5.11/2.03	YES
5.11/2.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.11/2.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.11/2.08	
5.11/2.08	
5.11/2.08	Termination w.r.t. Q of the given QTRS could be proven:
5.11/2.08	
5.11/2.08	(0) QTRS
5.11/2.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
5.11/2.08	(2) QTRS
5.11/2.08	(3) FlatCCProof [EQUIVALENT, 0 ms]
5.11/2.08	(4) QTRS
5.11/2.08	(5) RootLabelingProof [EQUIVALENT, 0 ms]
5.11/2.08	(6) QTRS
5.11/2.08	(7) QTRSRRRProof [EQUIVALENT, 12 ms]
5.11/2.08	(8) QTRS
5.11/2.08	(9) QTRSRRRProof [EQUIVALENT, 2 ms]
5.11/2.08	(10) QTRS
5.11/2.08	(11) RisEmptyProof [EQUIVALENT, 0 ms]
5.11/2.08	(12) YES
5.11/2.08	
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(0)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	The TRS R consists of the following rules:
5.11/2.08	
5.11/2.08	   b(b(b(x1))) -> a(x1)
5.11/2.08	   a(a(x1)) -> a(b(a(x1)))
5.11/2.08	   a(a(a(x1))) -> b(a(a(x1)))
5.11/2.08	
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(1) QTRS Reverse (EQUIVALENT)
5.11/2.08	We applied the QTRS Reverse Processor [REVERSE].
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(2)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	The TRS R consists of the following rules:
5.11/2.08	
5.11/2.08	   b(b(b(x1))) -> a(x1)
5.11/2.08	   a(a(x1)) -> a(b(a(x1)))
5.11/2.08	   a(a(a(x1))) -> a(a(b(x1)))
5.11/2.08	
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(3) FlatCCProof (EQUIVALENT)
5.11/2.08	We used flat context closure [ROOTLAB]
5.11/2.08	As Q is empty the flat context closure was sound AND complete.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(4)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	The TRS R consists of the following rules:
5.11/2.08	
5.11/2.08	   a(a(x1)) -> a(b(a(x1)))
5.11/2.08	   a(a(a(x1))) -> a(a(b(x1)))
5.11/2.08	   b(b(b(b(x1)))) -> b(a(x1))
5.11/2.08	   a(b(b(b(x1)))) -> a(a(x1))
5.11/2.08	
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(5) RootLabelingProof (EQUIVALENT)
5.11/2.08	We used plain root labeling [ROOTLAB] with the following heuristic:
5.11/2.08	LabelAll: All function symbols get labeled
5.11/2.08	
5.11/2.08	As Q is empty the root labeling was sound AND complete.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(6)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	The TRS R consists of the following rules:
5.11/2.08	
5.11/2.08	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.11/2.08	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.11/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.11/2.08	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.11/2.08	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
5.11/2.08	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
5.11/2.08	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
5.11/2.08	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
5.11/2.08	
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(7) QTRSRRRProof (EQUIVALENT)
5.11/2.08	Used ordering:
5.11/2.08	Polynomial interpretation [POLO]:
5.11/2.08	
5.11/2.08	   POL(a_{a_1}(x_1)) = 3 + x_1
5.11/2.08	   POL(a_{b_1}(x_1)) = x_1
5.11/2.08	   POL(b_{a_1}(x_1)) = 3 + x_1
5.11/2.08	   POL(b_{b_1}(x_1)) = 2 + x_1
5.11/2.08	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.11/2.08	
5.11/2.08	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.11/2.08	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.11/2.08	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
5.11/2.08	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
5.11/2.08	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
5.11/2.08	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
5.11/2.08	
5.11/2.08	
5.11/2.08	
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(8)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	The TRS R consists of the following rules:
5.11/2.08	
5.11/2.08	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.11/2.08	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.11/2.08	
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(9) QTRSRRRProof (EQUIVALENT)
5.11/2.08	Used ordering:
5.11/2.08	Polynomial interpretation [POLO]:
5.11/2.08	
5.11/2.08	   POL(a_{a_1}(x_1)) = 1 + x_1
5.11/2.08	   POL(a_{b_1}(x_1)) = x_1
5.11/2.08	   POL(b_{a_1}(x_1)) = x_1
5.11/2.08	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.11/2.08	
5.11/2.08	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.11/2.08	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.11/2.08	
5.11/2.08	
5.11/2.08	
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(10)
5.11/2.08	Obligation:
5.11/2.08	Q restricted rewrite system:
5.11/2.08	R is empty.
5.11/2.08	Q is empty.
5.11/2.08	
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(11) RisEmptyProof (EQUIVALENT)
5.11/2.08	The TRS R is empty. Hence, termination is trivially proven.
5.11/2.08	----------------------------------------
5.11/2.08	
5.11/2.08	(12)
5.11/2.08	YES
5.29/2.12	EOF