15.74/5.10	YES
16.10/5.12	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
16.10/5.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.10/5.12	
16.10/5.12	
16.10/5.12	Termination w.r.t. Q of the given QTRS could be proven:
16.10/5.12	
16.10/5.12	(0) QTRS
16.10/5.12	(1) FlatCCProof [EQUIVALENT, 0 ms]
16.10/5.12	(2) QTRS
16.10/5.12	(3) RootLabelingProof [EQUIVALENT, 0 ms]
16.10/5.12	(4) QTRS
16.10/5.12	(5) QTRSRRRProof [EQUIVALENT, 32 ms]
16.10/5.12	(6) QTRS
16.10/5.12	(7) DependencyPairsProof [EQUIVALENT, 129 ms]
16.10/5.12	(8) QDP
16.10/5.12	(9) QDPOrderProof [EQUIVALENT, 255 ms]
16.10/5.12	(10) QDP
16.10/5.12	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
16.10/5.12	(12) TRUE
16.10/5.12	
16.10/5.12	
16.10/5.12	----------------------------------------
16.10/5.12	
16.10/5.12	(0)
16.10/5.12	Obligation:
16.10/5.12	Q restricted rewrite system:
16.10/5.12	The TRS R consists of the following rules:
16.10/5.12	
16.10/5.12	   a(a(b(x1))) -> b(a(b(x1)))
16.10/5.12	   b(a(x1)) -> a(b(b(x1)))
16.10/5.12	   b(c(a(x1))) -> c(c(a(a(b(x1)))))
16.10/5.12	
16.10/5.12	Q is empty.
16.10/5.12	
16.10/5.12	----------------------------------------
16.10/5.12	
16.10/5.12	(1) FlatCCProof (EQUIVALENT)
16.10/5.12	We used flat context closure [ROOTLAB]
16.10/5.12	As Q is empty the flat context closure was sound AND complete.
16.10/5.12	
16.10/5.12	----------------------------------------
16.10/5.12	
16.10/5.12	(2)
16.10/5.12	Obligation:
16.10/5.12	Q restricted rewrite system:
16.10/5.12	The TRS R consists of the following rules:
16.10/5.12	
16.10/5.12	   a(a(a(b(x1)))) -> a(b(a(b(x1))))
16.10/5.12	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
16.10/5.12	   c(a(a(b(x1)))) -> c(b(a(b(x1))))
16.10/5.12	   a(b(a(x1))) -> a(a(b(b(x1))))
16.10/5.12	   b(b(a(x1))) -> b(a(b(b(x1))))
16.10/5.12	   c(b(a(x1))) -> c(a(b(b(x1))))
16.10/5.12	   a(b(c(a(x1)))) -> a(c(c(a(a(b(x1))))))
16.10/5.12	   b(b(c(a(x1)))) -> b(c(c(a(a(b(x1))))))
16.10/5.12	   c(b(c(a(x1)))) -> c(c(c(a(a(b(x1))))))
16.10/5.12	
16.10/5.12	Q is empty.
16.10/5.12	
16.10/5.12	----------------------------------------
16.10/5.12	
16.10/5.12	(3) RootLabelingProof (EQUIVALENT)
16.10/5.12	We used plain root labeling [ROOTLAB] with the following heuristic:
16.10/5.12	LabelAll: All function symbols get labeled
16.10/5.12	
16.10/5.12	As Q is empty the root labeling was sound AND complete.
16.10/5.12	
16.10/5.12	----------------------------------------
16.10/5.12	
16.10/5.12	(4)
16.10/5.12	Obligation:
16.10/5.12	Q restricted rewrite system:
16.10/5.12	The TRS R consists of the following rules:
16.10/5.12	
16.10/5.12	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.12	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.12	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.12	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	Q is empty.
16.10/5.13	
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(5) QTRSRRRProof (EQUIVALENT)
16.10/5.13	Used ordering:
16.10/5.13	Polynomial interpretation [POLO]:
16.10/5.13	
16.10/5.13	   POL(a_{a_1}(x_1)) = x_1
16.10/5.13	   POL(a_{b_1}(x_1)) = x_1
16.10/5.13	   POL(a_{c_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(b_{a_1}(x_1)) = x_1
16.10/5.13	   POL(b_{b_1}(x_1)) = x_1
16.10/5.13	   POL(b_{c_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(c_{a_1}(x_1)) = x_1
16.10/5.13	   POL(c_{b_1}(x_1)) = x_1
16.10/5.13	   POL(c_{c_1}(x_1)) = x_1
16.10/5.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
16.10/5.13	
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	
16.10/5.13	
16.10/5.13	
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(6)
16.10/5.13	Obligation:
16.10/5.13	Q restricted rewrite system:
16.10/5.13	The TRS R consists of the following rules:
16.10/5.13	
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	Q is empty.
16.10/5.13	
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(7) DependencyPairsProof (EQUIVALENT)
16.10/5.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(8)
16.10/5.13	Obligation:
16.10/5.13	Q DP problem:
16.10/5.13	The TRS P consists of the following rules:
16.10/5.13	
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.10/5.13	
16.10/5.13	The TRS R consists of the following rules:
16.10/5.13	
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	Q is empty.
16.10/5.13	We have to consider all minimal (P,Q,R)-chains.
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(9) QDPOrderProof (EQUIVALENT)
16.10/5.13	We use the reduction pair processor [LPAR04,JAR06].
16.10/5.13	
16.10/5.13	
16.10/5.13	The following pairs can be oriented strictly and are deleted.
16.10/5.13	
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   B_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.10/5.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.10/5.13	The remaining pairs can at least be oriented weakly.
16.10/5.13	Used ordering:  Polynomial interpretation [POLO]:
16.10/5.13	
16.10/5.13	   POL(A_{A_1}(x_1)) = x_1
16.10/5.13	   POL(A_{B_1}(x_1)) = x_1
16.10/5.13	   POL(B_{A_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(B_{B_1}(x_1)) = x_1
16.10/5.13	   POL(C_{A_1}(x_1)) = x_1
16.10/5.13	   POL(C_{B_1}(x_1)) = x_1
16.10/5.13	   POL(a_{a_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(a_{b_1}(x_1)) = x_1
16.10/5.13	   POL(a_{c_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(b_{a_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(b_{b_1}(x_1)) = x_1
16.10/5.13	   POL(b_{c_1}(x_1)) = 1 + x_1
16.10/5.13	   POL(c_{a_1}(x_1)) = x_1
16.10/5.13	   POL(c_{b_1}(x_1)) = 0
16.10/5.13	   POL(c_{c_1}(x_1)) = 0
16.10/5.13	
16.10/5.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
16.10/5.13	
16.10/5.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(10)
16.10/5.13	Obligation:
16.10/5.13	Q DP problem:
16.10/5.13	The TRS P consists of the following rules:
16.10/5.13	
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	
16.10/5.13	The TRS R consists of the following rules:
16.10/5.13	
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.10/5.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
16.10/5.13	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))))
16.10/5.13	
16.10/5.13	Q is empty.
16.10/5.13	We have to consider all minimal (P,Q,R)-chains.
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(11) DependencyGraphProof (EQUIVALENT)
16.10/5.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 21 less nodes.
16.10/5.13	----------------------------------------
16.10/5.13	
16.10/5.13	(12)
16.10/5.13	TRUE
16.42/5.27	EOF