5.37/2.12	YES
5.37/2.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.37/2.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.37/2.14	
5.37/2.14	
5.37/2.14	Termination w.r.t. Q of the given QTRS could be proven:
5.37/2.14	
5.37/2.14	(0) QTRS
5.37/2.14	(1) FlatCCProof [EQUIVALENT, 0 ms]
5.37/2.14	(2) QTRS
5.37/2.14	(3) RootLabelingProof [EQUIVALENT, 0 ms]
5.37/2.14	(4) QTRS
5.37/2.14	(5) QTRSRRRProof [EQUIVALENT, 3 ms]
5.37/2.14	(6) QTRS
5.37/2.14	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
5.37/2.14	(8) QTRS
5.37/2.14	(9) RisEmptyProof [EQUIVALENT, 2 ms]
5.37/2.14	(10) YES
5.37/2.14	
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(0)
5.37/2.14	Obligation:
5.37/2.14	Q restricted rewrite system:
5.37/2.14	The TRS R consists of the following rules:
5.37/2.14	
5.37/2.14	   a(a(x1)) -> a(b(a(x1)))
5.37/2.14	   b(b(x1)) -> a(a(x1))
5.37/2.14	   a(b(b(a(x1)))) -> x1
5.37/2.14	
5.37/2.14	Q is empty.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(1) FlatCCProof (EQUIVALENT)
5.37/2.14	We used flat context closure [ROOTLAB]
5.37/2.14	As Q is empty the flat context closure was sound AND complete.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(2)
5.37/2.14	Obligation:
5.37/2.14	Q restricted rewrite system:
5.37/2.14	The TRS R consists of the following rules:
5.37/2.14	
5.37/2.14	   a(a(x1)) -> a(b(a(x1)))
5.37/2.14	   a(b(b(x1))) -> a(a(a(x1)))
5.37/2.14	   b(b(b(x1))) -> b(a(a(x1)))
5.37/2.14	   a(a(b(b(a(x1))))) -> a(x1)
5.37/2.14	   b(a(b(b(a(x1))))) -> b(x1)
5.37/2.14	
5.37/2.14	Q is empty.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(3) RootLabelingProof (EQUIVALENT)
5.37/2.14	We used plain root labeling [ROOTLAB] with the following heuristic:
5.37/2.14	LabelAll: All function symbols get labeled
5.37/2.14	
5.37/2.14	As Q is empty the root labeling was sound AND complete.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(4)
5.37/2.14	Obligation:
5.37/2.14	Q restricted rewrite system:
5.37/2.14	The TRS R consists of the following rules:
5.37/2.14	
5.37/2.14	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.37/2.14	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.37/2.14	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.37/2.14	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.37/2.14	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.37/2.14	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.37/2.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(x1)
5.37/2.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(x1)
5.37/2.14	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(x1)
5.37/2.14	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(x1)
5.37/2.14	
5.37/2.14	Q is empty.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(5) QTRSRRRProof (EQUIVALENT)
5.37/2.14	Used ordering:
5.37/2.14	Polynomial interpretation [POLO]:
5.37/2.14	
5.37/2.14	   POL(a_{a_1}(x_1)) = 1 + x_1
5.37/2.14	   POL(a_{b_1}(x_1)) = 1 + x_1
5.37/2.14	   POL(b_{a_1}(x_1)) = x_1
5.37/2.14	   POL(b_{b_1}(x_1)) = 3 + x_1
5.37/2.14	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.14	
5.37/2.14	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.37/2.14	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.37/2.14	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.37/2.14	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.37/2.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(x1)
5.37/2.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(x1)
5.37/2.14	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(x1)
5.37/2.14	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(x1)
5.37/2.14	
5.37/2.14	
5.37/2.14	
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(6)
5.37/2.14	Obligation:
5.37/2.14	Q restricted rewrite system:
5.37/2.14	The TRS R consists of the following rules:
5.37/2.14	
5.37/2.14	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.37/2.14	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.37/2.14	
5.37/2.14	Q is empty.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(7) QTRSRRRProof (EQUIVALENT)
5.37/2.14	Used ordering:
5.37/2.14	Polynomial interpretation [POLO]:
5.37/2.14	
5.37/2.14	   POL(a_{a_1}(x_1)) = 1 + x_1
5.37/2.14	   POL(a_{b_1}(x_1)) = x_1
5.37/2.14	   POL(b_{a_1}(x_1)) = x_1
5.37/2.14	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.14	
5.37/2.14	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.37/2.14	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.37/2.14	
5.37/2.14	
5.37/2.14	
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(8)
5.37/2.14	Obligation:
5.37/2.14	Q restricted rewrite system:
5.37/2.14	R is empty.
5.37/2.14	Q is empty.
5.37/2.14	
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(9) RisEmptyProof (EQUIVALENT)
5.37/2.14	The TRS R is empty. Hence, termination is trivially proven.
5.37/2.14	----------------------------------------
5.37/2.14	
5.37/2.14	(10)
5.37/2.14	YES
5.37/2.17	EOF