11.42/3.72	YES
11.42/3.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
11.42/3.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.42/3.78	
11.42/3.78	
11.42/3.78	Termination w.r.t. Q of the given QTRS could be proven:
11.42/3.78	
11.42/3.78	(0) QTRS
11.42/3.78	(1) FlatCCProof [EQUIVALENT, 0 ms]
11.42/3.78	(2) QTRS
11.42/3.78	(3) RootLabelingProof [EQUIVALENT, 0 ms]
11.42/3.78	(4) QTRS
11.42/3.78	(5) QTRSRRRProof [EQUIVALENT, 55 ms]
11.42/3.78	(6) QTRS
11.42/3.78	(7) DependencyPairsProof [EQUIVALENT, 17 ms]
11.42/3.78	(8) QDP
11.42/3.78	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
11.42/3.78	(10) QDP
11.42/3.78	(11) QDPOrderProof [EQUIVALENT, 151 ms]
11.42/3.78	(12) QDP
11.42/3.78	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
11.42/3.78	(14) TRUE
11.42/3.78	
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(0)
11.42/3.78	Obligation:
11.42/3.78	Q restricted rewrite system:
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a(b(c(x1))) -> a(a(b(x1)))
11.42/3.78	   a(b(c(x1))) -> b(c(b(c(x1))))
11.42/3.78	   a(b(c(x1))) -> c(b(c(a(x1))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(1) FlatCCProof (EQUIVALENT)
11.42/3.78	We used flat context closure [ROOTLAB]
11.42/3.78	As Q is empty the flat context closure was sound AND complete.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(2)
11.42/3.78	Obligation:
11.42/3.78	Q restricted rewrite system:
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a(b(c(x1))) -> a(a(b(x1)))
11.42/3.78	   a(a(b(c(x1)))) -> a(b(c(b(c(x1)))))
11.42/3.78	   b(a(b(c(x1)))) -> b(b(c(b(c(x1)))))
11.42/3.78	   c(a(b(c(x1)))) -> c(b(c(b(c(x1)))))
11.42/3.78	   a(a(b(c(x1)))) -> a(c(b(c(a(x1)))))
11.42/3.78	   b(a(b(c(x1)))) -> b(c(b(c(a(x1)))))
11.42/3.78	   c(a(b(c(x1)))) -> c(c(b(c(a(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(3) RootLabelingProof (EQUIVALENT)
11.42/3.78	We used plain root labeling [ROOTLAB] with the following heuristic:
11.42/3.78	LabelAll: All function symbols get labeled
11.42/3.78	
11.42/3.78	As Q is empty the root labeling was sound AND complete.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(4)
11.42/3.78	Obligation:
11.42/3.78	Q restricted rewrite system:
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(5) QTRSRRRProof (EQUIVALENT)
11.42/3.78	Used ordering:
11.42/3.78	Polynomial interpretation [POLO]:
11.42/3.78	
11.42/3.78	   POL(a_{a_1}(x_1)) = x_1
11.42/3.78	   POL(a_{b_1}(x_1)) = 1 + x_1
11.42/3.78	   POL(a_{c_1}(x_1)) = x_1
11.42/3.78	   POL(b_{a_1}(x_1)) = x_1
11.42/3.78	   POL(b_{b_1}(x_1)) = x_1
11.42/3.78	   POL(b_{c_1}(x_1)) = x_1
11.42/3.78	   POL(c_{a_1}(x_1)) = x_1
11.42/3.78	   POL(c_{b_1}(x_1)) = x_1
11.42/3.78	   POL(c_{c_1}(x_1)) = x_1
11.42/3.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.42/3.78	
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))
11.42/3.78	
11.42/3.78	
11.42/3.78	
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(6)
11.42/3.78	Obligation:
11.42/3.78	Q restricted rewrite system:
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(7) DependencyPairsProof (EQUIVALENT)
11.42/3.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(8)
11.42/3.78	Obligation:
11.42/3.78	Q DP problem:
11.42/3.78	The TRS P consists of the following rules:
11.42/3.78	
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
11.42/3.78	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{B_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	We have to consider all minimal (P,Q,R)-chains.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(9) DependencyGraphProof (EQUIVALENT)
11.42/3.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(10)
11.42/3.78	Obligation:
11.42/3.78	Q DP problem:
11.42/3.78	The TRS P consists of the following rules:
11.42/3.78	
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	We have to consider all minimal (P,Q,R)-chains.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(11) QDPOrderProof (EQUIVALENT)
11.42/3.78	We use the reduction pair processor [LPAR04,JAR06].
11.42/3.78	
11.42/3.78	
11.42/3.78	The following pairs can be oriented strictly and are deleted.
11.42/3.78	
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(x1)
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
11.42/3.78	The remaining pairs can at least be oriented weakly.
11.42/3.78	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
11.42/3.78	
11.42/3.78	POL( A_{A_1}_1(x_1) ) = max{0, 2x_1 - 1}
11.42/3.78	POL( C_{A_1}_1(x_1) ) = 2x_1 + 1
11.42/3.78	POL( A_{B_1}_1(x_1) ) = max{0, 2x_1 - 1}
11.42/3.78	POL( b_{c_1}_1(x_1) ) = x_1
11.42/3.78	POL( c_{a_1}_1(x_1) ) = 2x_1 + 1
11.42/3.78	POL( c_{b_1}_1(x_1) ) = x_1 + 1
11.42/3.78	POL( a_{b_1}_1(x_1) ) = x_1 + 1
11.42/3.78	POL( a_{a_1}_1(x_1) ) = x_1 + 1
11.42/3.78	POL( b_{a_1}_1(x_1) ) = 2x_1
11.42/3.78	POL( b_{b_1}_1(x_1) ) = 0
11.42/3.78	POL( c_{c_1}_1(x_1) ) = x_1 + 1
11.42/3.78	POL( a_{c_1}_1(x_1) ) = 1
11.42/3.78	POL( B_{A_1}_1(x_1) ) = 2x_1
11.42/3.78	
11.42/3.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	
11.42/3.78	
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(12)
11.42/3.78	Obligation:
11.42/3.78	Q DP problem:
11.42/3.78	The TRS P consists of the following rules:
11.42/3.78	
11.42/3.78	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{A_1}(a_{b_1}(x1))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	
11.42/3.78	The TRS R consists of the following rules:
11.42/3.78	
11.42/3.78	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
11.42/3.78	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))))
11.42/3.78	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))
11.42/3.78	
11.42/3.78	Q is empty.
11.42/3.78	We have to consider all minimal (P,Q,R)-chains.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(13) DependencyGraphProof (EQUIVALENT)
11.42/3.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
11.42/3.78	----------------------------------------
11.42/3.78	
11.42/3.78	(14)
11.42/3.78	TRUE
11.83/3.90	EOF