5.44/2.19	YES
5.44/2.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.44/2.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.44/2.23	
5.44/2.23	
5.44/2.23	Termination w.r.t. Q of the given QTRS could be proven:
5.44/2.23	
5.44/2.23	(0) QTRS
5.44/2.23	(1) QTRSRRRProof [EQUIVALENT, 52 ms]
5.44/2.23	(2) QTRS
5.44/2.23	(3) QTRSRRRProof [EQUIVALENT, 2 ms]
5.44/2.23	(4) QTRS
5.44/2.23	(5) QTRSRRRProof [EQUIVALENT, 2 ms]
5.44/2.23	(6) QTRS
5.44/2.23	(7) RisEmptyProof [EQUIVALENT, 0 ms]
5.44/2.23	(8) YES
5.44/2.23	
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(0)
5.44/2.23	Obligation:
5.44/2.23	Q restricted rewrite system:
5.44/2.23	The TRS R consists of the following rules:
5.44/2.23	
5.44/2.23	   c(c(c(a(x1)))) -> d(d(x1))
5.44/2.23	   d(b(x1)) -> c(c(x1))
5.44/2.23	   c(x1) -> a(a(a(a(x1))))
5.44/2.23	   d(x1) -> b(b(b(b(x1))))
5.44/2.23	   b(d(x1)) -> c(c(x1))
5.44/2.23	   a(c(c(c(x1)))) -> d(d(x1))
5.44/2.23	
5.44/2.23	Q is empty.
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(1) QTRSRRRProof (EQUIVALENT)
5.44/2.23	Used ordering:
5.44/2.23	Polynomial interpretation [POLO]:
5.44/2.23	
5.44/2.23	   POL(a(x_1)) = 3 + x_1
5.44/2.23	   POL(b(x_1)) = 5 + x_1
5.44/2.23	   POL(c(x_1)) = 13 + x_1
5.44/2.23	   POL(d(x_1)) = 21 + x_1
5.44/2.23	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.44/2.23	
5.44/2.23	   c(x1) -> a(a(a(a(x1))))
5.44/2.23	   d(x1) -> b(b(b(b(x1))))
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(2)
5.44/2.23	Obligation:
5.44/2.23	Q restricted rewrite system:
5.44/2.23	The TRS R consists of the following rules:
5.44/2.23	
5.44/2.23	   c(c(c(a(x1)))) -> d(d(x1))
5.44/2.23	   d(b(x1)) -> c(c(x1))
5.44/2.23	   b(d(x1)) -> c(c(x1))
5.44/2.23	   a(c(c(c(x1)))) -> d(d(x1))
5.44/2.23	
5.44/2.23	Q is empty.
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(3) QTRSRRRProof (EQUIVALENT)
5.44/2.23	Used ordering:
5.44/2.23	Polynomial interpretation [POLO]:
5.44/2.23	
5.44/2.23	   POL(a(x_1)) = x_1
5.44/2.23	   POL(b(x_1)) = 1 + x_1
5.44/2.23	   POL(c(x_1)) = x_1
5.44/2.23	   POL(d(x_1)) = x_1
5.44/2.23	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.44/2.23	
5.44/2.23	   d(b(x1)) -> c(c(x1))
5.44/2.23	   b(d(x1)) -> c(c(x1))
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(4)
5.44/2.23	Obligation:
5.44/2.23	Q restricted rewrite system:
5.44/2.23	The TRS R consists of the following rules:
5.44/2.23	
5.44/2.23	   c(c(c(a(x1)))) -> d(d(x1))
5.44/2.23	   a(c(c(c(x1)))) -> d(d(x1))
5.44/2.23	
5.44/2.23	Q is empty.
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(5) QTRSRRRProof (EQUIVALENT)
5.44/2.23	Used ordering:
5.44/2.23	Polynomial interpretation [POLO]:
5.44/2.23	
5.44/2.23	   POL(a(x_1)) = 1 + x_1
5.44/2.23	   POL(c(x_1)) = x_1
5.44/2.23	   POL(d(x_1)) = x_1
5.44/2.23	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.44/2.23	
5.44/2.23	   c(c(c(a(x1)))) -> d(d(x1))
5.44/2.23	   a(c(c(c(x1)))) -> d(d(x1))
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(6)
5.44/2.23	Obligation:
5.44/2.23	Q restricted rewrite system:
5.44/2.23	R is empty.
5.44/2.23	Q is empty.
5.44/2.23	
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(7) RisEmptyProof (EQUIVALENT)
5.44/2.23	The TRS R is empty. Hence, termination is trivially proven.
5.44/2.23	----------------------------------------
5.44/2.23	
5.44/2.23	(8)
5.44/2.23	YES
5.82/2.30	EOF