17.59/5.42	YES
17.59/5.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.59/5.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.59/5.43	
17.59/5.43	
17.59/5.43	Termination w.r.t. Q of the given QTRS could be proven:
17.59/5.43	
17.59/5.43	(0) QTRS
17.59/5.43	(1) FlatCCProof [EQUIVALENT, 0 ms]
17.59/5.43	(2) QTRS
17.59/5.43	(3) RootLabelingProof [EQUIVALENT, 1 ms]
17.59/5.43	(4) QTRS
17.59/5.43	(5) DependencyPairsProof [EQUIVALENT, 87 ms]
17.59/5.43	(6) QDP
17.59/5.43	(7) DependencyGraphProof [EQUIVALENT, 7 ms]
17.59/5.43	(8) QDP
17.59/5.43	(9) QDPOrderProof [EQUIVALENT, 76 ms]
17.59/5.43	(10) QDP
17.59/5.43	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
17.59/5.43	(12) TRUE
17.59/5.43	
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(0)
17.59/5.43	Obligation:
17.59/5.43	Q restricted rewrite system:
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
17.59/5.43	   a(a(b(a(x1)))) -> a(a(a(b(x1))))
17.59/5.43	   b(a(a(a(x1)))) -> a(b(a(b(x1))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(1) FlatCCProof (EQUIVALENT)
17.59/5.43	We used flat context closure [ROOTLAB]
17.59/5.43	As Q is empty the flat context closure was sound AND complete.
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(2)
17.59/5.43	Obligation:
17.59/5.43	Q restricted rewrite system:
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   a(a(b(a(x1)))) -> a(a(a(b(x1))))
17.59/5.43	   b(b(b(b(b(x1))))) -> b(a(a(b(b(x1)))))
17.59/5.43	   a(b(b(b(b(x1))))) -> a(a(a(b(b(x1)))))
17.59/5.43	   b(b(a(a(a(x1))))) -> b(a(b(a(b(x1)))))
17.59/5.43	   a(b(a(a(a(x1))))) -> a(a(b(a(b(x1)))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(3) RootLabelingProof (EQUIVALENT)
17.59/5.43	We used plain root labeling [ROOTLAB] with the following heuristic:
17.59/5.43	LabelAll: All function symbols get labeled
17.59/5.43	
17.59/5.43	As Q is empty the root labeling was sound AND complete.
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(4)
17.59/5.43	Obligation:
17.59/5.43	Q restricted rewrite system:
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(5) DependencyPairsProof (EQUIVALENT)
17.59/5.43	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(6)
17.59/5.43	Obligation:
17.59/5.43	Q DP problem:
17.59/5.43	The TRS P consists of the following rules:
17.59/5.43	
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	We have to consider all minimal (P,Q,R)-chains.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(7) DependencyGraphProof (EQUIVALENT)
17.59/5.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(8)
17.59/5.43	Obligation:
17.59/5.43	Q DP problem:
17.59/5.43	The TRS P consists of the following rules:
17.59/5.43	
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	We have to consider all minimal (P,Q,R)-chains.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(9) QDPOrderProof (EQUIVALENT)
17.59/5.43	We use the reduction pair processor [LPAR04,JAR06].
17.59/5.43	
17.59/5.43	
17.59/5.43	The following pairs can be oriented strictly and are deleted.
17.59/5.43	
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.59/5.43	The remaining pairs can at least be oriented weakly.
17.59/5.43	Used ordering:  Polynomial interpretation [POLO]:
17.59/5.43	
17.59/5.43	   POL(A_{A_1}(x_1)) = 1 + x_1
17.59/5.43	   POL(A_{B_1}(x_1)) = 1 + x_1
17.59/5.43	   POL(B_{B_1}(x_1)) = x_1
17.59/5.43	   POL(a_{a_1}(x_1)) = 1 + x_1
17.59/5.43	   POL(a_{b_1}(x_1)) = 1 + x_1
17.59/5.43	   POL(b_{a_1}(x_1)) = 1 + x_1
17.59/5.43	   POL(b_{b_1}(x_1)) = 1 + x_1
17.59/5.43	
17.59/5.43	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.59/5.43	
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	
17.59/5.43	
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(10)
17.59/5.43	Obligation:
17.59/5.43	Q DP problem:
17.59/5.43	The TRS P consists of the following rules:
17.59/5.43	
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.59/5.43	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	
17.59/5.43	The TRS R consists of the following rules:
17.59/5.43	
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
17.59/5.43	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
17.59/5.43	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
17.59/5.43	
17.59/5.43	Q is empty.
17.59/5.43	We have to consider all minimal (P,Q,R)-chains.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(11) DependencyGraphProof (EQUIVALENT)
17.59/5.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.
17.59/5.43	----------------------------------------
17.59/5.43	
17.59/5.43	(12)
17.59/5.43	TRUE
17.75/5.49	EOF