5.72/2.31	YES
6.09/2.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.09/2.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.09/2.35	
6.09/2.35	
6.09/2.35	Termination w.r.t. Q of the given QTRS could be proven:
6.09/2.35	
6.09/2.35	(0) QTRS
6.09/2.35	(1) FlatCCProof [EQUIVALENT, 0 ms]
6.09/2.35	(2) QTRS
6.09/2.35	(3) RootLabelingProof [EQUIVALENT, 0 ms]
6.09/2.35	(4) QTRS
6.09/2.35	(5) QTRSRRRProof [EQUIVALENT, 32 ms]
6.09/2.35	(6) QTRS
6.09/2.35	(7) QTRSRRRProof [EQUIVALENT, 2 ms]
6.09/2.35	(8) QTRS
6.09/2.35	(9) RisEmptyProof [EQUIVALENT, 0 ms]
6.09/2.35	(10) YES
6.09/2.35	
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(0)
6.09/2.35	Obligation:
6.09/2.35	Q restricted rewrite system:
6.09/2.35	The TRS R consists of the following rules:
6.09/2.35	
6.09/2.35	   b(b(b(x1))) -> a(x1)
6.09/2.35	   a(a(a(x1))) -> b(b(x1))
6.09/2.35	   a(a(x1)) -> a(b(a(x1)))
6.09/2.35	
6.09/2.35	Q is empty.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(1) FlatCCProof (EQUIVALENT)
6.09/2.35	We used flat context closure [ROOTLAB]
6.09/2.35	As Q is empty the flat context closure was sound AND complete.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(2)
6.09/2.35	Obligation:
6.09/2.35	Q restricted rewrite system:
6.09/2.35	The TRS R consists of the following rules:
6.09/2.35	
6.09/2.35	   a(a(x1)) -> a(b(a(x1)))
6.09/2.35	   b(b(b(b(x1)))) -> b(a(x1))
6.09/2.35	   a(b(b(b(x1)))) -> a(a(x1))
6.09/2.35	   b(a(a(a(x1)))) -> b(b(b(x1)))
6.09/2.35	   a(a(a(a(x1)))) -> a(b(b(x1)))
6.09/2.35	
6.09/2.35	Q is empty.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(3) RootLabelingProof (EQUIVALENT)
6.09/2.35	We used plain root labeling [ROOTLAB] with the following heuristic:
6.09/2.35	LabelAll: All function symbols get labeled
6.09/2.35	
6.09/2.35	As Q is empty the root labeling was sound AND complete.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(4)
6.09/2.35	Obligation:
6.09/2.35	Q restricted rewrite system:
6.09/2.35	The TRS R consists of the following rules:
6.09/2.35	
6.09/2.35	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.09/2.35	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.09/2.35	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
6.09/2.35	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
6.09/2.35	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
6.09/2.35	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
6.09/2.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.09/2.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.09/2.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.09/2.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.09/2.35	
6.09/2.35	Q is empty.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(5) QTRSRRRProof (EQUIVALENT)
6.09/2.35	Used ordering:
6.09/2.35	Polynomial interpretation [POLO]:
6.09/2.35	
6.09/2.35	   POL(a_{a_1}(x_1)) = 7 + x_1
6.09/2.35	   POL(a_{b_1}(x_1)) = 7 + x_1
6.09/2.35	   POL(b_{a_1}(x_1)) = x_1
6.09/2.35	   POL(b_{b_1}(x_1)) = 5 + x_1
6.09/2.35	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.09/2.35	
6.09/2.35	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
6.09/2.35	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
6.09/2.35	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
6.09/2.35	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
6.09/2.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.09/2.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.09/2.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.09/2.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.09/2.35	
6.09/2.35	
6.09/2.35	
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(6)
6.09/2.35	Obligation:
6.09/2.35	Q restricted rewrite system:
6.09/2.35	The TRS R consists of the following rules:
6.09/2.35	
6.09/2.35	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.09/2.35	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.09/2.35	
6.09/2.35	Q is empty.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(7) QTRSRRRProof (EQUIVALENT)
6.09/2.35	Used ordering:
6.09/2.35	Polynomial interpretation [POLO]:
6.09/2.35	
6.09/2.35	   POL(a_{a_1}(x_1)) = 1 + x_1
6.09/2.35	   POL(a_{b_1}(x_1)) = x_1
6.09/2.35	   POL(b_{a_1}(x_1)) = x_1
6.09/2.35	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.09/2.35	
6.09/2.35	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
6.09/2.35	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
6.09/2.35	
6.09/2.35	
6.09/2.35	
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(8)
6.09/2.35	Obligation:
6.09/2.35	Q restricted rewrite system:
6.09/2.35	R is empty.
6.09/2.35	Q is empty.
6.09/2.35	
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(9) RisEmptyProof (EQUIVALENT)
6.09/2.35	The TRS R is empty. Hence, termination is trivially proven.
6.09/2.35	----------------------------------------
6.09/2.35	
6.09/2.35	(10)
6.09/2.35	YES
6.32/2.42	EOF