39.20/10.91	YES
39.87/11.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
39.87/11.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.87/11.03	
39.87/11.03	
39.87/11.03	Termination w.r.t. Q of the given QTRS could be proven:
39.87/11.03	
39.87/11.03	(0) QTRS
39.87/11.03	(1) FlatCCProof [EQUIVALENT, 0 ms]
39.87/11.03	(2) QTRS
39.87/11.03	(3) RootLabelingProof [EQUIVALENT, 0 ms]
39.87/11.03	(4) QTRS
39.87/11.03	(5) DependencyPairsProof [EQUIVALENT, 36 ms]
39.87/11.03	(6) QDP
39.87/11.03	(7) DependencyGraphProof [EQUIVALENT, 4 ms]
39.87/11.03	(8) QDP
39.87/11.03	(9) QDPOrderProof [EQUIVALENT, 141 ms]
39.87/11.03	(10) QDP
39.87/11.03	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
39.87/11.03	(12) QDP
39.87/11.03	(13) QDPOrderProof [EQUIVALENT, 255 ms]
39.87/11.03	(14) QDP
39.87/11.03	(15) PisEmptyProof [EQUIVALENT, 0 ms]
39.87/11.03	(16) YES
39.87/11.03	
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(0)
39.87/11.03	Obligation:
39.87/11.03	Q restricted rewrite system:
39.87/11.03	The TRS R consists of the following rules:
39.87/11.03	
39.87/11.03	   a(a(a(b(x1)))) -> b(b(a(a(x1))))
39.87/11.03	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
39.87/11.03	   b(a(b(a(x1)))) -> b(b(a(b(x1))))
39.87/11.03	
39.87/11.03	Q is empty.
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(1) FlatCCProof (EQUIVALENT)
39.87/11.03	We used flat context closure [ROOTLAB]
39.87/11.03	As Q is empty the flat context closure was sound AND complete.
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(2)
39.87/11.03	Obligation:
39.87/11.03	Q restricted rewrite system:
39.87/11.03	The TRS R consists of the following rules:
39.87/11.03	
39.87/11.03	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
39.87/11.03	   b(a(b(a(x1)))) -> b(b(a(b(x1))))
39.87/11.03	   a(a(a(a(b(x1))))) -> a(b(b(a(a(x1)))))
39.87/11.03	   b(a(a(a(b(x1))))) -> b(b(b(a(a(x1)))))
39.87/11.03	
39.87/11.03	Q is empty.
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(3) RootLabelingProof (EQUIVALENT)
39.87/11.03	We used plain root labeling [ROOTLAB] with the following heuristic:
39.87/11.03	LabelAll: All function symbols get labeled
39.87/11.03	
39.87/11.03	As Q is empty the root labeling was sound AND complete.
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(4)
39.87/11.03	Obligation:
39.87/11.03	Q restricted rewrite system:
39.87/11.03	The TRS R consists of the following rules:
39.87/11.03	
39.87/11.03	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.03	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.03	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.03	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.03	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.03	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.03	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.03	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.03	
39.87/11.03	Q is empty.
39.87/11.03	
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(5) DependencyPairsProof (EQUIVALENT)
39.87/11.03	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.87/11.03	----------------------------------------
39.87/11.03	
39.87/11.03	(6)
39.87/11.03	Obligation:
39.87/11.03	Q DP problem:
39.87/11.03	The TRS P consists of the following rules:
39.87/11.03	
39.87/11.03	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.03	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.03	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.03	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	
39.87/11.04	The TRS R consists of the following rules:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	Q is empty.
39.87/11.04	We have to consider all minimal (P,Q,R)-chains.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(7) DependencyGraphProof (EQUIVALENT)
39.87/11.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(8)
39.87/11.04	Obligation:
39.87/11.04	Q DP problem:
39.87/11.04	The TRS P consists of the following rules:
39.87/11.04	
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	
39.87/11.04	The TRS R consists of the following rules:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	Q is empty.
39.87/11.04	We have to consider all minimal (P,Q,R)-chains.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(9) QDPOrderProof (EQUIVALENT)
39.87/11.04	We use the reduction pair processor [LPAR04,JAR06].
39.87/11.04	
39.87/11.04	
39.87/11.04	The following pairs can be oriented strictly and are deleted.
39.87/11.04	
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.87/11.04	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
39.87/11.04	The remaining pairs can at least be oriented weakly.
39.87/11.04	Used ordering:  Polynomial interpretation [POLO]:
39.87/11.04	
39.87/11.04	   POL(A_{A_1}(x_1)) = x_1
39.87/11.04	   POL(A_{B_1}(x_1)) = x_1
39.87/11.04	   POL(B_{A_1}(x_1)) = x_1
39.87/11.04	   POL(a_{a_1}(x_1)) = 1 + x_1
39.87/11.04	   POL(a_{b_1}(x_1)) = 1 + x_1
39.87/11.04	   POL(b_{a_1}(x_1)) = 1 + x_1
39.87/11.04	   POL(b_{b_1}(x_1)) = 1 + x_1
39.87/11.04	
39.87/11.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(10)
39.87/11.04	Obligation:
39.87/11.04	Q DP problem:
39.87/11.04	The TRS P consists of the following rules:
39.87/11.04	
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	The TRS R consists of the following rules:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	Q is empty.
39.87/11.04	We have to consider all minimal (P,Q,R)-chains.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(11) DependencyGraphProof (EQUIVALENT)
39.87/11.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(12)
39.87/11.04	Obligation:
39.87/11.04	Q DP problem:
39.87/11.04	The TRS P consists of the following rules:
39.87/11.04	
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	
39.87/11.04	The TRS R consists of the following rules:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	Q is empty.
39.87/11.04	We have to consider all minimal (P,Q,R)-chains.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(13) QDPOrderProof (EQUIVALENT)
39.87/11.04	We use the reduction pair processor [LPAR04,JAR06].
39.87/11.04	
39.87/11.04	
39.87/11.04	The following pairs can be oriented strictly and are deleted.
39.87/11.04	
39.87/11.04	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	The remaining pairs can at least be oriented weakly.
39.87/11.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.87/11.04	
39.87/11.04	   <<<
39.87/11.04	 POL(A_{B_1}(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 1A]] 	* 	x_1
39.87/11.04	>>>
39.87/11.04	
39.87/11.04	   <<<
39.87/11.04	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
39.87/11.04	>>>
39.87/11.04	
39.87/11.04	   <<<
39.87/11.04	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
39.87/11.04	>>>
39.87/11.04	
39.87/11.04	   <<<
39.87/11.04	 POL(a_{b_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
39.87/11.04	>>>
39.87/11.04	
39.87/11.04	   <<<
39.87/11.04	 POL(a_{a_1}(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
39.87/11.04	>>>
39.87/11.04	
39.87/11.04	
39.87/11.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(14)
39.87/11.04	Obligation:
39.87/11.04	Q DP problem:
39.87/11.04	P is empty.
39.87/11.04	The TRS R consists of the following rules:
39.87/11.04	
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
39.87/11.04	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
39.87/11.04	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.87/11.04	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.87/11.04	
39.87/11.04	Q is empty.
39.87/11.04	We have to consider all minimal (P,Q,R)-chains.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(15) PisEmptyProof (EQUIVALENT)
39.87/11.04	The TRS P is empty. Hence, there is no (P,Q,R) chain.
39.87/11.04	----------------------------------------
39.87/11.04	
39.87/11.04	(16)
39.87/11.04	YES
40.02/11.13	EOF