/1.90	Polynomial interpretation [POLO]:
4.65/1.90	
4.65/1.90	   POL(a(x_1)) = 80 + x_1
4.65/1.90	   POL(b(x_1)) = 53 + x_1
4.65/1.90	   POL(c(x_1)) = 35 + x_1
4.65/1.90	   POL(d(x_1)) = 23 + x_1
4.65/1.90	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.65/1.90	
4.65/1.90	   a(a(x1)) -> b(b(b(x1)))
4.65/1.90	   b(b(x1)) -> c(c(c(x1)))
4.65/1.90	   c(c(x1)) -> d(d(d(x1)))
4.65/1.91	   b(x1) -> d(d(x1))
4.65/1.91	   c(d(d(x1))) -> a(x1)
4.65/1.91	
4.65/1.91	
4.65/1.91	
4.65/1.91	
4.65/1.91	----------------------------------------
4.65/1.91	
4.65/1.91	(2)
4.65/1.91	Obligation:
4.65/1.91	Q restricted rewrite system:
4.65/1.91	R is empty.
4.65/1.91	Q is empty.
4.65/1.91	
4.65/1.91	----------------------------------------
4.65/1.91	
4.65/1.91	(3) RisEmptyProof (EQUIVALENT)
4.65/1.91	The TRS R is empty. Hence, termination is trivially proven.
4.65/1.91	----------------------------------------
4.65/1.91	
4.65/1.91	(4)
4.65/1.91	YES
4.65/1.95	EOF