24.37/7.14	YES
25.81/7.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
25.81/7.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.81/7.51	
25.81/7.51	
25.81/7.51	Termination w.r.t. Q of the given QTRS could be proven:
25.81/7.51	
25.81/7.51	(0) QTRS
25.81/7.51	(1) QTRS Reverse [EQUIVALENT, 0 ms]
25.81/7.51	(2) QTRS
25.81/7.51	(3) FlatCCProof [EQUIVALENT, 0 ms]
25.81/7.51	(4) QTRS
25.81/7.51	(5) RootLabelingProof [EQUIVALENT, 0 ms]
25.81/7.51	(6) QTRS
25.81/7.51	(7) QTRSRRRProof [EQUIVALENT, 48 ms]
25.81/7.51	(8) QTRS
25.81/7.51	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
25.81/7.51	(10) QDP
25.81/7.51	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
25.81/7.51	(12) QDP
25.81/7.51	(13) QDPOrderProof [EQUIVALENT, 242 ms]
25.81/7.51	(14) QDP
25.81/7.51	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
25.81/7.51	(16) TRUE
25.81/7.51	
25.81/7.51	
25.81/7.51	----------------------------------------
25.81/7.51	
25.81/7.51	(0)
25.81/7.51	Obligation:
25.81/7.51	Q restricted rewrite system:
25.81/7.51	The TRS R consists of the following rules:
25.81/7.51	
25.81/7.51	   a(x1) -> b(b(c(x1)))
25.81/7.51	   a(b(x1)) -> x1
25.81/7.51	   a(c(b(x1))) -> a(a(a(x1)))
25.81/7.51	
25.81/7.51	Q is empty.
25.81/7.51	
25.81/7.51	----------------------------------------
25.81/7.51	
25.81/7.51	(1) QTRS Reverse (EQUIVALENT)
25.81/7.51	We applied the QTRS Reverse Processor [REVERSE].
25.81/7.51	----------------------------------------
25.81/7.51	
25.81/7.51	(2)
25.81/7.51	Obligation:
25.81/7.51	Q restricted rewrite system:
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a(x1) -> c(b(b(x1)))
25.81/7.52	   b(a(x1)) -> x1
25.81/7.52	   b(c(a(x1))) -> a(a(a(x1)))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(3) FlatCCProof (EQUIVALENT)
25.81/7.52	We used flat context closure [ROOTLAB]
25.81/7.52	As Q is empty the flat context closure was sound AND complete.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(4)
25.81/7.52	Obligation:
25.81/7.52	Q restricted rewrite system:
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a(a(x1)) -> a(c(b(b(x1))))
25.81/7.52	   c(a(x1)) -> c(c(b(b(x1))))
25.81/7.52	   b(a(x1)) -> b(c(b(b(x1))))
25.81/7.52	   a(b(a(x1))) -> a(x1)
25.81/7.52	   c(b(a(x1))) -> c(x1)
25.81/7.52	   b(b(a(x1))) -> b(x1)
25.81/7.52	   a(b(c(a(x1)))) -> a(a(a(a(x1))))
25.81/7.52	   c(b(c(a(x1)))) -> c(a(a(a(x1))))
25.81/7.52	   b(b(c(a(x1)))) -> b(a(a(a(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(5) RootLabelingProof (EQUIVALENT)
25.81/7.52	We used plain root labeling [ROOTLAB] with the following heuristic:
25.81/7.52	LabelAll: All function symbols get labeled
25.81/7.52	
25.81/7.52	As Q is empty the root labeling was sound AND complete.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(6)
25.81/7.52	Obligation:
25.81/7.52	Q restricted rewrite system:
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
25.81/7.52	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1)
25.81/7.52	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(7) QTRSRRRProof (EQUIVALENT)
25.81/7.52	Used ordering:
25.81/7.52	Polynomial interpretation [POLO]:
25.81/7.52	
25.81/7.52	   POL(a_{a_1}(x_1)) = x_1
25.81/7.52	   POL(a_{b_1}(x_1)) = 1 + x_1
25.81/7.52	   POL(a_{c_1}(x_1)) = x_1
25.81/7.52	   POL(b_{a_1}(x_1)) = x_1
25.81/7.52	   POL(b_{b_1}(x_1)) = x_1
25.81/7.52	   POL(b_{c_1}(x_1)) = x_1
25.81/7.52	   POL(c_{a_1}(x_1)) = x_1
25.81/7.52	   POL(c_{b_1}(x_1)) = x_1
25.81/7.52	   POL(c_{c_1}(x_1)) = x_1
25.81/7.52	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))
25.81/7.52	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
25.81/7.52	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1)
25.81/7.52	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	
25.81/7.52	
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(8)
25.81/7.52	Obligation:
25.81/7.52	Q restricted rewrite system:
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(9) DependencyPairsProof (EQUIVALENT)
25.81/7.52	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(10)
25.81/7.52	Obligation:
25.81/7.52	Q DP problem:
25.81/7.52	The TRS P consists of the following rules:
25.81/7.52	
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(x1)
25.81/7.52	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
25.81/7.52	
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	We have to consider all minimal (P,Q,R)-chains.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(11) DependencyGraphProof (EQUIVALENT)
25.81/7.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(12)
25.81/7.52	Obligation:
25.81/7.52	Q DP problem:
25.81/7.52	The TRS P consists of the following rules:
25.81/7.52	
25.81/7.52	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(x1)
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	We have to consider all minimal (P,Q,R)-chains.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(13) QDPOrderProof (EQUIVALENT)
25.81/7.52	We use the reduction pair processor [LPAR04,JAR06].
25.81/7.52	
25.81/7.52	
25.81/7.52	The following pairs can be oriented strictly and are deleted.
25.81/7.52	
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1)))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
25.81/7.52	The remaining pairs can at least be oriented weakly.
25.81/7.52	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
25.81/7.52	
25.81/7.52	POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
25.81/7.52	POL( C_{B_1}_1(x_1) ) = max{0, 2x_1 - 2}
25.81/7.52	POL( A_{A_1}_1(x_1) ) = 2x_1 + 2
25.81/7.52	POL( a_{c_1}_1(x_1) ) = x_1 + 2
25.81/7.52	POL( c_{b_1}_1(x_1) ) = x_1
25.81/7.52	POL( B_{A_1}_1(x_1) ) = 2x_1 + 2
25.81/7.52	POL( C_{A_1}_1(x_1) ) = 2x_1 + 2
25.81/7.52	POL( b_{a_1}_1(x_1) ) = x_1 + 1
25.81/7.52	POL( a_{a_1}_1(x_1) ) = x_1 + 1
25.81/7.52	POL( b_{c_1}_1(x_1) ) = x_1 + 2
25.81/7.52	POL( b_{b_1}_1(x_1) ) = max{0, x_1 - 1}
25.81/7.52	POL( c_{a_1}_1(x_1) ) = x_1 + 2
25.81/7.52	POL( a_{b_1}_1(x_1) ) = 0
25.81/7.52	POL( c_{c_1}_1(x_1) ) = max{0, -2}
25.81/7.52	
25.81/7.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.81/7.52	
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	
25.81/7.52	
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(14)
25.81/7.52	Obligation:
25.81/7.52	Q DP problem:
25.81/7.52	The TRS P consists of the following rules:
25.81/7.52	
25.81/7.52	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(x1)
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   A_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   B_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   A_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
25.81/7.52	
25.81/7.52	The TRS R consists of the following rules:
25.81/7.52	
25.81/7.52	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))
25.81/7.52	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1)
25.81/7.52	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
25.81/7.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1)
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
25.81/7.52	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
25.81/7.52	
25.81/7.52	Q is empty.
25.81/7.52	We have to consider all minimal (P,Q,R)-chains.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(15) DependencyGraphProof (EQUIVALENT)
25.81/7.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 10 less nodes.
25.81/7.52	----------------------------------------
25.81/7.52	
25.81/7.52	(16)
25.81/7.52	TRUE
26.03/10.88	EOF