18.10/5.51	YES
18.10/5.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
18.10/5.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.10/5.53	
18.10/5.53	
18.10/5.53	Termination w.r.t. Q of the given QTRS could be proven:
18.10/5.53	
18.10/5.53	(0) QTRS
18.10/5.53	(1) FlatCCProof [EQUIVALENT, 0 ms]
18.10/5.53	(2) QTRS
18.10/5.53	(3) RootLabelingProof [EQUIVALENT, 0 ms]
18.10/5.53	(4) QTRS
18.10/5.53	(5) QTRSRRRProof [EQUIVALENT, 5 ms]
18.10/5.53	(6) QTRS
18.10/5.53	(7) DependencyPairsProof [EQUIVALENT, 40 ms]
18.10/5.53	(8) QDP
18.10/5.53	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
18.10/5.53	(10) QDP
18.10/5.53	(11) QDPOrderProof [EQUIVALENT, 74 ms]
18.10/5.53	(12) QDP
18.10/5.53	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
18.10/5.53	(14) TRUE
18.10/5.53	
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(0)
18.10/5.53	Obligation:
18.10/5.53	Q restricted rewrite system:
18.10/5.53	The TRS R consists of the following rules:
18.10/5.53	
18.10/5.53	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
18.10/5.53	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
18.10/5.53	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
18.10/5.53	
18.10/5.53	Q is empty.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(1) FlatCCProof (EQUIVALENT)
18.10/5.53	We used flat context closure [ROOTLAB]
18.10/5.53	As Q is empty the flat context closure was sound AND complete.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(2)
18.10/5.53	Obligation:
18.10/5.53	Q restricted rewrite system:
18.10/5.53	The TRS R consists of the following rules:
18.10/5.53	
18.10/5.53	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
18.10/5.53	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
18.10/5.53	   a(b(a(b(a(x1))))) -> a(a(a(b(b(x1)))))
18.10/5.53	   b(b(a(b(a(x1))))) -> b(a(a(b(b(x1)))))
18.10/5.53	
18.10/5.53	Q is empty.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(3) RootLabelingProof (EQUIVALENT)
18.10/5.53	We used plain root labeling [ROOTLAB] with the following heuristic:
18.10/5.53	LabelAll: All function symbols get labeled
18.10/5.53	
18.10/5.53	As Q is empty the root labeling was sound AND complete.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(4)
18.10/5.53	Obligation:
18.10/5.53	Q restricted rewrite system:
18.10/5.53	The TRS R consists of the following rules:
18.10/5.53	
18.10/5.53	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
18.10/5.53	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.53	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.53	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.53	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.53	
18.10/5.53	Q is empty.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(5) QTRSRRRProof (EQUIVALENT)
18.10/5.53	Used ordering:
18.10/5.53	Polynomial interpretation [POLO]:
18.10/5.53	
18.10/5.53	   POL(a_{a_1}(x_1)) = x_1
18.10/5.53	   POL(a_{b_1}(x_1)) = 1 + x_1
18.10/5.53	   POL(b_{a_1}(x_1)) = x_1
18.10/5.53	   POL(b_{b_1}(x_1)) = 1 + x_1
18.10/5.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
18.10/5.53	
18.10/5.53	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
18.10/5.53	
18.10/5.53	
18.10/5.53	
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(6)
18.10/5.53	Obligation:
18.10/5.53	Q restricted rewrite system:
18.10/5.53	The TRS R consists of the following rules:
18.10/5.53	
18.10/5.53	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.53	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.53	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.53	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.53	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.53	
18.10/5.53	Q is empty.
18.10/5.53	
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(7) DependencyPairsProof (EQUIVALENT)
18.10/5.53	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
18.10/5.53	----------------------------------------
18.10/5.53	
18.10/5.53	(8)
18.10/5.53	Obligation:
18.10/5.53	Q DP problem:
18.10/5.53	The TRS P consists of the following rules:
18.10/5.53	
18.10/5.53	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.53	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	
18.10/5.54	The TRS R consists of the following rules:
18.10/5.54	
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	
18.10/5.54	Q is empty.
18.10/5.54	We have to consider all minimal (P,Q,R)-chains.
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(9) DependencyGraphProof (EQUIVALENT)
18.10/5.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(10)
18.10/5.54	Obligation:
18.10/5.54	Q DP problem:
18.10/5.54	The TRS P consists of the following rules:
18.10/5.54	
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	
18.10/5.54	The TRS R consists of the following rules:
18.10/5.54	
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	
18.10/5.54	Q is empty.
18.10/5.54	We have to consider all minimal (P,Q,R)-chains.
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(11) QDPOrderProof (EQUIVALENT)
18.10/5.54	We use the reduction pair processor [LPAR04,JAR06].
18.10/5.54	
18.10/5.54	
18.10/5.54	The following pairs can be oriented strictly and are deleted.
18.10/5.54	
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
18.10/5.54	   A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
18.10/5.54	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
18.10/5.54	The remaining pairs can at least be oriented weakly.
18.10/5.54	Used ordering:  Polynomial interpretation [POLO]:
18.10/5.54	
18.10/5.54	   POL(A_{B_1}(x_1)) = x_1
18.10/5.54	   POL(B_{A_1}(x_1)) = x_1
18.10/5.54	   POL(B_{B_1}(x_1)) = x_1
18.10/5.54	   POL(a_{a_1}(x_1)) = 1 + x_1
18.10/5.54	   POL(a_{b_1}(x_1)) = x_1
18.10/5.54	   POL(b_{a_1}(x_1)) = 1 + x_1
18.10/5.54	   POL(b_{b_1}(x_1)) = x_1
18.10/5.54	
18.10/5.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.10/5.54	
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	
18.10/5.54	
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(12)
18.10/5.54	Obligation:
18.10/5.54	Q DP problem:
18.10/5.54	The TRS P consists of the following rules:
18.10/5.54	
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
18.10/5.54	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
18.10/5.54	
18.10/5.54	The TRS R consists of the following rules:
18.10/5.54	
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
18.10/5.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
18.10/5.54	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
18.10/5.54	
18.10/5.54	Q is empty.
18.10/5.54	We have to consider all minimal (P,Q,R)-chains.
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(13) DependencyGraphProof (EQUIVALENT)
18.10/5.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
18.10/5.54	----------------------------------------
18.10/5.54	
18.10/5.54	(14)
18.10/5.54	TRUE
21.20/9.84	EOF