25.22/7.35	YES
25.22/7.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
25.22/7.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.22/7.39	
25.22/7.39	
25.22/7.39	Termination w.r.t. Q of the given QTRS could be proven:
25.22/7.39	
25.22/7.39	(0) QTRS
25.22/7.39	(1) FlatCCProof [EQUIVALENT, 0 ms]
25.22/7.39	(2) QTRS
25.22/7.39	(3) RootLabelingProof [EQUIVALENT, 0 ms]
25.22/7.39	(4) QTRS
25.22/7.39	(5) QTRSRRRProof [EQUIVALENT, 69 ms]
25.22/7.39	(6) QTRS
25.22/7.39	(7) DependencyPairsProof [EQUIVALENT, 16 ms]
25.22/7.39	(8) QDP
25.22/7.39	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
25.22/7.39	(10) QDP
25.22/7.39	(11) QDPOrderProof [EQUIVALENT, 62 ms]
25.22/7.39	(12) QDP
25.22/7.39	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
25.22/7.39	(14) TRUE
25.22/7.39	
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(0)
25.22/7.39	Obligation:
25.22/7.39	Q restricted rewrite system:
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a(x1) -> x1
25.22/7.39	   a(a(x1)) -> a(b(x1))
25.22/7.39	   b(x1) -> x1
25.22/7.39	   c(b(x1)) -> a(b(c(c(x1))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(1) FlatCCProof (EQUIVALENT)
25.22/7.39	We used flat context closure [ROOTLAB]
25.22/7.39	As Q is empty the flat context closure was sound AND complete.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(2)
25.22/7.39	Obligation:
25.22/7.39	Q restricted rewrite system:
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a(a(x1)) -> a(b(x1))
25.22/7.39	   a(a(x1)) -> a(x1)
25.22/7.39	   b(a(x1)) -> b(x1)
25.22/7.39	   c(a(x1)) -> c(x1)
25.22/7.39	   a(b(x1)) -> a(x1)
25.22/7.39	   b(b(x1)) -> b(x1)
25.22/7.39	   c(b(x1)) -> c(x1)
25.22/7.39	   a(c(b(x1))) -> a(a(b(c(c(x1)))))
25.22/7.39	   b(c(b(x1))) -> b(a(b(c(c(x1)))))
25.22/7.39	   c(c(b(x1))) -> c(a(b(c(c(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(3) RootLabelingProof (EQUIVALENT)
25.22/7.39	We used plain root labeling [ROOTLAB] with the following heuristic:
25.22/7.39	LabelAll: All function symbols get labeled
25.22/7.39	
25.22/7.39	As Q is empty the root labeling was sound AND complete.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(4)
25.22/7.39	Obligation:
25.22/7.39	Q restricted rewrite system:
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
25.22/7.39	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
25.22/7.39	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
25.22/7.39	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
25.22/7.39	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
25.22/7.39	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
25.22/7.39	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(5) QTRSRRRProof (EQUIVALENT)
25.22/7.39	Used ordering:
25.22/7.39	Polynomial interpretation [POLO]:
25.22/7.39	
25.22/7.39	   POL(a_{a_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(a_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(a_{c_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(b_{a_1}(x_1)) = x_1
25.22/7.39	   POL(b_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(b_{c_1}(x_1)) = x_1
25.22/7.39	   POL(c_{a_1}(x_1)) = x_1
25.22/7.39	   POL(c_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(c_{c_1}(x_1)) = x_1
25.22/7.39	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
25.22/7.39	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1))
25.22/7.39	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
25.22/7.39	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
25.22/7.39	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
25.22/7.39	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
25.22/7.39	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
25.22/7.39	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
25.22/7.39	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
25.22/7.39	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
25.22/7.39	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
25.22/7.39	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
25.22/7.39	
25.22/7.39	
25.22/7.39	
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(6)
25.22/7.39	Obligation:
25.22/7.39	Q restricted rewrite system:
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(7) DependencyPairsProof (EQUIVALENT)
25.22/7.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(8)
25.22/7.39	Obligation:
25.22/7.39	Q DP problem:
25.22/7.39	The TRS P consists of the following rules:
25.22/7.39	
25.22/7.39	   A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{b_1}(x1))
25.22/7.39	   A_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
25.22/7.39	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	We have to consider all minimal (P,Q,R)-chains.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(9) DependencyGraphProof (EQUIVALENT)
25.22/7.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 14 less nodes.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(10)
25.22/7.39	Obligation:
25.22/7.39	Q DP problem:
25.22/7.39	The TRS P consists of the following rules:
25.22/7.39	
25.22/7.39	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	We have to consider all minimal (P,Q,R)-chains.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(11) QDPOrderProof (EQUIVALENT)
25.22/7.39	We use the reduction pair processor [LPAR04,JAR06].
25.22/7.39	
25.22/7.39	
25.22/7.39	The following pairs can be oriented strictly and are deleted.
25.22/7.39	
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{C_1}(c_{a_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{C_1}(c_{b_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(c_{c_1}(x1)))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(c_{c_1}(x1))
25.22/7.39	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
25.22/7.39	The remaining pairs can at least be oriented weakly.
25.22/7.39	Used ordering:  Polynomial interpretation [POLO]:
25.22/7.39	
25.22/7.39	   POL(A_{B_1}(x_1)) = x_1
25.22/7.39	   POL(A_{C_1}(x_1)) = x_1
25.22/7.39	   POL(B_{C_1}(x_1)) = x_1
25.22/7.39	   POL(C_{C_1}(x_1)) = x_1
25.22/7.39	   POL(a_{a_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(a_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(a_{c_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(b_{a_1}(x_1)) = x_1
25.22/7.39	   POL(b_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(b_{c_1}(x_1)) = x_1
25.22/7.39	   POL(c_{a_1}(x_1)) = x_1
25.22/7.39	   POL(c_{b_1}(x_1)) = 1 + x_1
25.22/7.39	   POL(c_{c_1}(x_1)) = x_1
25.22/7.39	
25.22/7.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.22/7.39	
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	
25.22/7.39	
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(12)
25.22/7.39	Obligation:
25.22/7.39	Q DP problem:
25.22/7.39	The TRS P consists of the following rules:
25.22/7.39	
25.22/7.39	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
25.22/7.39	
25.22/7.39	The TRS R consists of the following rules:
25.22/7.39	
25.22/7.39	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
25.22/7.39	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
25.22/7.39	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
25.22/7.39	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
25.22/7.39	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
25.22/7.39	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
25.22/7.39	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))
25.22/7.39	
25.22/7.39	Q is empty.
25.22/7.39	We have to consider all minimal (P,Q,R)-chains.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(13) DependencyGraphProof (EQUIVALENT)
25.22/7.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
25.22/7.39	----------------------------------------
25.22/7.39	
25.22/7.39	(14)
25.22/7.39	TRUE
25.55/7.70	EOF