21.31/6.27	YES
21.72/6.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.72/6.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.72/6.39	
21.72/6.39	
21.72/6.39	Termination w.r.t. Q of the given QTRS could be proven:
21.72/6.39	
21.72/6.39	(0) QTRS
21.72/6.39	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.72/6.39	(2) QTRS
21.72/6.39	(3) FlatCCProof [EQUIVALENT, 0 ms]
21.72/6.39	(4) QTRS
21.72/6.39	(5) RootLabelingProof [EQUIVALENT, 1 ms]
21.72/6.39	(6) QTRS
21.72/6.39	(7) QTRSRRRProof [EQUIVALENT, 36 ms]
21.72/6.39	(8) QTRS
21.72/6.39	(9) DependencyPairsProof [EQUIVALENT, 22 ms]
21.72/6.39	(10) QDP
21.72/6.39	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
21.72/6.39	(12) QDP
21.72/6.39	(13) QDPOrderProof [EQUIVALENT, 329 ms]
21.72/6.39	(14) QDP
21.72/6.39	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
21.72/6.39	(16) TRUE
21.72/6.39	
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(0)
21.72/6.39	Obligation:
21.72/6.39	Q restricted rewrite system:
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   a(x1) -> x1
21.72/6.39	   a(a(x1)) -> b(x1)
21.72/6.39	   b(c(x1)) -> x1
21.72/6.39	   c(b(x1)) -> a(b(c(c(x1))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(1) QTRS Reverse (EQUIVALENT)
21.72/6.39	We applied the QTRS Reverse Processor [REVERSE].
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(2)
21.72/6.39	Obligation:
21.72/6.39	Q restricted rewrite system:
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   a(x1) -> x1
21.72/6.39	   a(a(x1)) -> b(x1)
21.72/6.39	   c(b(x1)) -> x1
21.72/6.39	   b(c(x1)) -> c(c(b(a(x1))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(3) FlatCCProof (EQUIVALENT)
21.72/6.39	We used flat context closure [ROOTLAB]
21.72/6.39	As Q is empty the flat context closure was sound AND complete.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(4)
21.72/6.39	Obligation:
21.72/6.39	Q restricted rewrite system:
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   a(a(x1)) -> a(x1)
21.72/6.39	   b(a(x1)) -> b(x1)
21.72/6.39	   c(a(x1)) -> c(x1)
21.72/6.39	   a(a(a(x1))) -> a(b(x1))
21.72/6.39	   b(a(a(x1))) -> b(b(x1))
21.72/6.39	   c(a(a(x1))) -> c(b(x1))
21.72/6.39	   a(c(b(x1))) -> a(x1)
21.72/6.39	   b(c(b(x1))) -> b(x1)
21.72/6.39	   c(c(b(x1))) -> c(x1)
21.72/6.39	   a(b(c(x1))) -> a(c(c(b(a(x1)))))
21.72/6.39	   b(b(c(x1))) -> b(c(c(b(a(x1)))))
21.72/6.39	   c(b(c(x1))) -> c(c(c(b(a(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(5) RootLabelingProof (EQUIVALENT)
21.72/6.39	We used plain root labeling [ROOTLAB] with the following heuristic:
21.72/6.39	LabelAll: All function symbols get labeled
21.72/6.39	
21.72/6.39	As Q is empty the root labeling was sound AND complete.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(6)
21.72/6.39	Obligation:
21.72/6.39	Q restricted rewrite system:
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
21.72/6.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
21.72/6.39	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
21.72/6.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.72/6.39	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
21.72/6.39	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1))
21.72/6.39	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1))
21.72/6.39	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1)
21.72/6.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1)
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(7) QTRSRRRProof (EQUIVALENT)
21.72/6.39	Used ordering:
21.72/6.39	Polynomial interpretation [POLO]:
21.72/6.39	
21.72/6.39	   POL(a_{a_1}(x_1)) = 1 + x_1
21.72/6.39	   POL(a_{b_1}(x_1)) = x_1
21.72/6.39	   POL(a_{c_1}(x_1)) = x_1
21.72/6.39	   POL(b_{a_1}(x_1)) = 1 + x_1
21.72/6.39	   POL(b_{b_1}(x_1)) = 1 + x_1
21.72/6.39	   POL(b_{c_1}(x_1)) = 1 + x_1
21.72/6.39	   POL(c_{a_1}(x_1)) = 1 + x_1
21.72/6.39	   POL(c_{b_1}(x_1)) = x_1
21.72/6.39	   POL(c_{c_1}(x_1)) = x_1
21.72/6.39	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
21.72/6.39	
21.72/6.39	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
21.72/6.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
21.72/6.39	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
21.72/6.39	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
21.72/6.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.72/6.39	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
21.72/6.39	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1))
21.72/6.39	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1))
21.72/6.39	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1))
21.72/6.39	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1)
21.72/6.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1)
21.72/6.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1)
21.72/6.39	
21.72/6.39	
21.72/6.39	
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(8)
21.72/6.39	Obligation:
21.72/6.39	Q restricted rewrite system:
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(9) DependencyPairsProof (EQUIVALENT)
21.72/6.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(10)
21.72/6.39	Obligation:
21.72/6.39	Q DP problem:
21.72/6.39	The TRS P consists of the following rules:
21.72/6.39	
21.72/6.39	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1)
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.72/6.39	   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{C_1}(x1)
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{C_1}(x1)
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{C_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{C_1}(x1)
21.72/6.39	
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	We have to consider all minimal (P,Q,R)-chains.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(11) DependencyGraphProof (EQUIVALENT)
21.72/6.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 18 less nodes.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(12)
21.72/6.39	Obligation:
21.72/6.39	Q DP problem:
21.72/6.39	The TRS P consists of the following rules:
21.72/6.39	
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1)
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	We have to consider all minimal (P,Q,R)-chains.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(13) QDPOrderProof (EQUIVALENT)
21.72/6.39	We use the reduction pair processor [LPAR04,JAR06].
21.72/6.39	
21.72/6.39	
21.72/6.39	The following pairs can be oriented strictly and are deleted.
21.72/6.39	
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1)))
21.72/6.39	   B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1))
21.72/6.39	The remaining pairs can at least be oriented weakly.
21.72/6.39	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
21.72/6.39	
21.72/6.39	POL( C_{B_1}_1(x_1) ) = max{0, 2x_1 - 2}
21.72/6.39	POL( B_{A_1}_1(x_1) ) = x_1 + 2
21.72/6.39	POL( a_{c_1}_1(x_1) ) = 2x_1 + 2
21.72/6.39	POL( c_{b_1}_1(x_1) ) = x_1
21.72/6.39	POL( c_{c_1}_1(x_1) ) = x_1 + 1
21.72/6.39	POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
21.72/6.39	POL( b_{c_1}_1(x_1) ) = 2x_1 + 2
21.72/6.39	POL( b_{a_1}_1(x_1) ) = x_1
21.72/6.39	POL( a_{a_1}_1(x_1) ) = 2x_1 + 2
21.72/6.39	POL( a_{b_1}_1(x_1) ) = 2x_1
21.72/6.39	POL( b_{b_1}_1(x_1) ) = 2x_1
21.72/6.39	POL( c_{a_1}_1(x_1) ) = x_1 + 1
21.72/6.39	POL( A_{B_1}_1(x_1) ) = 2x_1 + 1
21.72/6.39	
21.72/6.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.72/6.39	
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	
21.72/6.39	
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(14)
21.72/6.39	Obligation:
21.72/6.39	Q DP problem:
21.72/6.39	The TRS P consists of the following rules:
21.72/6.39	
21.72/6.39	   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
21.72/6.39	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1)
21.72/6.39	
21.72/6.39	The TRS R consists of the following rules:
21.72/6.39	
21.72/6.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.72/6.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.72/6.39	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1))
21.72/6.39	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1))
21.72/6.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1)
21.72/6.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1)
21.72/6.39	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.72/6.39	   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))))
21.72/6.39	
21.72/6.39	Q is empty.
21.72/6.39	We have to consider all minimal (P,Q,R)-chains.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(15) DependencyGraphProof (EQUIVALENT)
21.72/6.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
21.72/6.39	----------------------------------------
21.72/6.39	
21.72/6.39	(16)
21.72/6.39	TRUE
22.02/6.52	EOF