96.66/25.60	YES
97.18/25.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
97.18/25.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
97.18/25.69	
97.18/25.69	
97.18/25.69	Termination w.r.t. Q of the given QTRS could be proven:
97.18/25.69	
97.18/25.69	(0) QTRS
97.18/25.69	(1) FlatCCProof [EQUIVALENT, 0 ms]
97.18/25.69	(2) QTRS
97.18/25.69	(3) RootLabelingProof [EQUIVALENT, 0 ms]
97.18/25.69	(4) QTRS
97.18/25.69	(5) DependencyPairsProof [EQUIVALENT, 83 ms]
97.18/25.69	(6) QDP
97.18/25.69	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
97.18/25.69	(8) QDP
97.18/25.69	(9) QDPOrderProof [EQUIVALENT, 107 ms]
97.18/25.69	(10) QDP
97.18/25.69	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
97.18/25.69	(12) QDP
97.18/25.69	(13) QDPOrderProof [EQUIVALENT, 1604 ms]
97.18/25.69	(14) QDP
97.18/25.69	(15) QDPOrderProof [EQUIVALENT, 2615 ms]
97.18/25.69	(16) QDP
97.18/25.69	(17) PisEmptyProof [EQUIVALENT, 0 ms]
97.18/25.69	(18) YES
97.18/25.69	
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(0)
97.18/25.69	Obligation:
97.18/25.69	Q restricted rewrite system:
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a(a(b(b(x1)))) -> a(a(a(b(x1))))
97.18/25.69	   b(b(a(b(x1)))) -> a(a(b(a(x1))))
97.18/25.69	   b(a(a(a(x1)))) -> b(a(b(b(x1))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(1) FlatCCProof (EQUIVALENT)
97.18/25.69	We used flat context closure [ROOTLAB]
97.18/25.69	As Q is empty the flat context closure was sound AND complete.
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(2)
97.18/25.69	Obligation:
97.18/25.69	Q restricted rewrite system:
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a(a(b(b(x1)))) -> a(a(a(b(x1))))
97.18/25.69	   b(a(a(a(x1)))) -> b(a(b(b(x1))))
97.18/25.69	   a(b(b(a(b(x1))))) -> a(a(a(b(a(x1)))))
97.18/25.69	   b(b(b(a(b(x1))))) -> b(a(a(b(a(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(3) RootLabelingProof (EQUIVALENT)
97.18/25.69	We used plain root labeling [ROOTLAB] with the following heuristic:
97.18/25.69	LabelAll: All function symbols get labeled
97.18/25.69	
97.18/25.69	As Q is empty the root labeling was sound AND complete.
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(4)
97.18/25.69	Obligation:
97.18/25.69	Q restricted rewrite system:
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(5) DependencyPairsProof (EQUIVALENT)
97.18/25.69	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(6)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	The TRS P consists of the following rules:
97.18/25.69	
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(7) DependencyGraphProof (EQUIVALENT)
97.18/25.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 15 less nodes.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(8)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	The TRS P consists of the following rules:
97.18/25.69	
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(9) QDPOrderProof (EQUIVALENT)
97.18/25.69	We use the reduction pair processor [LPAR04,JAR06].
97.18/25.69	
97.18/25.69	
97.18/25.69	The following pairs can be oriented strictly and are deleted.
97.18/25.69	
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
97.18/25.69	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
97.18/25.69	The remaining pairs can at least be oriented weakly.
97.18/25.69	Used ordering:  Polynomial interpretation [POLO]:
97.18/25.69	
97.18/25.69	   POL(A_{A_1}(x_1)) = x_1
97.18/25.69	   POL(A_{B_1}(x_1)) = x_1
97.18/25.69	   POL(B_{A_1}(x_1)) = 1 + x_1
97.18/25.69	   POL(B_{B_1}(x_1)) = 1 + x_1
97.18/25.69	   POL(a_{a_1}(x_1)) = 1 + x_1
97.18/25.69	   POL(a_{b_1}(x_1)) = 1 + x_1
97.18/25.69	   POL(b_{a_1}(x_1)) = 1 + x_1
97.18/25.69	   POL(b_{b_1}(x_1)) = 1 + x_1
97.18/25.69	
97.18/25.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
97.18/25.69	
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(10)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	The TRS P consists of the following rules:
97.18/25.69	
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(11) DependencyGraphProof (EQUIVALENT)
97.18/25.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(12)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	The TRS P consists of the following rules:
97.18/25.69	
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(13) QDPOrderProof (EQUIVALENT)
97.18/25.69	We use the reduction pair processor [LPAR04,JAR06].
97.18/25.69	
97.18/25.69	
97.18/25.69	The following pairs can be oriented strictly and are deleted.
97.18/25.69	
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	The remaining pairs can at least be oriented weakly.
97.18/25.69	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(B_{A_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	
97.18/25.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
97.18/25.69	
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(14)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	The TRS P consists of the following rules:
97.18/25.69	
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(15) QDPOrderProof (EQUIVALENT)
97.18/25.69	We use the reduction pair processor [LPAR04,JAR06].
97.18/25.69	
97.18/25.69	
97.18/25.69	The following pairs can be oriented strictly and are deleted.
97.18/25.69	
97.18/25.69	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	The remaining pairs can at least be oriented weakly.
97.18/25.69	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(B_{A_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, 1A], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	   <<<
97.18/25.69	 POL(b_{a_1}(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
97.18/25.69	>>>
97.18/25.69	
97.18/25.69	
97.18/25.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
97.18/25.69	
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	
97.18/25.69	
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(16)
97.18/25.69	Obligation:
97.18/25.69	Q DP problem:
97.18/25.69	P is empty.
97.18/25.69	The TRS R consists of the following rules:
97.18/25.69	
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
97.18/25.69	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
97.18/25.69	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
97.18/25.69	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
97.18/25.69	
97.18/25.69	Q is empty.
97.18/25.69	We have to consider all minimal (P,Q,R)-chains.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(17) PisEmptyProof (EQUIVALENT)
97.18/25.69	The TRS P is empty. Hence, there is no (P,Q,R) chain.
97.18/25.69	----------------------------------------
97.18/25.69	
97.18/25.69	(18)
97.18/25.69	YES
97.55/25.89	EOF