18.72/5.76	YES
19.34/5.89	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
19.34/5.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.34/5.89	
19.34/5.89	
19.34/5.89	Termination w.r.t. Q of the given QTRS could be proven:
19.34/5.89	
19.34/5.89	(0) QTRS
19.34/5.89	(1) FlatCCProof [EQUIVALENT, 0 ms]
19.34/5.89	(2) QTRS
19.34/5.89	(3) RootLabelingProof [EQUIVALENT, 0 ms]
19.34/5.89	(4) QTRS
19.34/5.89	(5) QTRSRRRProof [EQUIVALENT, 35 ms]
19.34/5.89	(6) QTRS
19.34/5.89	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
19.34/5.89	(8) QDP
19.34/5.89	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
19.34/5.89	(10) QDP
19.34/5.89	(11) QDPOrderProof [EQUIVALENT, 101 ms]
19.34/5.89	(12) QDP
19.34/5.89	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
19.34/5.89	(14) TRUE
19.34/5.89	
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(0)
19.34/5.89	Obligation:
19.34/5.89	Q restricted rewrite system:
19.34/5.89	The TRS R consists of the following rules:
19.34/5.89	
19.34/5.89	   a(x1) -> x1
19.34/5.89	   a(a(x1)) -> a(b(c(x1)))
19.34/5.89	   b(x1) -> x1
19.34/5.89	   c(b(x1)) -> b(a(c(x1)))
19.34/5.89	
19.34/5.89	Q is empty.
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(1) FlatCCProof (EQUIVALENT)
19.34/5.89	We used flat context closure [ROOTLAB]
19.34/5.89	As Q is empty the flat context closure was sound AND complete.
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(2)
19.34/5.89	Obligation:
19.34/5.89	Q restricted rewrite system:
19.34/5.89	The TRS R consists of the following rules:
19.34/5.89	
19.34/5.89	   a(a(x1)) -> a(b(c(x1)))
19.34/5.89	   a(a(x1)) -> a(x1)
19.34/5.89	   b(a(x1)) -> b(x1)
19.34/5.89	   c(a(x1)) -> c(x1)
19.34/5.89	   a(b(x1)) -> a(x1)
19.34/5.89	   b(b(x1)) -> b(x1)
19.34/5.89	   c(b(x1)) -> c(x1)
19.34/5.89	   a(c(b(x1))) -> a(b(a(c(x1))))
19.34/5.89	   b(c(b(x1))) -> b(b(a(c(x1))))
19.34/5.89	   c(c(b(x1))) -> c(b(a(c(x1))))
19.34/5.89	
19.34/5.89	Q is empty.
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(3) RootLabelingProof (EQUIVALENT)
19.34/5.89	We used plain root labeling [ROOTLAB] with the following heuristic:
19.34/5.89	LabelAll: All function symbols get labeled
19.34/5.89	
19.34/5.89	As Q is empty the root labeling was sound AND complete.
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(4)
19.34/5.89	Obligation:
19.34/5.89	Q restricted rewrite system:
19.34/5.89	The TRS R consists of the following rules:
19.34/5.89	
19.34/5.89	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
19.34/5.89	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.89	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.89	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.89	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
19.34/5.89	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
19.34/5.89	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
19.34/5.89	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.89	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.89	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
19.34/5.89	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.89	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.89	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.89	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
19.34/5.89	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
19.34/5.89	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
19.34/5.89	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.89	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.89	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
19.34/5.89	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.89	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.89	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.89	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.89	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.89	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.89	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.89	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.89	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.89	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.89	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.89	
19.34/5.89	Q is empty.
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(5) QTRSRRRProof (EQUIVALENT)
19.34/5.89	Used ordering:
19.34/5.89	Polynomial interpretation [POLO]:
19.34/5.89	
19.34/5.89	   POL(a_{a_1}(x_1)) = 1 + x_1
19.34/5.89	   POL(a_{b_1}(x_1)) = x_1
19.34/5.89	   POL(a_{c_1}(x_1)) = x_1
19.34/5.89	   POL(b_{a_1}(x_1)) = 1 + x_1
19.34/5.89	   POL(b_{b_1}(x_1)) = 1 + x_1
19.34/5.89	   POL(b_{c_1}(x_1)) = 1 + x_1
19.34/5.89	   POL(c_{a_1}(x_1)) = x_1
19.34/5.89	   POL(c_{b_1}(x_1)) = x_1
19.34/5.89	   POL(c_{c_1}(x_1)) = x_1
19.34/5.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
19.34/5.89	
19.34/5.89	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
19.34/5.89	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.89	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
19.34/5.89	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
19.34/5.89	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
19.34/5.89	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
19.34/5.89	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
19.34/5.89	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
19.34/5.89	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
19.34/5.89	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.89	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.89	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
19.34/5.89	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.89	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.89	
19.34/5.89	
19.34/5.89	
19.34/5.89	
19.34/5.89	----------------------------------------
19.34/5.89	
19.34/5.89	(6)
19.34/5.89	Obligation:
19.34/5.89	Q restricted rewrite system:
19.34/5.89	The TRS R consists of the following rules:
19.34/5.89	
19.34/5.89	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.89	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.89	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.89	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.89	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.89	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.89	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.89	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.89	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.89	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.89	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	
19.34/5.90	Q is empty.
19.34/5.90	
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(7) DependencyPairsProof (EQUIVALENT)
19.34/5.90	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(8)
19.34/5.90	Obligation:
19.34/5.90	Q DP problem:
19.34/5.90	The TRS P consists of the following rules:
19.34/5.90	
19.34/5.90	   A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.90	   A_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(x1))
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{c_1}(x1))
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
19.34/5.90	   C_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	   A_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	
19.34/5.90	The TRS R consists of the following rules:
19.34/5.90	
19.34/5.90	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.90	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.90	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.90	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.90	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.90	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.90	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.90	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	
19.34/5.90	Q is empty.
19.34/5.90	We have to consider all minimal (P,Q,R)-chains.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(9) DependencyGraphProof (EQUIVALENT)
19.34/5.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(10)
19.34/5.90	Obligation:
19.34/5.90	Q DP problem:
19.34/5.90	The TRS P consists of the following rules:
19.34/5.90	
19.34/5.90	   A_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   A_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{c_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   C_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	
19.34/5.90	The TRS R consists of the following rules:
19.34/5.90	
19.34/5.90	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.90	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.90	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.90	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.90	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.90	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.90	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.90	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	
19.34/5.90	Q is empty.
19.34/5.90	We have to consider all minimal (P,Q,R)-chains.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(11) QDPOrderProof (EQUIVALENT)
19.34/5.90	We use the reduction pair processor [LPAR04,JAR06].
19.34/5.90	
19.34/5.90	
19.34/5.90	The following pairs can be oriented strictly and are deleted.
19.34/5.90	
19.34/5.90	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
19.34/5.90	   C_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1)
19.34/5.90	The remaining pairs can at least be oriented weakly.
19.34/5.90	Used ordering:  Polynomial interpretation [POLO]:
19.34/5.90	
19.34/5.90	   POL(A_{A_1}(x_1)) = x_1
19.34/5.90	   POL(A_{B_1}(x_1)) = x_1
19.34/5.90	   POL(A_{C_1}(x_1)) = x_1
19.34/5.90	   POL(B_{A_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(B_{C_1}(x_1)) = x_1
19.34/5.90	   POL(C_{A_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(C_{C_1}(x_1)) = x_1
19.34/5.90	   POL(a_{a_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(a_{b_1}(x_1)) = x_1
19.34/5.90	   POL(a_{c_1}(x_1)) = x_1
19.34/5.90	   POL(b_{a_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(b_{b_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(b_{c_1}(x_1)) = 1 + x_1
19.34/5.90	   POL(c_{a_1}(x_1)) = x_1
19.34/5.90	   POL(c_{b_1}(x_1)) = x_1
19.34/5.90	   POL(c_{c_1}(x_1)) = x_1
19.34/5.90	
19.34/5.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.34/5.90	
19.34/5.90	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.90	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.90	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.90	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.90	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.90	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.90	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	
19.34/5.90	
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(12)
19.34/5.90	Obligation:
19.34/5.90	Q DP problem:
19.34/5.90	The TRS P consists of the following rules:
19.34/5.90	
19.34/5.90	   A_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{c_1}(x1))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{c_1}(c_{a_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
19.34/5.90	   C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{c_1}(c_{b_1}(x1)))
19.34/5.90	   B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(c_{c_1}(x1)))
19.34/5.90	   A_{A_1}(a_{c_1}(x1)) -> C_{C_1}(x1)
19.34/5.90	
19.34/5.90	The TRS R consists of the following rules:
19.34/5.90	
19.34/5.90	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
19.34/5.90	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
19.34/5.90	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
19.34/5.90	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
19.34/5.90	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
19.34/5.90	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
19.34/5.90	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
19.34/5.90	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
19.34/5.90	   c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
19.34/5.90	
19.34/5.90	Q is empty.
19.34/5.90	We have to consider all minimal (P,Q,R)-chains.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(13) DependencyGraphProof (EQUIVALENT)
19.34/5.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 18 less nodes.
19.34/5.90	----------------------------------------
19.34/5.90	
19.34/5.90	(14)
19.34/5.90	TRUE
19.57/5.99	EOF