15.53/4.84	YES
15.81/4.91	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
15.81/4.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.81/4.91	
15.81/4.91	
15.81/4.91	Termination w.r.t. Q of the given QTRS could be proven:
15.81/4.91	
15.81/4.91	(0) QTRS
15.81/4.91	(1) QTRS Reverse [EQUIVALENT, 0 ms]
15.81/4.91	(2) QTRS
15.81/4.91	(3) FlatCCProof [EQUIVALENT, 0 ms]
15.81/4.91	(4) QTRS
15.81/4.91	(5) RootLabelingProof [EQUIVALENT, 0 ms]
15.81/4.91	(6) QTRS
15.81/4.91	(7) QTRSRRRProof [EQUIVALENT, 101 ms]
15.81/4.91	(8) QTRS
15.81/4.91	(9) DependencyPairsProof [EQUIVALENT, 80 ms]
15.81/4.91	(10) QDP
15.81/4.91	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
15.81/4.91	(12) QDP
15.81/4.91	(13) QDPOrderProof [EQUIVALENT, 136 ms]
15.81/4.91	(14) QDP
15.81/4.91	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
15.81/4.91	(16) TRUE
15.81/4.91	
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(0)
15.81/4.91	Obligation:
15.81/4.91	Q restricted rewrite system:
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   a(x1) -> x1
15.81/4.91	   a(a(x1)) -> b(x1)
15.81/4.91	   b(x1) -> x1
15.81/4.91	   b(c(x1)) -> c(c(c(b(a(x1)))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(1) QTRS Reverse (EQUIVALENT)
15.81/4.91	We applied the QTRS Reverse Processor [REVERSE].
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(2)
15.81/4.91	Obligation:
15.81/4.91	Q restricted rewrite system:
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   a(x1) -> x1
15.81/4.91	   a(a(x1)) -> b(x1)
15.81/4.91	   b(x1) -> x1
15.81/4.91	   c(b(x1)) -> a(b(c(c(c(x1)))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(3) FlatCCProof (EQUIVALENT)
15.81/4.91	We used flat context closure [ROOTLAB]
15.81/4.91	As Q is empty the flat context closure was sound AND complete.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(4)
15.81/4.91	Obligation:
15.81/4.91	Q restricted rewrite system:
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   a(a(x1)) -> a(x1)
15.81/4.91	   b(a(x1)) -> b(x1)
15.81/4.91	   c(a(x1)) -> c(x1)
15.81/4.91	   a(a(a(x1))) -> a(b(x1))
15.81/4.91	   b(a(a(x1))) -> b(b(x1))
15.81/4.91	   c(a(a(x1))) -> c(b(x1))
15.81/4.91	   a(b(x1)) -> a(x1)
15.81/4.91	   b(b(x1)) -> b(x1)
15.81/4.91	   c(b(x1)) -> c(x1)
15.81/4.91	   a(c(b(x1))) -> a(a(b(c(c(c(x1))))))
15.81/4.91	   b(c(b(x1))) -> b(a(b(c(c(c(x1))))))
15.81/4.91	   c(c(b(x1))) -> c(a(b(c(c(c(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(5) RootLabelingProof (EQUIVALENT)
15.81/4.91	We used plain root labeling [ROOTLAB] with the following heuristic:
15.81/4.91	LabelAll: All function symbols get labeled
15.81/4.91	
15.81/4.91	As Q is empty the root labeling was sound AND complete.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(6)
15.81/4.91	Obligation:
15.81/4.91	Q restricted rewrite system:
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
15.81/4.91	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
15.81/4.91	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
15.81/4.91	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1))
15.81/4.91	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1))
15.81/4.91	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1))
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
15.81/4.91	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
15.81/4.91	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
15.81/4.91	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(7) QTRSRRRProof (EQUIVALENT)
15.81/4.91	Used ordering:
15.81/4.91	Polynomial interpretation [POLO]:
15.81/4.91	
15.81/4.91	   POL(a_{a_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(a_{b_1}(x_1)) = x_1
15.81/4.91	   POL(a_{c_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{a_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{b_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{c_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(c_{a_1}(x_1)) = x_1
15.81/4.91	   POL(c_{b_1}(x_1)) = x_1
15.81/4.91	   POL(c_{c_1}(x_1)) = x_1
15.81/4.91	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
15.81/4.91	
15.81/4.91	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
15.81/4.91	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
15.81/4.91	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
15.81/4.91	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
15.81/4.91	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
15.81/4.91	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1))
15.81/4.91	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1))
15.81/4.91	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1))
15.81/4.91	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1))
15.81/4.91	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
15.81/4.91	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
15.81/4.91	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
15.81/4.91	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
15.81/4.91	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
15.81/4.91	
15.81/4.91	
15.81/4.91	
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(8)
15.81/4.91	Obligation:
15.81/4.91	Q restricted rewrite system:
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(9) DependencyPairsProof (EQUIVALENT)
15.81/4.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(10)
15.81/4.91	Obligation:
15.81/4.91	Q DP problem:
15.81/4.91	The TRS P consists of the following rules:
15.81/4.91	
15.81/4.91	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	We have to consider all minimal (P,Q,R)-chains.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(11) DependencyGraphProof (EQUIVALENT)
15.81/4.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(12)
15.81/4.91	Obligation:
15.81/4.91	Q DP problem:
15.81/4.91	The TRS P consists of the following rules:
15.81/4.91	
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	We have to consider all minimal (P,Q,R)-chains.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(13) QDPOrderProof (EQUIVALENT)
15.81/4.91	We use the reduction pair processor [LPAR04,JAR06].
15.81/4.91	
15.81/4.91	
15.81/4.91	The following pairs can be oriented strictly and are deleted.
15.81/4.91	
15.81/4.91	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1)))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
15.81/4.91	The remaining pairs can at least be oriented weakly.
15.81/4.91	Used ordering:  Polynomial interpretation [POLO]:
15.81/4.91	
15.81/4.91	   POL(A_{B_1}(x_1)) = x_1
15.81/4.91	   POL(A_{C_1}(x_1)) = x_1
15.81/4.91	   POL(B_{C_1}(x_1)) = x_1
15.81/4.91	   POL(C_{C_1}(x_1)) = x_1
15.81/4.91	   POL(a_{a_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(a_{b_1}(x_1)) = x_1
15.81/4.91	   POL(a_{c_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{a_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{b_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(b_{c_1}(x_1)) = 1 + x_1
15.81/4.91	   POL(c_{a_1}(x_1)) = x_1
15.81/4.91	   POL(c_{b_1}(x_1)) = x_1
15.81/4.91	   POL(c_{c_1}(x_1)) = x_1
15.81/4.91	
15.81/4.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
15.81/4.91	
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	
15.81/4.91	
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(14)
15.81/4.91	Obligation:
15.81/4.91	Q DP problem:
15.81/4.91	The TRS P consists of the following rules:
15.81/4.91	
15.81/4.91	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))
15.81/4.91	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))
15.81/4.91	
15.81/4.91	The TRS R consists of the following rules:
15.81/4.91	
15.81/4.91	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
15.81/4.91	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
15.81/4.91	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
15.81/4.91	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
15.81/4.91	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
15.81/4.91	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
15.81/4.91	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))
15.81/4.91	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))
15.81/4.91	
15.81/4.91	Q is empty.
15.81/4.91	We have to consider all minimal (P,Q,R)-chains.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(15) DependencyGraphProof (EQUIVALENT)
15.81/4.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 9 less nodes.
15.81/4.91	----------------------------------------
15.81/4.91	
15.81/4.91	(16)
15.81/4.91	TRUE
16.00/5.00	EOF