39.91/11.20	YES
39.91/11.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
39.91/11.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.91/11.23	
39.91/11.23	
39.91/11.23	Termination w.r.t. Q of the given QTRS could be proven:
39.91/11.23	
39.91/11.23	(0) QTRS
39.91/11.23	(1) QTRS Reverse [EQUIVALENT, 0 ms]
39.91/11.23	(2) QTRS
39.91/11.23	(3) FlatCCProof [EQUIVALENT, 0 ms]
39.91/11.23	(4) QTRS
39.91/11.23	(5) RootLabelingProof [EQUIVALENT, 0 ms]
39.91/11.23	(6) QTRS
39.91/11.23	(7) QTRSRRRProof [EQUIVALENT, 22 ms]
39.91/11.23	(8) QTRS
39.91/11.23	(9) DependencyPairsProof [EQUIVALENT, 3 ms]
39.91/11.23	(10) QDP
39.91/11.23	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
39.91/11.23	(12) QDP
39.91/11.23	(13) QDPOrderProof [EQUIVALENT, 70 ms]
39.91/11.23	(14) QDP
39.91/11.23	(15) QDPOrderProof [EQUIVALENT, 88 ms]
39.91/11.23	(16) QDP
39.91/11.23	(17) DependencyGraphProof [EQUIVALENT, 0 ms]
39.91/11.23	(18) QDP
39.91/11.23	(19) QDPOrderProof [EQUIVALENT, 43 ms]
39.91/11.23	(20) QDP
39.91/11.23	(21) QDPOrderProof [EQUIVALENT, 633 ms]
39.91/11.23	(22) QDP
39.91/11.23	(23) DependencyGraphProof [EQUIVALENT, 0 ms]
39.91/11.23	(24) TRUE
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(0)
39.91/11.23	Obligation:
39.91/11.23	Q restricted rewrite system:
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a(x1) -> b(c(x1))
39.91/11.23	   b(a(b(x1))) -> x1
39.91/11.23	   c(c(x1)) -> a(a(a(b(x1))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(1) QTRS Reverse (EQUIVALENT)
39.91/11.23	We applied the QTRS Reverse Processor [REVERSE].
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(2)
39.91/11.23	Obligation:
39.91/11.23	Q restricted rewrite system:
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a(x1) -> c(b(x1))
39.91/11.23	   b(a(b(x1))) -> x1
39.91/11.23	   c(c(x1)) -> b(a(a(a(x1))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(3) FlatCCProof (EQUIVALENT)
39.91/11.23	We used flat context closure [ROOTLAB]
39.91/11.23	As Q is empty the flat context closure was sound AND complete.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(4)
39.91/11.23	Obligation:
39.91/11.23	Q restricted rewrite system:
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a(a(x1)) -> a(c(b(x1)))
39.91/11.23	   c(a(x1)) -> c(c(b(x1)))
39.91/11.23	   b(a(x1)) -> b(c(b(x1)))
39.91/11.23	   a(b(a(b(x1)))) -> a(x1)
39.91/11.23	   c(b(a(b(x1)))) -> c(x1)
39.91/11.23	   b(b(a(b(x1)))) -> b(x1)
39.91/11.23	   a(c(c(x1))) -> a(b(a(a(a(x1)))))
39.91/11.23	   c(c(c(x1))) -> c(b(a(a(a(x1)))))
39.91/11.23	   b(c(c(x1))) -> b(b(a(a(a(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(5) RootLabelingProof (EQUIVALENT)
39.91/11.23	We used plain root labeling [ROOTLAB] with the following heuristic:
39.91/11.23	LabelAll: All function symbols get labeled
39.91/11.23	
39.91/11.23	As Q is empty the root labeling was sound AND complete.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(6)
39.91/11.23	Obligation:
39.91/11.23	Q restricted rewrite system:
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(7) QTRSRRRProof (EQUIVALENT)
39.91/11.23	Used ordering:
39.91/11.23	Polynomial interpretation [POLO]:
39.91/11.23	
39.91/11.23	   POL(a_{a_1}(x_1)) = 1 + x_1
39.91/11.23	   POL(a_{b_1}(x_1)) = 1 + x_1
39.91/11.23	   POL(a_{c_1}(x_1)) = 2 + x_1
39.91/11.23	   POL(b_{a_1}(x_1)) = x_1
39.91/11.23	   POL(b_{b_1}(x_1)) = x_1
39.91/11.23	   POL(b_{c_1}(x_1)) = 1 + x_1
39.91/11.23	   POL(c_{a_1}(x_1)) = 1 + x_1
39.91/11.23	   POL(c_{b_1}(x_1)) = x_1
39.91/11.23	   POL(c_{c_1}(x_1)) = 2 + x_1
39.91/11.23	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
39.91/11.23	
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
39.91/11.23	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
39.91/11.23	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
39.91/11.23	
39.91/11.23	
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(8)
39.91/11.23	Obligation:
39.91/11.23	Q restricted rewrite system:
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(9) DependencyPairsProof (EQUIVALENT)
39.91/11.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(10)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
39.91/11.23	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   B_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
39.91/11.23	   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(11) DependencyGraphProof (EQUIVALENT)
39.91/11.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(12)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(13) QDPOrderProof (EQUIVALENT)
39.91/11.23	We use the reduction pair processor [LPAR04,JAR06].
39.91/11.23	
39.91/11.23	
39.91/11.23	The following pairs can be oriented strictly and are deleted.
39.91/11.23	
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	The remaining pairs can at least be oriented weakly.
39.91/11.23	Used ordering:  Polynomial interpretation [POLO]:
39.91/11.23	
39.91/11.23	   POL(A_{A_1}(x_1)) = 1
39.91/11.23	   POL(A_{C_1}(x_1)) = x_1
39.91/11.23	   POL(B_{A_1}(x_1)) = 1
39.91/11.23	   POL(B_{C_1}(x_1)) = 1
39.91/11.23	   POL(C_{A_1}(x_1)) = 1
39.91/11.23	   POL(C_{B_1}(x_1)) = 1
39.91/11.23	   POL(a_{a_1}(x_1)) = 0
39.91/11.23	   POL(a_{b_1}(x_1)) = 0
39.91/11.23	   POL(a_{c_1}(x_1)) = 0
39.91/11.23	   POL(b_{a_1}(x_1)) = 1
39.91/11.23	   POL(b_{b_1}(x_1)) = 0
39.91/11.23	   POL(b_{c_1}(x_1)) = 0
39.91/11.23	   POL(c_{a_1}(x_1)) = 1
39.91/11.23	   POL(c_{b_1}(x_1)) = x_1
39.91/11.23	   POL(c_{c_1}(x_1)) = 1
39.91/11.23	
39.91/11.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.91/11.23	
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(14)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(15) QDPOrderProof (EQUIVALENT)
39.91/11.23	We use the reduction pair processor [LPAR04,JAR06].
39.91/11.23	
39.91/11.23	
39.91/11.23	The following pairs can be oriented strictly and are deleted.
39.91/11.23	
39.91/11.23	   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
39.91/11.23	The remaining pairs can at least be oriented weakly.
39.91/11.23	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
39.91/11.23	
39.91/11.23	POL( A_{A_1}_1(x_1) ) = x_1 + 1
39.91/11.23	POL( B_{A_1}_1(x_1) ) = x_1
39.91/11.23	POL( C_{B_1}_1(x_1) ) = x_1
39.91/11.23	POL( b_{a_1}_1(x_1) ) = x_1 + 1
39.91/11.23	POL( b_{b_1}_1(x_1) ) = max{0, x_1 - 2}
39.91/11.23	POL( A_{C_1}_1(x_1) ) = x_1
39.91/11.23	POL( c_{b_1}_1(x_1) ) = x_1
39.91/11.23	POL( B_{C_1}_1(x_1) ) = x_1
39.91/11.23	POL( a_{a_1}_1(x_1) ) = x_1 + 1
39.91/11.23	POL( a_{b_1}_1(x_1) ) = x_1
39.91/11.23	POL( a_{c_1}_1(x_1) ) = x_1 + 1
39.91/11.23	POL( b_{c_1}_1(x_1) ) = x_1 + 1
39.91/11.23	POL( c_{a_1}_1(x_1) ) = x_1 + 2
39.91/11.23	POL( c_{c_1}_1(x_1) ) = x_1 + 2
39.91/11.23	POL( C_{A_1}_1(x_1) ) = x_1 + 1
39.91/11.23	
39.91/11.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(16)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(17) DependencyGraphProof (EQUIVALENT)
39.91/11.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(18)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(19) QDPOrderProof (EQUIVALENT)
39.91/11.23	We use the reduction pair processor [LPAR04,JAR06].
39.91/11.23	
39.91/11.23	
39.91/11.23	The following pairs can be oriented strictly and are deleted.
39.91/11.23	
39.91/11.23	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	The remaining pairs can at least be oriented weakly.
39.91/11.23	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
39.91/11.23	
39.91/11.23	POL( B_{A_1}_1(x_1) ) = 1
39.91/11.23	POL( B_{C_1}_1(x_1) ) = max{0, 2x_1 - 1}
39.91/11.23	POL( b_{a_1}_1(x_1) ) = 1
39.91/11.23	POL( b_{b_1}_1(x_1) ) = max{0, -2}
39.91/11.23	POL( c_{b_1}_1(x_1) ) = max{0, 2x_1 - 1}
39.91/11.23	POL( a_{a_1}_1(x_1) ) = max{0, 2x_1 - 2}
39.91/11.23	POL( a_{b_1}_1(x_1) ) = 0
39.91/11.23	POL( a_{c_1}_1(x_1) ) = 2
39.91/11.23	POL( b_{c_1}_1(x_1) ) = max{0, -2}
39.91/11.23	POL( c_{a_1}_1(x_1) ) = 1
39.91/11.23	POL( c_{c_1}_1(x_1) ) = 1
39.91/11.23	
39.91/11.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.91/11.23	
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(20)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(21) QDPOrderProof (EQUIVALENT)
39.91/11.23	We use the reduction pair processor [LPAR04,JAR06].
39.91/11.23	
39.91/11.23	
39.91/11.23	The following pairs can be oriented strictly and are deleted.
39.91/11.23	
39.91/11.23	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
39.91/11.23	The remaining pairs can at least be oriented weakly.
39.91/11.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(B_{C_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(c_{c_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, 0A, -I], [-I, 1A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(c_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [-I, 1A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(B_{A_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(a_{a_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 1A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(a_{b_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(b_{a_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(a_{c_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [-I, -I, 0A]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(b_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(b_{c_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	   <<<
39.91/11.23	 POL(c_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [1A, -I, -I]] 	* 	x_1
39.91/11.23	>>>
39.91/11.23	
39.91/11.23	
39.91/11.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	
39.91/11.23	
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(22)
39.91/11.23	Obligation:
39.91/11.23	Q DP problem:
39.91/11.23	The TRS P consists of the following rules:
39.91/11.23	
39.91/11.23	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	
39.91/11.23	The TRS R consists of the following rules:
39.91/11.23	
39.91/11.23	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
39.91/11.23	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
39.91/11.23	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
39.91/11.23	   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
39.91/11.23	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
39.91/11.23	
39.91/11.23	Q is empty.
39.91/11.23	We have to consider all minimal (P,Q,R)-chains.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(23) DependencyGraphProof (EQUIVALENT)
39.91/11.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
39.91/11.23	----------------------------------------
39.91/11.23	
39.91/11.23	(24)
39.91/11.23	TRUE
40.31/11.32	EOF