20.91/6.08	YES
21.23/6.18	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.23/6.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.23/6.18	
21.23/6.18	
21.23/6.18	Termination w.r.t. Q of the given QTRS could be proven:
21.23/6.18	
21.23/6.18	(0) QTRS
21.23/6.18	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.23/6.18	(2) QTRS
21.23/6.18	(3) FlatCCProof [EQUIVALENT, 0 ms]
21.23/6.18	(4) QTRS
21.23/6.18	(5) RootLabelingProof [EQUIVALENT, 0 ms]
21.23/6.18	(6) QTRS
21.23/6.18	(7) QTRSRRRProof [EQUIVALENT, 43 ms]
21.23/6.18	(8) QTRS
21.23/6.18	(9) DependencyPairsProof [EQUIVALENT, 47 ms]
21.23/6.18	(10) QDP
21.23/6.18	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
21.23/6.18	(12) QDP
21.23/6.18	(13) QDPOrderProof [EQUIVALENT, 159 ms]
21.23/6.18	(14) QDP
21.23/6.18	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
21.23/6.18	(16) TRUE
21.23/6.18	
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(0)
21.23/6.18	Obligation:
21.23/6.18	Q restricted rewrite system:
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a(x1) -> b(x1)
21.23/6.18	   a(b(x1)) -> x1
21.23/6.18	   b(a(x1)) -> a(c(a(b(x1))))
21.23/6.18	   c(c(x1)) -> x1
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(1) QTRS Reverse (EQUIVALENT)
21.23/6.18	We applied the QTRS Reverse Processor [REVERSE].
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(2)
21.23/6.18	Obligation:
21.23/6.18	Q restricted rewrite system:
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a(x1) -> b(x1)
21.23/6.18	   b(a(x1)) -> x1
21.23/6.18	   a(b(x1)) -> b(a(c(a(x1))))
21.23/6.18	   c(c(x1)) -> x1
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(3) FlatCCProof (EQUIVALENT)
21.23/6.18	We used flat context closure [ROOTLAB]
21.23/6.18	As Q is empty the flat context closure was sound AND complete.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(4)
21.23/6.18	Obligation:
21.23/6.18	Q restricted rewrite system:
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a(a(x1)) -> a(b(x1))
21.23/6.18	   b(a(x1)) -> b(b(x1))
21.23/6.18	   c(a(x1)) -> c(b(x1))
21.23/6.18	   a(b(a(x1))) -> a(x1)
21.23/6.18	   b(b(a(x1))) -> b(x1)
21.23/6.18	   c(b(a(x1))) -> c(x1)
21.23/6.18	   a(a(b(x1))) -> a(b(a(c(a(x1)))))
21.23/6.18	   b(a(b(x1))) -> b(b(a(c(a(x1)))))
21.23/6.18	   c(a(b(x1))) -> c(b(a(c(a(x1)))))
21.23/6.18	   a(c(c(x1))) -> a(x1)
21.23/6.18	   b(c(c(x1))) -> b(x1)
21.23/6.18	   c(c(c(x1))) -> c(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(5) RootLabelingProof (EQUIVALENT)
21.23/6.18	We used plain root labeling [ROOTLAB] with the following heuristic:
21.23/6.18	LabelAll: All function symbols get labeled
21.23/6.18	
21.23/6.18	As Q is empty the root labeling was sound AND complete.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(6)
21.23/6.18	Obligation:
21.23/6.18	Q restricted rewrite system:
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(7) QTRSRRRProof (EQUIVALENT)
21.23/6.18	Used ordering:
21.23/6.18	Polynomial interpretation [POLO]:
21.23/6.18	
21.23/6.18	   POL(a_{a_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(a_{b_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(a_{c_1}(x_1)) = x_1
21.23/6.18	   POL(b_{a_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(b_{b_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(b_{c_1}(x_1)) = x_1
21.23/6.18	   POL(c_{a_1}(x_1)) = x_1
21.23/6.18	   POL(c_{b_1}(x_1)) = x_1
21.23/6.18	   POL(c_{c_1}(x_1)) = 1 + x_1
21.23/6.18	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
21.23/6.18	
21.23/6.18	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
21.23/6.18	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	
21.23/6.18	
21.23/6.18	
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(8)
21.23/6.18	Obligation:
21.23/6.18	Q restricted rewrite system:
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(9) DependencyPairsProof (EQUIVALENT)
21.23/6.18	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(10)
21.23/6.18	Obligation:
21.23/6.18	Q DP problem:
21.23/6.18	The TRS P consists of the following rules:
21.23/6.18	
21.23/6.18	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
21.23/6.18	
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	We have to consider all minimal (P,Q,R)-chains.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(11) DependencyGraphProof (EQUIVALENT)
21.23/6.18	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(12)
21.23/6.18	Obligation:
21.23/6.18	Q DP problem:
21.23/6.18	The TRS P consists of the following rules:
21.23/6.18	
21.23/6.18	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	We have to consider all minimal (P,Q,R)-chains.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(13) QDPOrderProof (EQUIVALENT)
21.23/6.18	We use the reduction pair processor [LPAR04,JAR06].
21.23/6.18	
21.23/6.18	
21.23/6.18	The following pairs can be oriented strictly and are deleted.
21.23/6.18	
21.23/6.18	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
21.23/6.18	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
21.23/6.18	   A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
21.23/6.18	The remaining pairs can at least be oriented weakly.
21.23/6.18	Used ordering:  Polynomial interpretation [POLO]:
21.23/6.18	
21.23/6.18	   POL(A_{A_1}(x_1)) = x_1
21.23/6.18	   POL(A_{C_1}(x_1)) = x_1
21.23/6.18	   POL(B_{A_1}(x_1)) = x_1
21.23/6.18	   POL(B_{C_1}(x_1)) = x_1
21.23/6.18	   POL(C_{A_1}(x_1)) = x_1
21.23/6.18	   POL(a_{a_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(a_{b_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(a_{c_1}(x_1)) = x_1
21.23/6.18	   POL(b_{a_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(b_{b_1}(x_1)) = 1 + x_1
21.23/6.18	   POL(b_{c_1}(x_1)) = x_1
21.23/6.18	   POL(c_{a_1}(x_1)) = x_1
21.23/6.18	   POL(c_{b_1}(x_1)) = x_1
21.23/6.18	   POL(c_{c_1}(x_1)) = 1 + x_1
21.23/6.18	
21.23/6.18	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	
21.23/6.18	
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(14)
21.23/6.18	Obligation:
21.23/6.18	Q DP problem:
21.23/6.18	The TRS P consists of the following rules:
21.23/6.18	
21.23/6.18	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
21.23/6.18	
21.23/6.18	The TRS R consists of the following rules:
21.23/6.18	
21.23/6.18	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
21.23/6.18	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
21.23/6.18	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
21.23/6.18	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
21.23/6.18	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
21.23/6.18	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1))
21.23/6.18	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1))
21.23/6.18	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1))
21.23/6.18	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
21.23/6.18	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
21.23/6.18	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
21.23/6.18	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
21.23/6.18	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
21.23/6.18	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
21.23/6.18	
21.23/6.18	Q is empty.
21.23/6.18	We have to consider all minimal (P,Q,R)-chains.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(15) DependencyGraphProof (EQUIVALENT)
21.23/6.18	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
21.23/6.18	----------------------------------------
21.23/6.18	
21.23/6.18	(16)
21.23/6.18	TRUE
21.47/6.24	EOF