20.75/6.12	YES
20.75/6.13	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.75/6.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.75/6.13	
20.75/6.13	
20.75/6.13	Termination w.r.t. Q of the given QTRS could be proven:
20.75/6.13	
20.75/6.13	(0) QTRS
20.75/6.13	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.75/6.13	(2) QTRS
20.75/6.13	(3) FlatCCProof [EQUIVALENT, 0 ms]
20.75/6.13	(4) QTRS
20.75/6.13	(5) RootLabelingProof [EQUIVALENT, 0 ms]
20.75/6.13	(6) QTRS
20.75/6.13	(7) QTRSRRRProof [EQUIVALENT, 18 ms]
20.75/6.13	(8) QTRS
20.75/6.13	(9) QTRSRRRProof [EQUIVALENT, 4 ms]
20.75/6.13	(10) QTRS
20.75/6.13	(11) DependencyPairsProof [EQUIVALENT, 19 ms]
20.75/6.13	(12) QDP
20.75/6.13	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
20.75/6.13	(14) QDP
20.75/6.13	(15) QDPOrderProof [EQUIVALENT, 147 ms]
20.75/6.13	(16) QDP
20.75/6.13	(17) DependencyGraphProof [EQUIVALENT, 0 ms]
20.75/6.13	(18) TRUE
20.75/6.13	
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(0)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   a(x1) -> x1
20.75/6.13	   a(x1) -> b(b(x1))
20.75/6.13	   a(b(x1)) -> a(c(a(c(x1))))
20.75/6.13	   c(c(x1)) -> x1
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(1) QTRS Reverse (EQUIVALENT)
20.75/6.13	We applied the QTRS Reverse Processor [REVERSE].
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(2)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   a(x1) -> x1
20.75/6.13	   a(x1) -> b(b(x1))
20.75/6.13	   b(a(x1)) -> c(a(c(a(x1))))
20.75/6.13	   c(c(x1)) -> x1
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(3) FlatCCProof (EQUIVALENT)
20.75/6.13	We used flat context closure [ROOTLAB]
20.75/6.13	As Q is empty the flat context closure was sound AND complete.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(4)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   a(a(x1)) -> a(x1)
20.75/6.13	   b(a(x1)) -> b(x1)
20.75/6.13	   c(a(x1)) -> c(x1)
20.75/6.13	   a(a(x1)) -> a(b(b(x1)))
20.75/6.13	   b(a(x1)) -> b(b(b(x1)))
20.75/6.13	   c(a(x1)) -> c(b(b(x1)))
20.75/6.13	   a(b(a(x1))) -> a(c(a(c(a(x1)))))
20.75/6.13	   b(b(a(x1))) -> b(c(a(c(a(x1)))))
20.75/6.13	   c(b(a(x1))) -> c(c(a(c(a(x1)))))
20.75/6.13	   a(c(c(x1))) -> a(x1)
20.75/6.13	   b(c(c(x1))) -> b(x1)
20.75/6.13	   c(c(c(x1))) -> c(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(5) RootLabelingProof (EQUIVALENT)
20.75/6.13	We used plain root labeling [ROOTLAB] with the following heuristic:
20.75/6.13	LabelAll: All function symbols get labeled
20.75/6.13	
20.75/6.13	As Q is empty the root labeling was sound AND complete.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(6)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
20.75/6.13	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
20.75/6.13	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
20.75/6.13	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
20.75/6.13	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
20.75/6.13	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
20.75/6.13	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
20.75/6.13	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(7) QTRSRRRProof (EQUIVALENT)
20.75/6.13	Used ordering:
20.75/6.13	Polynomial interpretation [POLO]:
20.75/6.13	
20.75/6.13	   POL(a_{a_1}(x_1)) = 2 + x_1
20.75/6.13	   POL(a_{b_1}(x_1)) = 3 + x_1
20.75/6.13	   POL(a_{c_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(b_{a_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(b_{b_1}(x_1)) = x_1
20.75/6.13	   POL(b_{c_1}(x_1)) = x_1
20.75/6.13	   POL(c_{a_1}(x_1)) = x_1
20.75/6.13	   POL(c_{b_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(c_{c_1}(x_1)) = 1 + x_1
20.75/6.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
20.75/6.13	
20.75/6.13	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
20.75/6.13	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
20.75/6.13	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
20.75/6.13	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
20.75/6.13	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
20.75/6.13	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
20.75/6.13	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
20.75/6.13	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1)
20.75/6.13	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1)
20.75/6.13	
20.75/6.13	
20.75/6.13	
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(8)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(9) QTRSRRRProof (EQUIVALENT)
20.75/6.13	Used ordering:
20.75/6.13	Polynomial interpretation [POLO]:
20.75/6.13	
20.75/6.13	   POL(a_{a_1}(x_1)) = 2 + x_1
20.75/6.13	   POL(a_{b_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(a_{c_1}(x_1)) = x_1
20.75/6.13	   POL(b_{a_1}(x_1)) = 2 + x_1
20.75/6.13	   POL(b_{b_1}(x_1)) = x_1
20.75/6.13	   POL(b_{c_1}(x_1)) = x_1
20.75/6.13	   POL(c_{a_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(c_{b_1}(x_1)) = 1 + x_1
20.75/6.13	   POL(c_{c_1}(x_1)) = 1 + x_1
20.75/6.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
20.75/6.13	
20.75/6.13	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1)
20.75/6.13	
20.75/6.13	
20.75/6.13	
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(10)
20.75/6.13	Obligation:
20.75/6.13	Q restricted rewrite system:
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(11) DependencyPairsProof (EQUIVALENT)
20.75/6.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(12)
20.75/6.13	Obligation:
20.75/6.13	Q DP problem:
20.75/6.13	The TRS P consists of the following rules:
20.75/6.13	
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	We have to consider all minimal (P,Q,R)-chains.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(13) DependencyGraphProof (EQUIVALENT)
20.75/6.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(14)
20.75/6.13	Obligation:
20.75/6.13	Q DP problem:
20.75/6.13	The TRS P consists of the following rules:
20.75/6.13	
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	We have to consider all minimal (P,Q,R)-chains.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(15) QDPOrderProof (EQUIVALENT)
20.75/6.13	We use the reduction pair processor [LPAR04,JAR06].
20.75/6.13	
20.75/6.13	
20.75/6.13	The following pairs can be oriented strictly and are deleted.
20.75/6.13	
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   C_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   C_{A_1}(a_{c_1}(x1)) -> B_{B_1}(b_{c_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1))
20.75/6.13	The remaining pairs can at least be oriented weakly.
20.75/6.13	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
20.75/6.13	
20.75/6.13	POL( C_{A_1}_1(x_1) ) = 2x_1 + 1
20.75/6.13	POL( a_{c_1}_1(x_1) ) = x_1
20.75/6.13	POL( C_{B_1}_1(x_1) ) = max{0, 2x_1 - 1}
20.75/6.13	POL( b_{c_1}_1(x_1) ) = x_1
20.75/6.13	POL( c_{a_1}_1(x_1) ) = x_1 + 1
20.75/6.13	POL( a_{a_1}_1(x_1) ) = x_1 + 2
20.75/6.13	POL( c_{b_1}_1(x_1) ) = x_1 + 1
20.75/6.13	POL( b_{b_1}_1(x_1) ) = x_1
20.75/6.13	POL( b_{a_1}_1(x_1) ) = x_1 + 2
20.75/6.13	POL( c_{c_1}_1(x_1) ) = x_1 + 1
20.75/6.13	POL( a_{b_1}_1(x_1) ) = 1
20.75/6.13	POL( B_{B_1}_1(x_1) ) = 2x_1
20.75/6.13	
20.75/6.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	
20.75/6.13	
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(16)
20.75/6.13	Obligation:
20.75/6.13	Q DP problem:
20.75/6.13	The TRS P consists of the following rules:
20.75/6.13	
20.75/6.13	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
20.75/6.13	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
20.75/6.13	
20.75/6.13	The TRS R consists of the following rules:
20.75/6.13	
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
20.75/6.13	   c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.75/6.13	   c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))))
20.75/6.13	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))))
20.75/6.13	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1)
20.75/6.13	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1)
20.75/6.13	
20.75/6.13	Q is empty.
20.75/6.13	We have to consider all minimal (P,Q,R)-chains.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(17) DependencyGraphProof (EQUIVALENT)
20.75/6.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
20.75/6.13	----------------------------------------
20.75/6.13	
20.75/6.13	(18)
20.75/6.13	TRUE
20.75/6.17	EOF