7.49/2.77	YES
7.78/2.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.78/2.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.78/2.86	
7.78/2.86	
7.78/2.86	Termination w.r.t. Q of the given QTRS could be proven:
7.78/2.86	
7.78/2.86	(0) QTRS
7.78/2.86	(1) QTRS Reverse [EQUIVALENT, 0 ms]
7.78/2.86	(2) QTRS
7.78/2.86	(3) FlatCCProof [EQUIVALENT, 0 ms]
7.78/2.86	(4) QTRS
7.78/2.86	(5) RootLabelingProof [EQUIVALENT, 0 ms]
7.78/2.86	(6) QTRS
7.78/2.86	(7) QTRSRRRProof [EQUIVALENT, 8 ms]
7.78/2.86	(8) QTRS
7.78/2.86	(9) QTRSRRRProof [EQUIVALENT, 2 ms]
7.78/2.86	(10) QTRS
7.78/2.86	(11) QTRSRRRProof [EQUIVALENT, 2 ms]
7.78/2.86	(12) QTRS
7.78/2.86	(13) RisEmptyProof [EQUIVALENT, 0 ms]
7.78/2.86	(14) YES
7.78/2.86	
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(0)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b(b(x1)) -> c(c(c(c(x1))))
7.78/2.86	   c(x1) -> x1
7.78/2.86	   b(c(b(x1))) -> b(b(b(x1)))
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(1) QTRS Reverse (EQUIVALENT)
7.78/2.86	We applied the QTRS Reverse Processor [REVERSE].
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(2)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b(b(x1)) -> c(c(c(c(x1))))
7.78/2.86	   c(x1) -> x1
7.78/2.86	   b(c(b(x1))) -> b(b(b(x1)))
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(3) FlatCCProof (EQUIVALENT)
7.78/2.86	We used flat context closure [ROOTLAB]
7.78/2.86	As Q is empty the flat context closure was sound AND complete.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(4)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b(c(b(x1))) -> b(b(b(x1)))
7.78/2.86	   b(b(b(x1))) -> b(c(c(c(c(x1)))))
7.78/2.86	   c(b(b(x1))) -> c(c(c(c(c(x1)))))
7.78/2.86	   b(c(x1)) -> b(x1)
7.78/2.86	   c(c(x1)) -> c(x1)
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(5) RootLabelingProof (EQUIVALENT)
7.78/2.86	We used plain root labeling [ROOTLAB] with the following heuristic:
7.78/2.86	LabelAll: All function symbols get labeled
7.78/2.86	
7.78/2.86	As Q is empty the root labeling was sound AND complete.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(6)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
7.78/2.86	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
7.78/2.86	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
7.78/2.86	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
7.78/2.86	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
7.78/2.86	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
7.78/2.86	   b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1)
7.78/2.86	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
7.78/2.86	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
7.78/2.86	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(7) QTRSRRRProof (EQUIVALENT)
7.78/2.86	Used ordering:
7.78/2.86	Polynomial interpretation [POLO]:
7.78/2.86	
7.78/2.86	   POL(b_{b_1}(x_1)) = 1 + x_1
7.78/2.86	   POL(b_{c_1}(x_1)) = x_1
7.78/2.86	   POL(c_{b_1}(x_1)) = 2 + x_1
7.78/2.86	   POL(c_{c_1}(x_1)) = x_1
7.78/2.86	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.78/2.86	
7.78/2.86	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
7.78/2.86	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
7.78/2.86	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
7.78/2.86	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
7.78/2.86	   b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1)
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(8)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
7.78/2.86	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
7.78/2.86	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
7.78/2.86	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
7.78/2.86	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(9) QTRSRRRProof (EQUIVALENT)
7.78/2.86	Used ordering:
7.78/2.86	Polynomial interpretation [POLO]:
7.78/2.86	
7.78/2.86	   POL(b_{b_1}(x_1)) = x_1
7.78/2.86	   POL(b_{c_1}(x_1)) = x_1
7.78/2.86	   POL(c_{b_1}(x_1)) = 1 + x_1
7.78/2.86	   POL(c_{c_1}(x_1)) = x_1
7.78/2.86	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.78/2.86	
7.78/2.86	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
7.78/2.86	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(10)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	The TRS R consists of the following rules:
7.78/2.86	
7.78/2.86	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
7.78/2.86	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
7.78/2.86	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
7.78/2.86	
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(11) QTRSRRRProof (EQUIVALENT)
7.78/2.86	Used ordering:
7.78/2.86	Polynomial interpretation [POLO]:
7.78/2.86	
7.78/2.86	   POL(b_{c_1}(x_1)) = x_1
7.78/2.86	   POL(c_{b_1}(x_1)) = x_1
7.78/2.86	   POL(c_{c_1}(x_1)) = 1 + x_1
7.78/2.86	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.78/2.86	
7.78/2.86	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
7.78/2.86	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
7.78/2.86	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(12)
7.78/2.86	Obligation:
7.78/2.86	Q restricted rewrite system:
7.78/2.86	R is empty.
7.78/2.86	Q is empty.
7.78/2.86	
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(13) RisEmptyProof (EQUIVALENT)
7.78/2.86	The TRS R is empty. Hence, termination is trivially proven.
7.78/2.86	----------------------------------------
7.78/2.86	
7.78/2.86	(14)
7.78/2.86	YES
7.91/2.91	EOF