4.62/2.04	YES
4.94/2.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.94/2.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.94/2.08	
4.94/2.08	
4.94/2.08	Termination w.r.t. Q of the given QTRS could be proven:
4.94/2.08	
4.94/2.08	(0) QTRS
4.94/2.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
4.94/2.08	(2) QTRS
4.94/2.08	(3) FlatCCProof [EQUIVALENT, 0 ms]
4.94/2.08	(4) QTRS
4.94/2.08	(5) RootLabelingProof [EQUIVALENT, 0 ms]
4.94/2.08	(6) QTRS
4.94/2.08	(7) QTRSRRRProof [EQUIVALENT, 4 ms]
4.94/2.08	(8) QTRS
4.94/2.08	(9) RisEmptyProof [EQUIVALENT, 0 ms]
4.94/2.08	(10) YES
4.94/2.08	
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(0)
4.94/2.08	Obligation:
4.94/2.08	Q restricted rewrite system:
4.94/2.08	The TRS R consists of the following rules:
4.94/2.08	
4.94/2.08	   b(a(b(x1))) -> a(x1)
4.94/2.08	   a(a(a(x1))) -> b(x1)
4.94/2.08	   b(b(x1)) -> b(a(b(x1)))
4.94/2.08	
4.94/2.08	Q is empty.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(1) QTRS Reverse (EQUIVALENT)
4.94/2.08	We applied the QTRS Reverse Processor [REVERSE].
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(2)
4.94/2.08	Obligation:
4.94/2.08	Q restricted rewrite system:
4.94/2.08	The TRS R consists of the following rules:
4.94/2.08	
4.94/2.08	   b(a(b(x1))) -> a(x1)
4.94/2.08	   a(a(a(x1))) -> b(x1)
4.94/2.08	   b(b(x1)) -> b(a(b(x1)))
4.94/2.08	
4.94/2.08	Q is empty.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(3) FlatCCProof (EQUIVALENT)
4.94/2.08	We used flat context closure [ROOTLAB]
4.94/2.08	As Q is empty the flat context closure was sound AND complete.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(4)
4.94/2.08	Obligation:
4.94/2.08	Q restricted rewrite system:
4.94/2.08	The TRS R consists of the following rules:
4.94/2.08	
4.94/2.08	   b(b(x1)) -> b(a(b(x1)))
4.94/2.08	   b(b(a(b(x1)))) -> b(a(x1))
4.94/2.08	   a(b(a(b(x1)))) -> a(a(x1))
4.94/2.08	   b(a(a(a(x1)))) -> b(b(x1))
4.94/2.08	   a(a(a(a(x1)))) -> a(b(x1))
4.94/2.08	
4.94/2.08	Q is empty.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(5) RootLabelingProof (EQUIVALENT)
4.94/2.08	We used plain root labeling [ROOTLAB] with the following heuristic:
4.94/2.08	LabelAll: All function symbols get labeled
4.94/2.08	
4.94/2.08	As Q is empty the root labeling was sound AND complete.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(6)
4.94/2.08	Obligation:
4.94/2.08	Q restricted rewrite system:
4.94/2.08	The TRS R consists of the following rules:
4.94/2.08	
4.94/2.08	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
4.94/2.08	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
4.94/2.08	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
4.94/2.08	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
4.94/2.08	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
4.94/2.08	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
4.94/2.08	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
4.94/2.08	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
4.94/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
4.94/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
4.94/2.08	
4.94/2.08	Q is empty.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(7) QTRSRRRProof (EQUIVALENT)
4.94/2.08	Used ordering:
4.94/2.08	Polynomial interpretation [POLO]:
4.94/2.08	
4.94/2.08	   POL(a_{a_1}(x_1)) = 5 + x_1
4.94/2.08	   POL(a_{b_1}(x_1)) = 6 + x_1
4.94/2.08	   POL(b_{a_1}(x_1)) = x_1
4.94/2.08	   POL(b_{b_1}(x_1)) = 7 + x_1
4.94/2.08	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.94/2.08	
4.94/2.08	   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
4.94/2.08	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
4.94/2.08	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
4.94/2.08	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
4.94/2.08	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
4.94/2.08	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
4.94/2.08	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
4.94/2.08	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
4.94/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
4.94/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
4.94/2.08	
4.94/2.08	
4.94/2.08	
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(8)
4.94/2.08	Obligation:
4.94/2.08	Q restricted rewrite system:
4.94/2.08	R is empty.
4.94/2.08	Q is empty.
4.94/2.08	
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(9) RisEmptyProof (EQUIVALENT)
4.94/2.08	The TRS R is empty. Hence, termination is trivially proven.
4.94/2.08	----------------------------------------
4.94/2.08	
4.94/2.08	(10)
4.94/2.08	YES
4.94/2.11	EOF