16.91/5.25	YES
17.57/5.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.57/5.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.57/5.42	
17.57/5.42	
17.57/5.42	Termination w.r.t. Q of the given QTRS could be proven:
17.57/5.42	
17.57/5.42	(0) QTRS
17.57/5.42	(1) QTRS Reverse [EQUIVALENT, 0 ms]
17.57/5.42	(2) QTRS
17.57/5.42	(3) FlatCCProof [EQUIVALENT, 0 ms]
17.57/5.42	(4) QTRS
17.57/5.42	(5) RootLabelingProof [EQUIVALENT, 0 ms]
17.57/5.42	(6) QTRS
17.57/5.42	(7) QTRSRRRProof [EQUIVALENT, 40 ms]
17.57/5.42	(8) QTRS
17.57/5.42	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
17.57/5.42	(10) QDP
17.57/5.42	(11) DependencyGraphProof [EQUIVALENT, 4 ms]
17.57/5.42	(12) QDP
17.57/5.42	(13) QDPOrderProof [EQUIVALENT, 200 ms]
17.57/5.42	(14) QDP
17.57/5.42	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
17.57/5.42	(16) TRUE
17.57/5.42	
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(0)
17.57/5.42	Obligation:
17.57/5.42	Q restricted rewrite system:
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a(x1) -> x1
17.57/5.42	   a(b(x1)) -> b(c(a(x1)))
17.57/5.42	   c(c(a(x1))) -> a(b(a(x1)))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(1) QTRS Reverse (EQUIVALENT)
17.57/5.42	We applied the QTRS Reverse Processor [REVERSE].
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(2)
17.57/5.42	Obligation:
17.57/5.42	Q restricted rewrite system:
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a(x1) -> x1
17.57/5.42	   b(a(x1)) -> a(c(b(x1)))
17.57/5.42	   a(c(c(x1))) -> a(b(a(x1)))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(3) FlatCCProof (EQUIVALENT)
17.57/5.42	We used flat context closure [ROOTLAB]
17.57/5.42	As Q is empty the flat context closure was sound AND complete.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(4)
17.57/5.42	Obligation:
17.57/5.42	Q restricted rewrite system:
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a(c(c(x1))) -> a(b(a(x1)))
17.57/5.42	   a(a(x1)) -> a(x1)
17.57/5.42	   b(a(x1)) -> b(x1)
17.57/5.42	   c(a(x1)) -> c(x1)
17.57/5.42	   a(b(a(x1))) -> a(a(c(b(x1))))
17.57/5.42	   b(b(a(x1))) -> b(a(c(b(x1))))
17.57/5.42	   c(b(a(x1))) -> c(a(c(b(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(5) RootLabelingProof (EQUIVALENT)
17.57/5.42	We used plain root labeling [ROOTLAB] with the following heuristic:
17.57/5.42	LabelAll: All function symbols get labeled
17.57/5.42	
17.57/5.42	As Q is empty the root labeling was sound AND complete.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(6)
17.57/5.42	Obligation:
17.57/5.42	Q restricted rewrite system:
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
17.57/5.42	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
17.57/5.42	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
17.57/5.42	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
17.57/5.42	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
17.57/5.42	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
17.57/5.42	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(7) QTRSRRRProof (EQUIVALENT)
17.57/5.42	Used ordering:
17.57/5.42	Polynomial interpretation [POLO]:
17.57/5.42	
17.57/5.42	   POL(a_{a_1}(x_1)) = 1 + x_1
17.57/5.42	   POL(a_{b_1}(x_1)) = x_1
17.57/5.42	   POL(a_{c_1}(x_1)) = x_1
17.57/5.42	   POL(b_{a_1}(x_1)) = 1 + x_1
17.57/5.42	   POL(b_{b_1}(x_1)) = x_1
17.57/5.42	   POL(b_{c_1}(x_1)) = x_1
17.57/5.42	   POL(c_{a_1}(x_1)) = 1 + x_1
17.57/5.42	   POL(c_{b_1}(x_1)) = x_1
17.57/5.42	   POL(c_{c_1}(x_1)) = 1 + x_1
17.57/5.42	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
17.57/5.42	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
17.57/5.42	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
17.57/5.42	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
17.57/5.42	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
17.57/5.42	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
17.57/5.42	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
17.57/5.42	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
17.57/5.42	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
17.57/5.42	
17.57/5.42	
17.57/5.42	
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(8)
17.57/5.42	Obligation:
17.57/5.42	Q restricted rewrite system:
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(9) DependencyPairsProof (EQUIVALENT)
17.57/5.42	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(10)
17.57/5.42	Obligation:
17.57/5.42	Q DP problem:
17.57/5.42	The TRS P consists of the following rules:
17.57/5.42	
17.57/5.42	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	We have to consider all minimal (P,Q,R)-chains.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(11) DependencyGraphProof (EQUIVALENT)
17.57/5.42	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(12)
17.57/5.42	Obligation:
17.57/5.42	Q DP problem:
17.57/5.42	The TRS P consists of the following rules:
17.57/5.42	
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	We have to consider all minimal (P,Q,R)-chains.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(13) QDPOrderProof (EQUIVALENT)
17.57/5.42	We use the reduction pair processor [LPAR04,JAR06].
17.57/5.42	
17.57/5.42	
17.57/5.42	The following pairs can be oriented strictly and are deleted.
17.57/5.42	
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1))
17.57/5.42	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	The remaining pairs can at least be oriented weakly.
17.57/5.42	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
17.57/5.42	
17.57/5.42	POL( A_{C_1}_1(x_1) ) = x_1 + 2
17.57/5.42	POL( C_{B_1}_1(x_1) ) = max{0, 2x_1 - 2}
17.57/5.42	POL( a_{c_1}_1(x_1) ) = x_1 + 1
17.57/5.42	POL( b_{a_1}_1(x_1) ) = 2x_1
17.57/5.42	POL( A_{B_1}_1(x_1) ) = x_1 + 1
17.57/5.42	POL( a_{a_1}_1(x_1) ) = 2x_1 + 1
17.57/5.42	POL( c_{b_1}_1(x_1) ) = x_1
17.57/5.42	POL( c_{a_1}_1(x_1) ) = 2x_1
17.57/5.42	POL( b_{c_1}_1(x_1) ) = 0
17.57/5.42	POL( a_{b_1}_1(x_1) ) = x_1 + 1
17.57/5.42	POL( b_{b_1}_1(x_1) ) = x_1
17.57/5.42	POL( c_{c_1}_1(x_1) ) = 2x_1 + 2
17.57/5.42	POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
17.57/5.42	
17.57/5.42	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.57/5.42	
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	
17.57/5.42	
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(14)
17.57/5.42	Obligation:
17.57/5.42	Q DP problem:
17.57/5.42	The TRS P consists of the following rules:
17.57/5.42	
17.57/5.42	   C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
17.57/5.42	   C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
17.57/5.42	
17.57/5.42	The TRS R consists of the following rules:
17.57/5.42	
17.57/5.42	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
17.57/5.42	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
17.57/5.42	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
17.57/5.42	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
17.57/5.42	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
17.57/5.42	
17.57/5.42	Q is empty.
17.57/5.42	We have to consider all minimal (P,Q,R)-chains.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(15) DependencyGraphProof (EQUIVALENT)
17.57/5.42	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
17.57/5.42	----------------------------------------
17.57/5.42	
17.57/5.42	(16)
17.57/5.42	TRUE
18.11/5.52	EOF