26.30/7.61	YES
27.43/7.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.43/7.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.43/7.85	
27.43/7.85	
27.43/7.85	Termination w.r.t. Q of the given QTRS could be proven:
27.43/7.85	
27.43/7.85	(0) QTRS
27.43/7.85	(1) QTRS Reverse [EQUIVALENT, 0 ms]
27.43/7.85	(2) QTRS
27.43/7.85	(3) FlatCCProof [EQUIVALENT, 0 ms]
27.43/7.85	(4) QTRS
27.43/7.85	(5) RootLabelingProof [EQUIVALENT, 0 ms]
27.43/7.85	(6) QTRS
27.43/7.85	(7) QTRSRRRProof [EQUIVALENT, 72 ms]
27.43/7.85	(8) QTRS
27.43/7.85	(9) DependencyPairsProof [EQUIVALENT, 67 ms]
27.43/7.85	(10) QDP
27.43/7.85	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
27.43/7.85	(12) QDP
27.43/7.85	(13) QDPOrderProof [EQUIVALENT, 482 ms]
27.43/7.85	(14) QDP
27.43/7.85	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
27.43/7.85	(16) TRUE
27.43/7.85	
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(0)
27.43/7.85	Obligation:
27.43/7.85	Q restricted rewrite system:
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   a(x1) -> x1
27.43/7.85	   a(b(x1)) -> b(b(c(a(x1))))
27.43/7.85	   b(b(x1)) -> a(x1)
27.43/7.85	   c(c(x1)) -> x1
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(1) QTRS Reverse (EQUIVALENT)
27.43/7.85	We applied the QTRS Reverse Processor [REVERSE].
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(2)
27.43/7.85	Obligation:
27.43/7.85	Q restricted rewrite system:
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   a(x1) -> x1
27.43/7.85	   b(a(x1)) -> a(c(b(b(x1))))
27.43/7.85	   b(b(x1)) -> a(x1)
27.43/7.85	   c(c(x1)) -> x1
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(3) FlatCCProof (EQUIVALENT)
27.43/7.85	We used flat context closure [ROOTLAB]
27.43/7.85	As Q is empty the flat context closure was sound AND complete.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(4)
27.43/7.85	Obligation:
27.43/7.85	Q restricted rewrite system:
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   a(a(x1)) -> a(x1)
27.43/7.85	   b(a(x1)) -> b(x1)
27.43/7.85	   c(a(x1)) -> c(x1)
27.43/7.85	   a(b(a(x1))) -> a(a(c(b(b(x1)))))
27.43/7.85	   b(b(a(x1))) -> b(a(c(b(b(x1)))))
27.43/7.85	   c(b(a(x1))) -> c(a(c(b(b(x1)))))
27.43/7.85	   a(b(b(x1))) -> a(a(x1))
27.43/7.85	   b(b(b(x1))) -> b(a(x1))
27.43/7.85	   c(b(b(x1))) -> c(a(x1))
27.43/7.85	   a(c(c(x1))) -> a(x1)
27.43/7.85	   b(c(c(x1))) -> b(x1)
27.43/7.85	   c(c(c(x1))) -> c(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(5) RootLabelingProof (EQUIVALENT)
27.43/7.85	We used plain root labeling [ROOTLAB] with the following heuristic:
27.43/7.85	LabelAll: All function symbols get labeled
27.43/7.85	
27.43/7.85	As Q is empty the root labeling was sound AND complete.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(6)
27.43/7.85	Obligation:
27.43/7.85	Q restricted rewrite system:
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
27.43/7.85	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
27.43/7.85	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
27.43/7.85	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
27.43/7.85	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
27.43/7.85	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
27.43/7.85	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
27.43/7.85	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(7) QTRSRRRProof (EQUIVALENT)
27.43/7.85	Used ordering:
27.43/7.85	Polynomial interpretation [POLO]:
27.43/7.85	
27.43/7.85	   POL(a_{a_1}(x_1)) = 2 + x_1
27.43/7.85	   POL(a_{b_1}(x_1)) = 2 + x_1
27.43/7.85	   POL(a_{c_1}(x_1)) = 1 + x_1
27.43/7.85	   POL(b_{a_1}(x_1)) = 1 + x_1
27.43/7.85	   POL(b_{b_1}(x_1)) = 1 + x_1
27.43/7.85	   POL(b_{c_1}(x_1)) = x_1
27.43/7.85	   POL(c_{a_1}(x_1)) = x_1
27.43/7.85	   POL(c_{b_1}(x_1)) = x_1
27.43/7.85	   POL(c_{c_1}(x_1)) = 1 + x_1
27.43/7.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
27.43/7.85	
27.43/7.85	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
27.43/7.85	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
27.43/7.85	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
27.43/7.85	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
27.43/7.85	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
27.43/7.85	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
27.43/7.85	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
27.43/7.85	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
27.43/7.85	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
27.43/7.85	
27.43/7.85	
27.43/7.85	
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(8)
27.43/7.85	Obligation:
27.43/7.85	Q restricted rewrite system:
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(9) DependencyPairsProof (EQUIVALENT)
27.43/7.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(10)
27.43/7.85	Obligation:
27.43/7.85	Q DP problem:
27.43/7.85	The TRS P consists of the following rules:
27.43/7.85	
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
27.43/7.85	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
27.43/7.85	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
27.43/7.85	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	We have to consider all minimal (P,Q,R)-chains.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(11) DependencyGraphProof (EQUIVALENT)
27.43/7.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(12)
27.43/7.85	Obligation:
27.43/7.85	Q DP problem:
27.43/7.85	The TRS P consists of the following rules:
27.43/7.85	
27.43/7.85	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	We have to consider all minimal (P,Q,R)-chains.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(13) QDPOrderProof (EQUIVALENT)
27.43/7.85	We use the reduction pair processor [LPAR04,JAR06].
27.43/7.85	
27.43/7.85	
27.43/7.85	The following pairs can be oriented strictly and are deleted.
27.43/7.85	
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(b_{b_1}(x1)))
27.43/7.85	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
27.43/7.85	   B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
27.43/7.85	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
27.43/7.85	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	   A_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
27.43/7.85	The remaining pairs can at least be oriented weakly.
27.43/7.85	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
27.43/7.85	
27.43/7.85	POL( B_{B_1}_1(x_1) ) = max{0, 2x_1 - 2}
27.43/7.85	POL( C_{B_1}_1(x_1) ) = max{0, 2x_1 - 2}
27.43/7.85	POL( A_{C_1}_1(x_1) ) = 2x_1
27.43/7.85	POL( c_{b_1}_1(x_1) ) = max{0, x_1 - 1}
27.43/7.85	POL( b_{b_1}_1(x_1) ) = x_1 + 1
27.43/7.85	POL( b_{a_1}_1(x_1) ) = x_1 + 1
27.43/7.85	POL( a_{a_1}_1(x_1) ) = x_1 + 2
27.43/7.85	POL( a_{c_1}_1(x_1) ) = x_1 + 2
27.43/7.85	POL( a_{b_1}_1(x_1) ) = x_1 + 2
27.43/7.85	POL( b_{c_1}_1(x_1) ) = x_1 + 1
27.43/7.85	POL( c_{a_1}_1(x_1) ) = max{0, x_1 - 1}
27.43/7.85	POL( c_{c_1}_1(x_1) ) = x_1 + 1
27.43/7.85	POL( A_{B_1}_1(x_1) ) = 2x_1
27.43/7.85	POL( B_{C_1}_1(x_1) ) = 2x_1 + 2
27.43/7.85	
27.43/7.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.43/7.85	
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	
27.43/7.85	
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(14)
27.43/7.85	Obligation:
27.43/7.85	Q DP problem:
27.43/7.85	The TRS P consists of the following rules:
27.43/7.85	
27.43/7.85	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
27.43/7.85	
27.43/7.85	The TRS R consists of the following rules:
27.43/7.85	
27.43/7.85	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
27.43/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))))
27.43/7.85	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
27.43/7.85	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
27.43/7.85	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
27.43/7.85	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
27.43/7.85	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
27.43/7.85	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
27.43/7.85	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
27.43/7.85	
27.43/7.85	Q is empty.
27.43/7.85	We have to consider all minimal (P,Q,R)-chains.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(15) DependencyGraphProof (EQUIVALENT)
27.43/7.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
27.43/7.85	----------------------------------------
27.43/7.85	
27.43/7.85	(16)
27.43/7.85	TRUE
28.73/8.26	EOF