5.17/2.23	YES
5.52/2.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.52/2.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.52/2.30	
5.52/2.30	
5.52/2.30	Termination w.r.t. Q of the given QTRS could be proven:
5.52/2.30	
5.52/2.30	(0) QTRS
5.52/2.30	(1) QTRSRRRProof [EQUIVALENT, 43 ms]
5.52/2.30	(2) QTRS
5.52/2.30	(3) RisEmptyProof [EQUIVALENT, 0 ms]
5.52/2.30	(4) YES
5.52/2.30	
5.52/2.30	
5.52/2.30	----------------------------------------
5.52/2.30	
5.52/2.30	(0)
5.52/2.30	Obligation:
5.52/2.30	Q restricted rewrite system:
5.52/2.30	The TRS R consists of the following rules:
5.52/2.30	
5.52/2.30	   a(a(x1)) -> b(c(x1))
5.52/2.30	   b(b(x1)) -> c(d(x1))
5.52/2.30	   b(x1) -> a(x1)
5.52/2.30	   c(c(x1)) -> d(f(x1))
5.52/2.30	   d(d(x1)) -> f(f(f(x1)))
5.52/2.30	   d(x1) -> b(x1)
5.52/2.30	   f(f(x1)) -> g(a(x1))
5.52/2.30	   g(g(x1)) -> a(x1)
5.52/2.30	
5.52/2.30	Q is empty.
5.52/2.30	
5.52/2.30	----------------------------------------
5.52/2.30	
5.52/2.30	(1) QTRSRRRProof (EQUIVALENT)
5.52/2.30	Used ordering:
5.52/2.30	Polynomial interpretation [POLO]:
5.52/2.30	
5.52/2.30	   POL(a(x_1)) = 155 + x_1
5.52/2.30	   POL(b(x_1)) = 162 + x_1
5.52/2.30	   POL(c(x_1)) = 147 + x_1
5.52/2.30	   POL(d(x_1)) = 176 + x_1
5.52/2.30	   POL(f(x_1)) = 117 + x_1
5.52/2.30	   POL(g(x_1)) = 78 + x_1
5.52/2.30	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.52/2.30	
5.52/2.30	   a(a(x1)) -> b(c(x1))
5.52/2.30	   b(b(x1)) -> c(d(x1))
5.52/2.30	   b(x1) -> a(x1)
5.52/2.30	   c(c(x1)) -> d(f(x1))
5.52/2.30	   d(d(x1)) -> f(f(f(x1)))
5.52/2.30	   d(x1) -> b(x1)
5.52/2.30	   f(f(x1)) -> g(a(x1))
5.52/2.30	   g(g(x1)) -> a(x1)
5.52/2.30	
5.52/2.30	
5.52/2.30	
5.52/2.30	
5.52/2.30	----------------------------------------
5.52/2.30	
5.52/2.30	(2)
5.52/2.30	Obligation:
5.52/2.30	Q restricted rewrite system:
5.52/2.30	R is empty.
5.52/2.30	Q is empty.
5.52/2.30	
5.52/2.30	----------------------------------------
5.52/2.30	
5.52/2.30	(3) RisEmptyProof (EQUIVALENT)
5.52/2.30	The TRS R is empty. Hence, termination is trivially proven.
5.52/2.30	----------------------------------------
5.52/2.30	
5.52/2.30	(4)
5.52/2.30	YES
5.88/2.37	EOF