16.22/5.01	YES
16.22/5.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.22/5.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.22/5.04	
16.22/5.04	
16.22/5.04	Termination w.r.t. Q of the given QTRS could be proven:
16.22/5.04	
16.22/5.04	(0) QTRS
16.22/5.04	(1) FlatCCProof [EQUIVALENT, 0 ms]
16.22/5.04	(2) QTRS
16.22/5.04	(3) RootLabelingProof [EQUIVALENT, 0 ms]
16.22/5.04	(4) QTRS
16.22/5.04	(5) QTRSRRRProof [EQUIVALENT, 84 ms]
16.22/5.04	(6) QTRS
16.22/5.04	(7) DependencyPairsProof [EQUIVALENT, 59 ms]
16.22/5.04	(8) QDP
16.22/5.04	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
16.22/5.04	(10) QDP
16.22/5.04	(11) UsableRulesProof [EQUIVALENT, 20 ms]
16.22/5.04	(12) QDP
16.22/5.04	(13) QDPOrderProof [EQUIVALENT, 80 ms]
16.22/5.04	(14) QDP
16.22/5.04	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
16.22/5.04	(16) TRUE
16.22/5.04	
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(0)
16.22/5.04	Obligation:
16.22/5.04	Q restricted rewrite system:
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   a(a(b(x1))) -> b(a(b(c(a(x1)))))
16.22/5.04	   b(a(x1)) -> a(b(b(x1)))
16.22/5.04	   b(c(a(x1))) -> c(a(b(x1)))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(1) FlatCCProof (EQUIVALENT)
16.22/5.04	We used flat context closure [ROOTLAB]
16.22/5.04	As Q is empty the flat context closure was sound AND complete.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(2)
16.22/5.04	Obligation:
16.22/5.04	Q restricted rewrite system:
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   a(a(a(b(x1)))) -> a(b(a(b(c(a(x1))))))
16.22/5.04	   b(a(a(b(x1)))) -> b(b(a(b(c(a(x1))))))
16.22/5.04	   c(a(a(b(x1)))) -> c(b(a(b(c(a(x1))))))
16.22/5.04	   a(b(a(x1))) -> a(a(b(b(x1))))
16.22/5.04	   b(b(a(x1))) -> b(a(b(b(x1))))
16.22/5.04	   c(b(a(x1))) -> c(a(b(b(x1))))
16.22/5.04	   a(b(c(a(x1)))) -> a(c(a(b(x1))))
16.22/5.04	   b(b(c(a(x1)))) -> b(c(a(b(x1))))
16.22/5.04	   c(b(c(a(x1)))) -> c(c(a(b(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(3) RootLabelingProof (EQUIVALENT)
16.22/5.04	We used plain root labeling [ROOTLAB] with the following heuristic:
16.22/5.04	LabelAll: All function symbols get labeled
16.22/5.04	
16.22/5.04	As Q is empty the root labeling was sound AND complete.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(4)
16.22/5.04	Obligation:
16.22/5.04	Q restricted rewrite system:
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(5) QTRSRRRProof (EQUIVALENT)
16.22/5.04	Used ordering:
16.22/5.04	Polynomial interpretation [POLO]:
16.22/5.04	
16.22/5.04	   POL(a_{a_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(a_{b_1}(x_1)) = x_1
16.22/5.04	   POL(a_{c_1}(x_1)) = x_1
16.22/5.04	   POL(b_{a_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(b_{b_1}(x_1)) = x_1
16.22/5.04	   POL(b_{c_1}(x_1)) = x_1
16.22/5.04	   POL(c_{a_1}(x_1)) = x_1
16.22/5.04	   POL(c_{b_1}(x_1)) = x_1
16.22/5.04	   POL(c_{c_1}(x_1)) = x_1
16.22/5.04	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
16.22/5.04	
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	
16.22/5.04	
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(6)
16.22/5.04	Obligation:
16.22/5.04	Q restricted rewrite system:
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(7) DependencyPairsProof (EQUIVALENT)
16.22/5.04	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(8)
16.22/5.04	Obligation:
16.22/5.04	Q DP problem:
16.22/5.04	The TRS P consists of the following rules:
16.22/5.04	
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{A_1}(a_{c_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	We have to consider all minimal (P,Q,R)-chains.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(9) DependencyGraphProof (EQUIVALENT)
16.22/5.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(10)
16.22/5.04	Obligation:
16.22/5.04	Q DP problem:
16.22/5.04	The TRS P consists of the following rules:
16.22/5.04	
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	We have to consider all minimal (P,Q,R)-chains.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(11) UsableRulesProof (EQUIVALENT)
16.22/5.04	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(12)
16.22/5.04	Obligation:
16.22/5.04	Q DP problem:
16.22/5.04	The TRS P consists of the following rules:
16.22/5.04	
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	
16.22/5.04	The TRS R consists of the following rules:
16.22/5.04	
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	Q is empty.
16.22/5.04	We have to consider all minimal (P,Q,R)-chains.
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(13) QDPOrderProof (EQUIVALENT)
16.22/5.04	We use the reduction pair processor [LPAR04,JAR06].
16.22/5.04	
16.22/5.04	
16.22/5.04	The following pairs can be oriented strictly and are deleted.
16.22/5.04	
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1)))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1)))
16.22/5.04	   B_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(x1))
16.22/5.04	The remaining pairs can at least be oriented weakly.
16.22/5.04	Used ordering:  Polynomial interpretation [POLO]:
16.22/5.04	
16.22/5.04	   POL(A_{B_1}(x_1)) = x_1
16.22/5.04	   POL(B_{B_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(C_{A_1}(x_1)) = x_1
16.22/5.04	   POL(a_{a_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(a_{b_1}(x_1)) = x_1
16.22/5.04	   POL(a_{c_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(b_{a_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(b_{b_1}(x_1)) = x_1
16.22/5.04	   POL(b_{c_1}(x_1)) = 1 + x_1
16.22/5.04	   POL(c_{a_1}(x_1)) = x_1
16.22/5.04	   POL(c_{b_1}(x_1)) = 0
16.22/5.04	
16.22/5.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
16.22/5.04	
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.04	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.04	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.04	
16.22/5.04	
16.22/5.04	----------------------------------------
16.22/5.04	
16.22/5.04	(14)
16.22/5.04	Obligation:
16.22/5.04	Q DP problem:
16.22/5.04	The TRS P consists of the following rules:
16.22/5.04	
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
16.22/5.04	   C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
16.22/5.04	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
16.22/5.05	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
16.22/5.05	   C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
16.22/5.05	   A_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
16.22/5.05	
16.22/5.05	The TRS R consists of the following rules:
16.22/5.05	
16.22/5.05	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.05	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.05	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.05	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))))
16.22/5.05	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))))
16.22/5.05	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))))
16.22/5.05	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.05	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.05	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.05	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
16.22/5.05	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
16.22/5.05	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
16.22/5.05	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
16.22/5.05	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
16.22/5.05	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
16.22/5.05	
16.22/5.05	Q is empty.
16.22/5.05	We have to consider all minimal (P,Q,R)-chains.
16.22/5.05	----------------------------------------
16.22/5.05	
16.22/5.05	(15) DependencyGraphProof (EQUIVALENT)
16.22/5.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes.
16.22/5.05	----------------------------------------
16.22/5.05	
16.22/5.05	(16)
16.22/5.05	TRUE
16.54/5.10	EOF