15.43/4.81	YES
15.79/4.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.79/4.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.79/4.83	
15.79/4.83	
15.79/4.83	Termination w.r.t. Q of the given QTRS could be proven:
15.79/4.83	
15.79/4.83	(0) QTRS
15.79/4.83	(1) FlatCCProof [EQUIVALENT, 0 ms]
15.79/4.83	(2) QTRS
15.79/4.83	(3) RootLabelingProof [EQUIVALENT, 0 ms]
15.79/4.83	(4) QTRS
15.79/4.83	(5) QTRSRRRProof [EQUIVALENT, 34 ms]
15.79/4.83	(6) QTRS
15.79/4.83	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
15.79/4.83	(8) QDP
15.79/4.83	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
15.79/4.83	(10) QDP
15.79/4.83	(11) QDPOrderProof [EQUIVALENT, 23 ms]
15.79/4.83	(12) QDP
15.79/4.83	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
15.79/4.83	(14) TRUE
15.79/4.83	
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(0)
15.79/4.83	Obligation:
15.79/4.83	Q restricted rewrite system:
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a(a(a(a(x1)))) -> a(b(b(b(x1))))
15.79/4.83	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
15.79/4.83	   b(a(a(b(x1)))) -> b(a(b(b(x1))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(1) FlatCCProof (EQUIVALENT)
15.79/4.83	We used flat context closure [ROOTLAB]
15.79/4.83	As Q is empty the flat context closure was sound AND complete.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(2)
15.79/4.83	Obligation:
15.79/4.83	Q restricted rewrite system:
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a(a(a(a(x1)))) -> a(b(b(b(x1))))
15.79/4.83	   b(a(a(b(x1)))) -> b(a(b(b(x1))))
15.79/4.83	   a(a(b(b(b(x1))))) -> a(b(a(b(a(x1)))))
15.79/4.83	   b(a(b(b(b(x1))))) -> b(b(a(b(a(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(3) RootLabelingProof (EQUIVALENT)
15.79/4.83	We used plain root labeling [ROOTLAB] with the following heuristic:
15.79/4.83	LabelAll: All function symbols get labeled
15.79/4.83	
15.79/4.83	As Q is empty the root labeling was sound AND complete.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(4)
15.79/4.83	Obligation:
15.79/4.83	Q restricted rewrite system:
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(5) QTRSRRRProof (EQUIVALENT)
15.79/4.83	Used ordering:
15.79/4.83	Polynomial interpretation [POLO]:
15.79/4.83	
15.79/4.83	   POL(a_{a_1}(x_1)) = 1 + x_1
15.79/4.83	   POL(a_{b_1}(x_1)) = x_1
15.79/4.83	   POL(b_{a_1}(x_1)) = 1 + x_1
15.79/4.83	   POL(b_{b_1}(x_1)) = 1 + x_1
15.79/4.83	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
15.79/4.83	
15.79/4.83	
15.79/4.83	
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(6)
15.79/4.83	Obligation:
15.79/4.83	Q restricted rewrite system:
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(7) DependencyPairsProof (EQUIVALENT)
15.79/4.83	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(8)
15.79/4.83	Obligation:
15.79/4.83	Q DP problem:
15.79/4.83	The TRS P consists of the following rules:
15.79/4.83	
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
15.79/4.83	
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	We have to consider all minimal (P,Q,R)-chains.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(9) DependencyGraphProof (EQUIVALENT)
15.79/4.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(10)
15.79/4.83	Obligation:
15.79/4.83	Q DP problem:
15.79/4.83	The TRS P consists of the following rules:
15.79/4.83	
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	We have to consider all minimal (P,Q,R)-chains.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(11) QDPOrderProof (EQUIVALENT)
15.79/4.83	We use the reduction pair processor [LPAR04,JAR06].
15.79/4.83	
15.79/4.83	
15.79/4.83	The following pairs can be oriented strictly and are deleted.
15.79/4.83	
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
15.79/4.83	The remaining pairs can at least be oriented weakly.
15.79/4.83	Used ordering:  Polynomial interpretation [POLO]:
15.79/4.83	
15.79/4.83	   POL(B_{A_1}(x_1)) = x_1
15.79/4.83	   POL(a_{a_1}(x_1)) = 1 + x_1
15.79/4.83	   POL(a_{b_1}(x_1)) = 1 + x_1
15.79/4.83	   POL(b_{a_1}(x_1)) = 1 + x_1
15.79/4.83	   POL(b_{b_1}(x_1)) = 1 + x_1
15.79/4.83	
15.79/4.83	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	
15.79/4.83	
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(12)
15.79/4.83	Obligation:
15.79/4.83	Q DP problem:
15.79/4.83	The TRS P consists of the following rules:
15.79/4.83	
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	
15.79/4.83	The TRS R consists of the following rules:
15.79/4.83	
15.79/4.83	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
15.79/4.83	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
15.79/4.83	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
15.79/4.83	
15.79/4.83	Q is empty.
15.79/4.83	We have to consider all minimal (P,Q,R)-chains.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(13) DependencyGraphProof (EQUIVALENT)
15.79/4.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
15.79/4.83	----------------------------------------
15.79/4.83	
15.79/4.83	(14)
15.79/4.83	TRUE
15.96/4.97	EOF