8.00/3.02	YES
8.40/3.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.40/3.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.40/3.09	
8.40/3.09	
8.40/3.09	Termination w.r.t. Q of the given QTRS could be proven:
8.40/3.09	
8.40/3.09	(0) QTRS
8.40/3.09	(1) FlatCCProof [EQUIVALENT, 0 ms]
8.40/3.09	(2) QTRS
8.40/3.09	(3) RootLabelingProof [EQUIVALENT, 0 ms]
8.40/3.09	(4) QTRS
8.40/3.09	(5) QTRSRRRProof [EQUIVALENT, 26 ms]
8.40/3.09	(6) QTRS
8.40/3.09	(7) QTRSRRRProof [EQUIVALENT, 4 ms]
8.40/3.09	(8) QTRS
8.40/3.09	(9) QTRSRRRProof [EQUIVALENT, 2 ms]
8.40/3.09	(10) QTRS
8.40/3.09	(11) RisEmptyProof [EQUIVALENT, 0 ms]
8.40/3.09	(12) YES
8.40/3.09	
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(0)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	The TRS R consists of the following rules:
8.40/3.09	
8.40/3.09	   a(a(a(x1))) -> b(x1)
8.40/3.09	   b(b(x1)) -> b(a(b(x1)))
8.40/3.09	   b(b(x1)) -> a(a(x1))
8.40/3.09	
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(1) FlatCCProof (EQUIVALENT)
8.40/3.09	We used flat context closure [ROOTLAB]
8.40/3.09	As Q is empty the flat context closure was sound AND complete.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(2)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	The TRS R consists of the following rules:
8.40/3.09	
8.40/3.09	   b(b(x1)) -> b(a(b(x1)))
8.40/3.09	   a(a(a(a(x1)))) -> a(b(x1))
8.40/3.09	   b(a(a(a(x1)))) -> b(b(x1))
8.40/3.09	   a(b(b(x1))) -> a(a(a(x1)))
8.40/3.09	   b(b(b(x1))) -> b(a(a(x1)))
8.40/3.09	
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(3) RootLabelingProof (EQUIVALENT)
8.40/3.09	We used plain root labeling [ROOTLAB] with the following heuristic:
8.40/3.09	LabelAll: All function symbols get labeled
8.40/3.09	
8.40/3.09	As Q is empty the root labeling was sound AND complete.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(4)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	The TRS R consists of the following rules:
8.40/3.09	
8.40/3.09	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
8.40/3.09	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
8.40/3.09	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
8.40/3.09	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
8.40/3.09	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
8.40/3.09	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
8.40/3.09	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
8.40/3.09	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
8.40/3.09	
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(5) QTRSRRRProof (EQUIVALENT)
8.40/3.09	Used ordering:
8.40/3.09	Polynomial interpretation [POLO]:
8.40/3.09	
8.40/3.09	   POL(a_{a_1}(x_1)) = 1 + x_1
8.40/3.09	   POL(a_{b_1}(x_1)) = 2 + x_1
8.40/3.09	   POL(b_{a_1}(x_1)) = x_1
8.40/3.09	   POL(b_{b_1}(x_1)) = 2 + x_1
8.40/3.09	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
8.40/3.09	
8.40/3.09	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
8.40/3.09	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
8.40/3.09	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
8.40/3.09	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
8.40/3.09	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
8.40/3.09	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(6)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	The TRS R consists of the following rules:
8.40/3.09	
8.40/3.09	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
8.40/3.09	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
8.40/3.09	
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(7) QTRSRRRProof (EQUIVALENT)
8.40/3.09	Used ordering:
8.40/3.09	Polynomial interpretation [POLO]:
8.40/3.09	
8.40/3.09	   POL(a_{a_1}(x_1)) = 1 + x_1
8.40/3.09	   POL(a_{b_1}(x_1)) = 1 + x_1
8.40/3.09	   POL(b_{a_1}(x_1)) = x_1
8.40/3.09	   POL(b_{b_1}(x_1)) = 1 + x_1
8.40/3.09	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
8.40/3.09	
8.40/3.09	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(8)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	The TRS R consists of the following rules:
8.40/3.09	
8.40/3.09	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
8.40/3.09	
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(9) QTRSRRRProof (EQUIVALENT)
8.40/3.09	Used ordering:
8.40/3.09	Polynomial interpretation [POLO]:
8.40/3.09	
8.40/3.09	   POL(a_{b_1}(x_1)) = x_1
8.40/3.09	   POL(b_{a_1}(x_1)) = x_1
8.40/3.09	   POL(b_{b_1}(x_1)) = 1 + x_1
8.40/3.09	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
8.40/3.09	
8.40/3.09	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
8.40/3.09	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(10)
8.40/3.09	Obligation:
8.40/3.09	Q restricted rewrite system:
8.40/3.09	R is empty.
8.40/3.09	Q is empty.
8.40/3.09	
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(11) RisEmptyProof (EQUIVALENT)
8.40/3.09	The TRS R is empty. Hence, termination is trivially proven.
8.40/3.09	----------------------------------------
8.40/3.09	
8.40/3.09	(12)
8.40/3.09	YES
8.55/3.14	EOF