6.42/2.42	YES
6.42/2.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.42/2.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.42/2.47	
6.42/2.47	
6.42/2.47	Termination w.r.t. Q of the given QTRS could be proven:
6.42/2.47	
6.42/2.47	(0) QTRS
6.42/2.47	(1) FlatCCProof [EQUIVALENT, 0 ms]
6.42/2.47	(2) QTRS
6.42/2.47	(3) RootLabelingProof [EQUIVALENT, 0 ms]
6.42/2.47	(4) QTRS
6.42/2.47	(5) QTRSRRRProof [EQUIVALENT, 3 ms]
6.42/2.47	(6) QTRS
6.42/2.47	(7) QTRSRRRProof [EQUIVALENT, 7 ms]
6.42/2.47	(8) QTRS
6.42/2.47	(9) QTRSRRRProof [EQUIVALENT, 3 ms]
6.42/2.47	(10) QTRS
6.42/2.47	(11) RisEmptyProof [EQUIVALENT, 0 ms]
6.42/2.47	(12) YES
6.42/2.47	
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(0)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	The TRS R consists of the following rules:
6.42/2.47	
6.42/2.47	   a(b(a(x1))) -> b(a(x1))
6.42/2.47	   b(b(b(x1))) -> b(a(b(x1)))
6.42/2.47	   a(a(x1)) -> b(b(b(x1)))
6.42/2.47	
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(1) FlatCCProof (EQUIVALENT)
6.42/2.47	We used flat context closure [ROOTLAB]
6.42/2.47	As Q is empty the flat context closure was sound AND complete.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(2)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	The TRS R consists of the following rules:
6.42/2.47	
6.42/2.47	   b(b(b(x1))) -> b(a(b(x1)))
6.42/2.47	   a(a(b(a(x1)))) -> a(b(a(x1)))
6.42/2.47	   b(a(b(a(x1)))) -> b(b(a(x1)))
6.42/2.47	   a(a(a(x1))) -> a(b(b(b(x1))))
6.42/2.47	   b(a(a(x1))) -> b(b(b(b(x1))))
6.42/2.47	
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(3) RootLabelingProof (EQUIVALENT)
6.42/2.47	We used plain root labeling [ROOTLAB] with the following heuristic:
6.42/2.47	LabelAll: All function symbols get labeled
6.42/2.47	
6.42/2.47	As Q is empty the root labeling was sound AND complete.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(4)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	The TRS R consists of the following rules:
6.42/2.47	
6.42/2.47	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
6.42/2.47	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
6.42/2.47	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
6.42/2.47	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
6.42/2.47	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
6.42/2.47	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
6.42/2.47	
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(5) QTRSRRRProof (EQUIVALENT)
6.42/2.47	Used ordering:
6.42/2.47	Polynomial interpretation [POLO]:
6.42/2.47	
6.42/2.47	   POL(a_{a_1}(x_1)) = 1 + x_1
6.42/2.47	   POL(a_{b_1}(x_1)) = x_1
6.42/2.47	   POL(b_{a_1}(x_1)) = x_1
6.42/2.47	   POL(b_{b_1}(x_1)) = x_1
6.42/2.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.42/2.47	
6.42/2.47	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
6.42/2.47	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
6.42/2.47	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
6.42/2.47	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(6)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	The TRS R consists of the following rules:
6.42/2.47	
6.42/2.47	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
6.42/2.47	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(7) QTRSRRRProof (EQUIVALENT)
6.42/2.47	Used ordering:
6.42/2.47	Polynomial interpretation [POLO]:
6.42/2.47	
6.42/2.47	   POL(a_{a_1}(x_1)) = x_1
6.42/2.47	   POL(a_{b_1}(x_1)) = 1 + x_1
6.42/2.47	   POL(b_{a_1}(x_1)) = x_1
6.42/2.47	   POL(b_{b_1}(x_1)) = 1 + x_1
6.42/2.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.42/2.47	
6.42/2.47	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
6.42/2.47	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(8)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	The TRS R consists of the following rules:
6.42/2.47	
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(9) QTRSRRRProof (EQUIVALENT)
6.42/2.47	Used ordering:
6.42/2.47	Polynomial interpretation [POLO]:
6.42/2.47	
6.42/2.47	   POL(a_{a_1}(x_1)) = x_1
6.42/2.47	   POL(a_{b_1}(x_1)) = 1 + x_1
6.42/2.47	   POL(b_{a_1}(x_1)) = x_1
6.42/2.47	   POL(b_{b_1}(x_1)) = x_1
6.42/2.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.42/2.47	
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.42/2.47	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(10)
6.42/2.47	Obligation:
6.42/2.47	Q restricted rewrite system:
6.42/2.47	R is empty.
6.42/2.47	Q is empty.
6.42/2.47	
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(11) RisEmptyProof (EQUIVALENT)
6.42/2.47	The TRS R is empty. Hence, termination is trivially proven.
6.42/2.47	----------------------------------------
6.42/2.47	
6.42/2.47	(12)
6.42/2.47	YES
6.72/2.54	EOF