5.72/2.39	YES
6.01/2.45	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.01/2.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.01/2.45	
6.01/2.45	
6.01/2.45	Termination w.r.t. Q of the given QTRS could be proven:
6.01/2.45	
6.01/2.45	(0) QTRS
6.01/2.45	(1) FlatCCProof [EQUIVALENT, 0 ms]
6.01/2.45	(2) QTRS
6.01/2.45	(3) RootLabelingProof [EQUIVALENT, 13 ms]
6.01/2.45	(4) QTRS
6.01/2.45	(5) QTRSRRRProof [EQUIVALENT, 172 ms]
6.01/2.45	(6) QTRS
6.01/2.45	(7) QTRSRRRProof [EQUIVALENT, 3 ms]
6.01/2.45	(8) QTRS
6.01/2.45	(9) RisEmptyProof [EQUIVALENT, 0 ms]
6.01/2.45	(10) YES
6.01/2.45	
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(0)
6.01/2.45	Obligation:
6.01/2.45	Q restricted rewrite system:
6.01/2.45	The TRS R consists of the following rules:
6.01/2.45	
6.01/2.45	   a(x1) -> g(d(x1))
6.01/2.45	   b(b(b(x1))) -> c(d(c(x1)))
6.01/2.45	   b(b(x1)) -> a(g(g(x1)))
6.01/2.45	   c(d(x1)) -> g(g(x1))
6.01/2.45	   g(g(g(x1))) -> b(b(x1))
6.01/2.45	
6.01/2.45	Q is empty.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(1) FlatCCProof (EQUIVALENT)
6.01/2.45	We used flat context closure [ROOTLAB]
6.01/2.45	As Q is empty the flat context closure was sound AND complete.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(2)
6.01/2.45	Obligation:
6.01/2.45	Q restricted rewrite system:
6.01/2.45	The TRS R consists of the following rules:
6.01/2.45	
6.01/2.45	   a(a(x1)) -> a(g(d(x1)))
6.01/2.45	   g(a(x1)) -> g(g(d(x1)))
6.01/2.45	   d(a(x1)) -> d(g(d(x1)))
6.01/2.45	   b(a(x1)) -> b(g(d(x1)))
6.01/2.45	   c(a(x1)) -> c(g(d(x1)))
6.01/2.45	   a(b(b(b(x1)))) -> a(c(d(c(x1))))
6.01/2.45	   g(b(b(b(x1)))) -> g(c(d(c(x1))))
6.01/2.45	   d(b(b(b(x1)))) -> d(c(d(c(x1))))
6.01/2.45	   b(b(b(b(x1)))) -> b(c(d(c(x1))))
6.01/2.45	   c(b(b(b(x1)))) -> c(c(d(c(x1))))
6.01/2.45	   a(b(b(x1))) -> a(a(g(g(x1))))
6.01/2.45	   g(b(b(x1))) -> g(a(g(g(x1))))
6.01/2.45	   d(b(b(x1))) -> d(a(g(g(x1))))
6.01/2.45	   b(b(b(x1))) -> b(a(g(g(x1))))
6.01/2.45	   c(b(b(x1))) -> c(a(g(g(x1))))
6.01/2.45	   a(c(d(x1))) -> a(g(g(x1)))
6.01/2.45	   g(c(d(x1))) -> g(g(g(x1)))
6.01/2.45	   d(c(d(x1))) -> d(g(g(x1)))
6.01/2.45	   b(c(d(x1))) -> b(g(g(x1)))
6.01/2.45	   c(c(d(x1))) -> c(g(g(x1)))
6.01/2.45	   a(g(g(g(x1)))) -> a(b(b(x1)))
6.01/2.45	   g(g(g(g(x1)))) -> g(b(b(x1)))
6.01/2.45	   d(g(g(g(x1)))) -> d(b(b(x1)))
6.01/2.45	   b(g(g(g(x1)))) -> b(b(b(x1)))
6.01/2.45	   c(g(g(g(x1)))) -> c(b(b(x1)))
6.01/2.45	
6.01/2.45	Q is empty.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(3) RootLabelingProof (EQUIVALENT)
6.01/2.45	We used plain root labeling [ROOTLAB] with the following heuristic:
6.01/2.45	LabelAll: All function symbols get labeled
6.01/2.45	
6.01/2.45	As Q is empty the root labeling was sound AND complete.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(4)
6.01/2.45	Obligation:
6.01/2.45	Q restricted rewrite system:
6.01/2.45	The TRS R consists of the following rules:
6.01/2.45	
6.01/2.45	   a_{a_1}(a_{a_1}(x1)) -> a_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   a_{a_1}(a_{g_1}(x1)) -> a_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   a_{a_1}(a_{d_1}(x1)) -> a_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   a_{a_1}(a_{b_1}(x1)) -> a_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   a_{a_1}(a_{c_1}(x1)) -> a_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   g_{a_1}(a_{a_1}(x1)) -> g_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   g_{a_1}(a_{g_1}(x1)) -> g_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   g_{a_1}(a_{d_1}(x1)) -> g_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   g_{a_1}(a_{b_1}(x1)) -> g_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   g_{a_1}(a_{c_1}(x1)) -> g_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   d_{a_1}(a_{a_1}(x1)) -> d_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   d_{a_1}(a_{g_1}(x1)) -> d_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   d_{a_1}(a_{d_1}(x1)) -> d_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   d_{a_1}(a_{b_1}(x1)) -> d_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   d_{a_1}(a_{c_1}(x1)) -> d_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   b_{a_1}(a_{a_1}(x1)) -> b_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   b_{a_1}(a_{g_1}(x1)) -> b_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   b_{a_1}(a_{d_1}(x1)) -> b_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   b_{a_1}(a_{b_1}(x1)) -> b_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   b_{a_1}(a_{c_1}(x1)) -> b_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   c_{a_1}(a_{a_1}(x1)) -> c_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   c_{a_1}(a_{g_1}(x1)) -> c_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   c_{a_1}(a_{d_1}(x1)) -> c_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   c_{a_1}(a_{b_1}(x1)) -> c_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   c_{a_1}(a_{c_1}(x1)) -> c_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{g_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{d_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{a_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{g_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{d_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{c_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{a_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{g_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{d_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{c_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{g_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{g_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{d_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{g_1}(x1))) -> a_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{a_1}(x1))) -> g_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{g_1}(x1))) -> g_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{d_1}(x1))) -> g_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{b_1}(x1))) -> g_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{c_1}(x1))) -> g_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{g_1}(x1))) -> d_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{g_1}(x1))) -> b_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{g_1}(x1))) -> c_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	
6.01/2.45	Q is empty.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(5) QTRSRRRProof (EQUIVALENT)
6.01/2.45	Used ordering:
6.01/2.45	Polynomial interpretation [POLO]:
6.01/2.45	
6.01/2.45	   POL(a_{a_1}(x_1)) = 21 + x_1
6.01/2.45	   POL(a_{b_1}(x_1)) = 52 + x_1
6.01/2.45	   POL(a_{c_1}(x_1)) = 70 + x_1
6.01/2.45	   POL(a_{d_1}(x_1)) = 20 + x_1
6.01/2.45	   POL(a_{g_1}(x_1)) = 38 + x_1
6.01/2.45	   POL(b_{a_1}(x_1)) = 4 + x_1
6.01/2.45	   POL(b_{b_1}(x_1)) = 35 + x_1
6.01/2.45	   POL(b_{c_1}(x_1)) = 54 + x_1
6.01/2.45	   POL(b_{d_1}(x_1)) = x_1
6.01/2.45	   POL(b_{g_1}(x_1)) = 22 + x_1
6.01/2.45	   POL(c_{a_1}(x_1)) = 4 + x_1
6.01/2.45	   POL(c_{b_1}(x_1)) = 35 + x_1
6.01/2.45	   POL(c_{c_1}(x_1)) = 54 + x_1
6.01/2.45	   POL(c_{d_1}(x_1)) = x_1
6.01/2.45	   POL(c_{g_1}(x_1)) = 22 + x_1
6.01/2.45	   POL(d_{a_1}(x_1)) = x_1
6.01/2.45	   POL(d_{b_1}(x_1)) = 31 + x_1
6.01/2.45	   POL(d_{c_1}(x_1)) = 50 + x_1
6.01/2.45	   POL(d_{d_1}(x_1)) = x_1
6.01/2.45	   POL(d_{g_1}(x_1)) = 17 + x_1
6.01/2.45	   POL(g_{a_1}(x_1)) = 7 + x_1
6.01/2.45	   POL(g_{b_1}(x_1)) = 38 + x_1
6.01/2.45	   POL(g_{c_1}(x_1)) = 57 + x_1
6.01/2.45	   POL(g_{d_1}(x_1)) = 2 + x_1
6.01/2.45	   POL(g_{g_1}(x_1)) = 24 + x_1
6.01/2.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.01/2.45	
6.01/2.45	   a_{a_1}(a_{a_1}(x1)) -> a_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   a_{a_1}(a_{g_1}(x1)) -> a_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   a_{a_1}(a_{d_1}(x1)) -> a_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   a_{a_1}(a_{b_1}(x1)) -> a_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   a_{a_1}(a_{c_1}(x1)) -> a_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   g_{a_1}(a_{a_1}(x1)) -> g_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   g_{a_1}(a_{g_1}(x1)) -> g_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   g_{a_1}(a_{d_1}(x1)) -> g_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   g_{a_1}(a_{b_1}(x1)) -> g_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   g_{a_1}(a_{c_1}(x1)) -> g_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   d_{a_1}(a_{a_1}(x1)) -> d_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   d_{a_1}(a_{g_1}(x1)) -> d_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   d_{a_1}(a_{d_1}(x1)) -> d_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   d_{a_1}(a_{b_1}(x1)) -> d_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   d_{a_1}(a_{c_1}(x1)) -> d_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   b_{a_1}(a_{a_1}(x1)) -> b_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   b_{a_1}(a_{g_1}(x1)) -> b_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   b_{a_1}(a_{b_1}(x1)) -> b_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   c_{a_1}(a_{a_1}(x1)) -> c_{g_1}(g_{d_1}(d_{a_1}(x1)))
6.01/2.45	   c_{a_1}(a_{g_1}(x1)) -> c_{g_1}(g_{d_1}(d_{g_1}(x1)))
6.01/2.45	   c_{a_1}(a_{b_1}(x1)) -> c_{g_1}(g_{d_1}(d_{b_1}(x1)))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> g_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> d_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{a_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{g_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{g_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{d_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{b_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{d_1}(d_{c_1}(c_{c_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{g_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{d_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{a_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{g_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{d_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{b_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   g_{b_1}(b_{b_1}(b_{c_1}(x1))) -> g_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{a_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{g_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{d_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{b_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   d_{b_1}(b_{b_1}(b_{c_1}(x1))) -> d_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{g_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{a_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{g_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{g_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{d_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{d_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{b_1}(x1))))
6.01/2.45	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{g_1}(g_{g_1}(g_{c_1}(x1))))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{g_1}(x1))) -> a_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{a_1}(x1))) -> g_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{g_1}(x1))) -> g_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{d_1}(x1))) -> g_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{b_1}(x1))) -> g_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   g_{c_1}(c_{d_1}(d_{c_1}(x1))) -> g_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{g_1}(x1))) -> d_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{g_1}(x1))) -> b_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{g_1}(g_{g_1}(g_{a_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{g_1}(x1))) -> c_{g_1}(g_{g_1}(g_{g_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{g_1}(g_{g_1}(g_{d_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{g_1}(g_{g_1}(g_{b_1}(x1)))
6.01/2.45	   c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{g_1}(g_{g_1}(g_{c_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   a_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   g_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> g_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   d_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   b_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{g_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{g_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{d_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
6.01/2.45	   c_{g_1}(g_{g_1}(g_{g_1}(g_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
6.01/2.45	
6.01/2.45	
6.01/2.45	
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(6)
6.01/2.45	Obligation:
6.01/2.45	Q restricted rewrite system:
6.01/2.45	The TRS R consists of the following rules:
6.01/2.45	
6.01/2.45	   b_{a_1}(a_{d_1}(x1)) -> b_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   b_{a_1}(a_{c_1}(x1)) -> b_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   c_{a_1}(a_{d_1}(x1)) -> c_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   c_{a_1}(a_{c_1}(x1)) -> c_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	
6.01/2.45	Q is empty.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(7) QTRSRRRProof (EQUIVALENT)
6.01/2.45	Used ordering:
6.01/2.45	Polynomial interpretation [POLO]:
6.01/2.45	
6.01/2.45	   POL(a_{c_1}(x_1)) = x_1
6.01/2.45	   POL(a_{d_1}(x_1)) = x_1
6.01/2.45	   POL(b_{a_1}(x_1)) = 1 + x_1
6.01/2.45	   POL(b_{g_1}(x_1)) = x_1
6.01/2.45	   POL(c_{a_1}(x_1)) = 1 + x_1
6.01/2.45	   POL(c_{g_1}(x_1)) = x_1
6.01/2.45	   POL(d_{c_1}(x_1)) = x_1
6.01/2.45	   POL(d_{d_1}(x_1)) = x_1
6.01/2.45	   POL(g_{d_1}(x_1)) = x_1
6.01/2.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.01/2.45	
6.01/2.45	   b_{a_1}(a_{d_1}(x1)) -> b_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   b_{a_1}(a_{c_1}(x1)) -> b_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	   c_{a_1}(a_{d_1}(x1)) -> c_{g_1}(g_{d_1}(d_{d_1}(x1)))
6.01/2.45	   c_{a_1}(a_{c_1}(x1)) -> c_{g_1}(g_{d_1}(d_{c_1}(x1)))
6.01/2.45	
6.01/2.45	
6.01/2.45	
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(8)
6.01/2.45	Obligation:
6.01/2.45	Q restricted rewrite system:
6.01/2.45	R is empty.
6.01/2.45	Q is empty.
6.01/2.45	
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(9) RisEmptyProof (EQUIVALENT)
6.01/2.45	The TRS R is empty. Hence, termination is trivially proven.
6.01/2.45	----------------------------------------
6.01/2.45	
6.01/2.45	(10)
6.01/2.45	YES
6.30/2.55	EOF