230.73/59.46	YES
231.09/59.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
231.09/59.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
231.09/59.57	
231.09/59.57	
231.09/59.57	Termination w.r.t. Q of the given QTRS could be proven:
231.09/59.57	
231.09/59.57	(0) QTRS
231.09/59.57	(1) RootLabelingProof [EQUIVALENT, 0 ms]
231.09/59.57	(2) QTRS
231.09/59.57	(3) QTRSRRRProof [EQUIVALENT, 14 ms]
231.09/59.57	(4) QTRS
231.09/59.57	(5) DependencyPairsProof [EQUIVALENT, 71 ms]
231.09/59.57	(6) QDP
231.09/59.57	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
231.09/59.57	(8) QDP
231.09/59.57	(9) QDPOrderProof [EQUIVALENT, 1717 ms]
231.09/59.57	(10) QDP
231.09/59.57	(11) QDPOrderProof [EQUIVALENT, 3673 ms]
231.09/59.57	(12) QDP
231.09/59.57	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
231.09/59.57	(14) TRUE
231.09/59.57	
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(0)
231.09/59.57	Obligation:
231.09/59.57	Q restricted rewrite system:
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a(a(a(a(x1)))) -> a(a(b(b(b(b(b(b(x1))))))))
231.09/59.57	   b(b(a(a(x1)))) -> b(b(b(b(c(c(x1))))))
231.09/59.57	   a(a(b(b(b(b(c(c(x1)))))))) -> a(a(a(a(a(a(b(b(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(1) RootLabelingProof (EQUIVALENT)
231.09/59.57	We used plain root labeling [ROOTLAB] with the following heuristic:
231.09/59.57	LabelAll: All function symbols get labeled
231.09/59.57	
231.09/59.57	As Q is empty the root labeling was sound AND complete.
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(2)
231.09/59.57	Obligation:
231.09/59.57	Q restricted rewrite system:
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(3) QTRSRRRProof (EQUIVALENT)
231.09/59.57	Used ordering:
231.09/59.57	Polynomial interpretation [POLO]:
231.09/59.57	
231.09/59.57	   POL(a_{a_1}(x_1)) = x_1
231.09/59.57	   POL(a_{b_1}(x_1)) = x_1
231.09/59.57	   POL(a_{c_1}(x_1)) = 1 + x_1
231.09/59.57	   POL(b_{a_1}(x_1)) = x_1
231.09/59.57	   POL(b_{b_1}(x_1)) = x_1
231.09/59.57	   POL(b_{c_1}(x_1)) = x_1
231.09/59.57	   POL(c_{a_1}(x_1)) = x_1
231.09/59.57	   POL(c_{b_1}(x_1)) = x_1
231.09/59.57	   POL(c_{c_1}(x_1)) = x_1
231.09/59.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))
231.09/59.57	
231.09/59.57	
231.09/59.57	
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(4)
231.09/59.57	Obligation:
231.09/59.57	Q restricted rewrite system:
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(5) DependencyPairsProof (EQUIVALENT)
231.09/59.57	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(6)
231.09/59.57	Obligation:
231.09/59.57	Q DP problem:
231.09/59.57	The TRS P consists of the following rules:
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))
231.09/59.57	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> B_{B_1}(b_{a_1}(x1))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> B_{B_1}(b_{b_1}(x1))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> B_{B_1}(x1)
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> B_{B_1}(b_{c_1}(x1))
231.09/59.57	
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	We have to consider all minimal (P,Q,R)-chains.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(7) DependencyGraphProof (EQUIVALENT)
231.09/59.57	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 23 less nodes.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(8)
231.09/59.57	Obligation:
231.09/59.57	Q DP problem:
231.09/59.57	The TRS P consists of the following rules:
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
231.09/59.57	
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	We have to consider all minimal (P,Q,R)-chains.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(9) QDPOrderProof (EQUIVALENT)
231.09/59.57	We use the reduction pair processor [LPAR04,JAR06].
231.09/59.57	
231.09/59.57	
231.09/59.57	The following pairs can be oriented strictly and are deleted.
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
231.09/59.57	The remaining pairs can at least be oriented weakly.
231.09/59.57	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
231.09/59.57	
231.09/59.57	POL( A_{A_1}_1(x_1) ) = 2x_1
231.09/59.57	POL( a_{b_1}_1(x_1) ) = x_1
231.09/59.57	POL( b_{b_1}_1(x_1) ) = max{0, x_1 - 1}
231.09/59.57	POL( b_{a_1}_1(x_1) ) = x_1 + 2
231.09/59.57	POL( a_{a_1}_1(x_1) ) = x_1
231.09/59.57	POL( b_{c_1}_1(x_1) ) = x_1 + 2
231.09/59.57	POL( c_{c_1}_1(x_1) ) = x_1 + 2
231.09/59.57	POL( c_{a_1}_1(x_1) ) = x_1
231.09/59.57	POL( c_{b_1}_1(x_1) ) = x_1
231.09/59.57	
231.09/59.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
231.09/59.57	
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(10)
231.09/59.57	Obligation:
231.09/59.57	Q DP problem:
231.09/59.57	The TRS P consists of the following rules:
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
231.09/59.57	
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	We have to consider all minimal (P,Q,R)-chains.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(11) QDPOrderProof (EQUIVALENT)
231.09/59.57	We use the reduction pair processor [LPAR04,JAR06].
231.09/59.57	
231.09/59.57	
231.09/59.57	The following pairs can be oriented strictly and are deleted.
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
231.09/59.57	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
231.09/59.57	The remaining pairs can at least be oriented weakly.
231.09/59.57	Used ordering:  Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(A_{A_1}(x_1)) =  	[[0]] 	 +  	[[1, 0, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(a_{b_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[0, 1, 0], [1, 1, 1], [0, 1, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(b_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[0, 1, 0], [0, 0, 1], [1, 0, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(b_{c_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[0, 0, 0], [1, 0, 0], [0, 1, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(c_{c_1}(x_1)) =  	[[1], [0], [0]] 	 +  	[[1, 1, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(c_{a_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[0, 1, 0], [0, 0, 0], [1, 1, 1]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(a_{a_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 1, 1], [0, 0, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[0, 0, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	   <<<
231.09/59.57	 POL(c_{b_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 1, 1], [0, 1, 0], [1, 1, 1]] 	* 	x_1
231.09/59.57	>>>
231.09/59.57	
231.09/59.57	
231.09/59.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
231.09/59.57	
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	
231.09/59.57	
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(12)
231.09/59.57	Obligation:
231.09/59.57	Q DP problem:
231.09/59.57	The TRS P consists of the following rules:
231.09/59.57	
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	
231.09/59.57	The TRS R consists of the following rules:
231.09/59.57	
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))
231.09/59.57	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
231.09/59.57	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))
231.09/59.57	
231.09/59.57	Q is empty.
231.09/59.57	We have to consider all minimal (P,Q,R)-chains.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(13) DependencyGraphProof (EQUIVALENT)
231.09/59.57	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
231.09/59.57	----------------------------------------
231.09/59.57	
231.09/59.57	(14)
231.09/59.57	TRUE
231.48/59.67	EOF