5.62/2.14	YES
5.62/2.17	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.62/2.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.62/2.17	
5.62/2.17	
5.62/2.17	Termination w.r.t. Q of the given QTRS could be proven:
5.62/2.17	
5.62/2.17	(0) QTRS
5.62/2.17	(1) QTRS Reverse [EQUIVALENT, 0 ms]
5.62/2.17	(2) QTRS
5.62/2.17	(3) FlatCCProof [EQUIVALENT, 0 ms]
5.62/2.17	(4) QTRS
5.62/2.17	(5) RootLabelingProof [EQUIVALENT, 0 ms]
5.62/2.17	(6) QTRS
5.62/2.17	(7) QTRSRRRProof [EQUIVALENT, 9 ms]
5.62/2.17	(8) QTRS
5.62/2.17	(9) QTRSRRRProof [EQUIVALENT, 1 ms]
5.62/2.17	(10) QTRS
5.62/2.17	(11) RisEmptyProof [EQUIVALENT, 0 ms]
5.62/2.17	(12) YES
5.62/2.17	
5.62/2.17	
5.62/2.17	----------------------------------------
5.62/2.17	
5.62/2.17	(0)
5.62/2.17	Obligation:
5.62/2.17	Q restricted rewrite system:
5.62/2.17	The TRS R consists of the following rules:
5.62/2.17	
5.62/2.17	   a(b(a(x1))) -> b(x1)
5.62/2.17	   b(a(b(x1))) -> b(b(b(x1)))
5.62/2.17	   b(b(x1)) -> a(a(a(x1)))
5.62/2.17	
5.62/2.17	Q is empty.
5.62/2.17	
5.62/2.17	----------------------------------------
5.62/2.17	
5.62/2.17	(1) QTRS Reverse (EQUIVALENT)
5.62/2.17	We applied the QTRS Reverse Processor [REVERSE].
5.62/2.17	----------------------------------------
5.62/2.17	
5.62/2.17	(2)
5.62/2.17	Obligation:
5.62/2.17	Q restricted rewrite system:
5.62/2.17	The TRS R consists of the following rules:
5.62/2.17	
5.62/2.17	   a(b(a(x1))) -> b(x1)
5.62/2.17	   b(a(b(x1))) -> b(b(b(x1)))
5.62/2.18	   b(b(x1)) -> a(a(a(x1)))
5.62/2.18	
5.62/2.18	Q is empty.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(3) FlatCCProof (EQUIVALENT)
5.62/2.18	We used flat context closure [ROOTLAB]
5.62/2.18	As Q is empty the flat context closure was sound AND complete.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(4)
5.62/2.18	Obligation:
5.62/2.18	Q restricted rewrite system:
5.62/2.18	The TRS R consists of the following rules:
5.62/2.18	
5.62/2.18	   b(a(b(x1))) -> b(b(b(x1)))
5.62/2.18	   a(a(b(a(x1)))) -> a(b(x1))
5.62/2.18	   b(a(b(a(x1)))) -> b(b(x1))
5.62/2.18	   a(b(b(x1))) -> a(a(a(a(x1))))
5.62/2.18	   b(b(b(x1))) -> b(a(a(a(x1))))
5.62/2.18	
5.62/2.18	Q is empty.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(5) RootLabelingProof (EQUIVALENT)
5.62/2.18	We used plain root labeling [ROOTLAB] with the following heuristic:
5.62/2.18	LabelAll: All function symbols get labeled
5.62/2.18	
5.62/2.18	As Q is empty the root labeling was sound AND complete.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(6)
5.62/2.18	Obligation:
5.62/2.18	Q restricted rewrite system:
5.62/2.18	The TRS R consists of the following rules:
5.62/2.18	
5.62/2.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
5.62/2.18	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
5.62/2.18	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
5.62/2.18	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.62/2.18	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.62/2.18	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.62/2.18	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.62/2.18	
5.62/2.18	Q is empty.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(7) QTRSRRRProof (EQUIVALENT)
5.62/2.18	Used ordering:
5.62/2.18	Polynomial interpretation [POLO]:
5.62/2.18	
5.62/2.18	   POL(a_{a_1}(x_1)) = 1 + x_1
5.62/2.18	   POL(a_{b_1}(x_1)) = 6 + x_1
5.62/2.18	   POL(b_{a_1}(x_1)) = x_1
5.62/2.18	   POL(b_{b_1}(x_1)) = 3 + x_1
5.62/2.18	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.62/2.18	
5.62/2.18	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
5.62/2.18	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
5.62/2.18	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.62/2.18	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.62/2.18	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.62/2.18	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.62/2.18	
5.62/2.18	
5.62/2.18	
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(8)
5.62/2.18	Obligation:
5.62/2.18	Q restricted rewrite system:
5.62/2.18	The TRS R consists of the following rules:
5.62/2.18	
5.62/2.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
5.62/2.18	
5.62/2.18	Q is empty.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(9) QTRSRRRProof (EQUIVALENT)
5.62/2.18	Used ordering:
5.62/2.18	Polynomial interpretation [POLO]:
5.62/2.18	
5.62/2.18	   POL(a_{b_1}(x_1)) = 1 + x_1
5.62/2.18	   POL(b_{a_1}(x_1)) = x_1
5.62/2.18	   POL(b_{b_1}(x_1)) = x_1
5.62/2.18	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.62/2.18	
5.62/2.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
5.62/2.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
5.62/2.18	
5.62/2.18	
5.62/2.18	
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(10)
5.62/2.18	Obligation:
5.62/2.18	Q restricted rewrite system:
5.62/2.18	R is empty.
5.62/2.18	Q is empty.
5.62/2.18	
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(11) RisEmptyProof (EQUIVALENT)
5.62/2.18	The TRS R is empty. Hence, termination is trivially proven.
5.62/2.18	----------------------------------------
5.62/2.18	
5.62/2.18	(12)
5.62/2.18	YES
5.88/2.24	EOF