7.64/2.82	YES
7.83/2.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.83/2.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.83/2.88	
7.83/2.88	
7.83/2.88	Termination w.r.t. Q of the given QTRS could be proven:
7.83/2.88	
7.83/2.88	(0) QTRS
7.83/2.88	(1) FlatCCProof [EQUIVALENT, 0 ms]
7.83/2.88	(2) QTRS
7.83/2.88	(3) RootLabelingProof [EQUIVALENT, 13 ms]
7.83/2.88	(4) QTRS
7.83/2.88	(5) QTRSRRRProof [EQUIVALENT, 151 ms]
7.83/2.88	(6) QTRS
7.83/2.88	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
7.83/2.88	(8) QTRS
7.83/2.88	(9) QTRSRRRProof [EQUIVALENT, 2 ms]
7.83/2.88	(10) QTRS
7.83/2.88	(11) RisEmptyProof [EQUIVALENT, 0 ms]
7.83/2.88	(12) YES
7.83/2.88	
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(0)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	The TRS R consists of the following rules:
7.83/2.88	
7.83/2.88	   a(a(x1)) -> b(c(x1))
7.83/2.88	   b(b(x1)) -> c(d(x1))
7.83/2.88	   c(c(x1)) -> d(d(d(x1)))
7.83/2.88	   d(c(x1)) -> b(f(x1))
7.83/2.88	   d(d(d(x1))) -> a(c(x1))
7.83/2.88	   f(f(x1)) -> f(b(x1))
7.83/2.88	
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(1) FlatCCProof (EQUIVALENT)
7.83/2.88	We used flat context closure [ROOTLAB]
7.83/2.88	As Q is empty the flat context closure was sound AND complete.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(2)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	The TRS R consists of the following rules:
7.83/2.88	
7.83/2.88	   f(f(x1)) -> f(b(x1))
7.83/2.88	   a(a(a(x1))) -> a(b(c(x1)))
7.83/2.88	   b(a(a(x1))) -> b(b(c(x1)))
7.83/2.88	   c(a(a(x1))) -> c(b(c(x1)))
7.83/2.88	   d(a(a(x1))) -> d(b(c(x1)))
7.83/2.88	   f(a(a(x1))) -> f(b(c(x1)))
7.83/2.88	   a(b(b(x1))) -> a(c(d(x1)))
7.83/2.88	   b(b(b(x1))) -> b(c(d(x1)))
7.83/2.88	   c(b(b(x1))) -> c(c(d(x1)))
7.83/2.88	   d(b(b(x1))) -> d(c(d(x1)))
7.83/2.88	   f(b(b(x1))) -> f(c(d(x1)))
7.83/2.88	   a(c(c(x1))) -> a(d(d(d(x1))))
7.83/2.88	   b(c(c(x1))) -> b(d(d(d(x1))))
7.83/2.88	   c(c(c(x1))) -> c(d(d(d(x1))))
7.83/2.88	   d(c(c(x1))) -> d(d(d(d(x1))))
7.83/2.88	   f(c(c(x1))) -> f(d(d(d(x1))))
7.83/2.88	   a(d(c(x1))) -> a(b(f(x1)))
7.83/2.88	   b(d(c(x1))) -> b(b(f(x1)))
7.83/2.88	   c(d(c(x1))) -> c(b(f(x1)))
7.83/2.88	   d(d(c(x1))) -> d(b(f(x1)))
7.83/2.88	   f(d(c(x1))) -> f(b(f(x1)))
7.83/2.88	   a(d(d(d(x1)))) -> a(a(c(x1)))
7.83/2.88	   b(d(d(d(x1)))) -> b(a(c(x1)))
7.83/2.88	   c(d(d(d(x1)))) -> c(a(c(x1)))
7.83/2.88	   d(d(d(d(x1)))) -> d(a(c(x1)))
7.83/2.88	   f(d(d(d(x1)))) -> f(a(c(x1)))
7.83/2.88	
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(3) RootLabelingProof (EQUIVALENT)
7.83/2.88	We used plain root labeling [ROOTLAB] with the following heuristic:
7.83/2.88	LabelAll: All function symbols get labeled
7.83/2.88	
7.83/2.88	As Q is empty the root labeling was sound AND complete.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(4)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	The TRS R consists of the following rules:
7.83/2.88	
7.83/2.88	   f_{f_1}(f_{f_1}(x1)) -> f_{b_1}(b_{f_1}(x1))
7.83/2.88	   f_{f_1}(f_{b_1}(x1)) -> f_{b_1}(b_{b_1}(x1))
7.83/2.88	   f_{f_1}(f_{a_1}(x1)) -> f_{b_1}(b_{a_1}(x1))
7.83/2.88	   f_{f_1}(f_{c_1}(x1)) -> f_{b_1}(b_{c_1}(x1))
7.83/2.88	   f_{f_1}(f_{d_1}(x1)) -> f_{b_1}(b_{d_1}(x1))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{f_1}(x1))) -> a_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{d_1}(x1))) -> a_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{f_1}(x1))) -> b_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{d_1}(x1))) -> b_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{f_1}(x1))) -> c_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{d_1}(x1))) -> c_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{f_1}(x1))) -> d_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{b_1}(x1))) -> d_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{a_1}(x1))) -> d_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{c_1}(x1))) -> d_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{d_1}(x1))) -> d_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{f_1}(x1))) -> f_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{b_1}(x1))) -> f_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{a_1}(x1))) -> f_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{c_1}(x1))) -> f_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{d_1}(x1))) -> f_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{f_1}(x1))) -> a_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{d_1}(x1))) -> a_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{f_1}(x1))) -> b_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{f_1}(x1))) -> c_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{d_1}(x1))) -> c_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{f_1}(x1))) -> d_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{a_1}(x1))) -> d_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{c_1}(x1))) -> d_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{f_1}(x1))) -> f_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{b_1}(x1))) -> f_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{a_1}(x1))) -> f_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{c_1}(x1))) -> f_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{d_1}(x1))) -> f_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{f_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{f_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{f_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{f_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{f_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{b_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{a_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{c_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{d_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{f_1}(x1))) -> a_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{d_1}(x1))) -> a_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{f_1}(x1))) -> b_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{d_1}(x1))) -> b_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{f_1}(x1))) -> c_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{f_1}(x1))) -> d_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{b_1}(x1))) -> d_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{a_1}(x1))) -> d_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{c_1}(x1))) -> d_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{d_1}(x1))) -> d_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{f_1}(x1))) -> f_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{b_1}(x1))) -> f_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{a_1}(x1))) -> f_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{c_1}(x1))) -> f_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{d_1}(x1))) -> f_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(5) QTRSRRRProof (EQUIVALENT)
7.83/2.88	Used ordering:
7.83/2.88	Polynomial interpretation [POLO]:
7.83/2.88	
7.83/2.88	   POL(a_{a_1}(x_1)) = 93 + x_1
7.83/2.88	   POL(a_{b_1}(x_1)) = 91 + x_1
7.83/2.88	   POL(a_{c_1}(x_1)) = 92 + x_1
7.83/2.88	   POL(a_{d_1}(x_1)) = 92 + x_1
7.83/2.88	   POL(a_{f_1}(x_1)) = 177 + x_1
7.83/2.88	   POL(b_{a_1}(x_1)) = 83 + x_1
7.83/2.88	   POL(b_{b_1}(x_1)) = 83 + x_1
7.83/2.88	   POL(b_{c_1}(x_1)) = 84 + x_1
7.83/2.88	   POL(b_{d_1}(x_1)) = 84 + x_1
7.83/2.88	   POL(b_{f_1}(x_1)) = 165 + x_1
7.83/2.88	   POL(c_{a_1}(x_1)) = 100 + x_1
7.83/2.88	   POL(c_{b_1}(x_1)) = 99 + x_1
7.83/2.88	   POL(c_{c_1}(x_1)) = 100 + x_1
7.83/2.88	   POL(c_{d_1}(x_1)) = 99 + x_1
7.83/2.88	   POL(c_{f_1}(x_1)) = 185 + x_1
7.83/2.88	   POL(d_{a_1}(x_1)) = 65 + x_1
7.83/2.88	   POL(d_{b_1}(x_1)) = 65 + x_1
7.83/2.88	   POL(d_{c_1}(x_1)) = 66 + x_1
7.83/2.88	   POL(d_{d_1}(x_1)) = 66 + x_1
7.83/2.88	   POL(d_{f_1}(x_1)) = 147 + x_1
7.83/2.88	   POL(f_{a_1}(x_1)) = x_1
7.83/2.88	   POL(f_{b_1}(x_1)) = x_1
7.83/2.88	   POL(f_{c_1}(x_1)) = x_1
7.83/2.88	   POL(f_{d_1}(x_1)) = x_1
7.83/2.88	   POL(f_{f_1}(x_1)) = 85 + x_1
7.83/2.88	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.83/2.88	
7.83/2.88	   f_{f_1}(f_{f_1}(x1)) -> f_{b_1}(b_{f_1}(x1))
7.83/2.88	   f_{f_1}(f_{b_1}(x1)) -> f_{b_1}(b_{b_1}(x1))
7.83/2.88	   f_{f_1}(f_{a_1}(x1)) -> f_{b_1}(b_{a_1}(x1))
7.83/2.88	   f_{f_1}(f_{c_1}(x1)) -> f_{b_1}(b_{c_1}(x1))
7.83/2.88	   f_{f_1}(f_{d_1}(x1)) -> f_{b_1}(b_{d_1}(x1))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{f_1}(x1))) -> a_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   a_{a_1}(a_{a_1}(a_{d_1}(x1))) -> a_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{f_1}(x1))) -> b_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   b_{a_1}(a_{a_1}(a_{d_1}(x1))) -> b_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{f_1}(x1))) -> c_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   c_{a_1}(a_{a_1}(a_{d_1}(x1))) -> c_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{f_1}(x1))) -> d_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{b_1}(x1))) -> d_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{a_1}(x1))) -> d_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{c_1}(x1))) -> d_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   d_{a_1}(a_{a_1}(a_{d_1}(x1))) -> d_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{f_1}(x1))) -> f_{b_1}(b_{c_1}(c_{f_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{b_1}(x1))) -> f_{b_1}(b_{c_1}(c_{b_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{a_1}(x1))) -> f_{b_1}(b_{c_1}(c_{a_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{c_1}(x1))) -> f_{b_1}(b_{c_1}(c_{c_1}(x1)))
7.83/2.88	   f_{a_1}(a_{a_1}(a_{d_1}(x1))) -> f_{b_1}(b_{c_1}(c_{d_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{f_1}(x1))) -> a_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   a_{b_1}(b_{b_1}(b_{d_1}(x1))) -> a_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{f_1}(x1))) -> b_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{f_1}(x1))) -> c_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   c_{b_1}(b_{b_1}(b_{d_1}(x1))) -> c_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{f_1}(x1))) -> d_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{a_1}(x1))) -> d_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{c_1}(x1))) -> d_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   d_{b_1}(b_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{f_1}(x1))) -> f_{c_1}(c_{d_1}(d_{f_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{b_1}(x1))) -> f_{c_1}(c_{d_1}(d_{b_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{a_1}(x1))) -> f_{c_1}(c_{d_1}(d_{a_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{c_1}(x1))) -> f_{c_1}(c_{d_1}(d_{c_1}(x1)))
7.83/2.88	   f_{b_1}(b_{b_1}(b_{d_1}(x1))) -> f_{c_1}(c_{d_1}(d_{d_1}(x1)))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{f_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   a_{c_1}(c_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{f_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   b_{c_1}(c_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{f_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   c_{c_1}(c_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{f_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   d_{c_1}(c_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{f_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{b_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{a_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{c_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1))))
7.83/2.88	   f_{c_1}(c_{c_1}(c_{d_1}(x1))) -> f_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1))))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{f_1}(x1))) -> a_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   a_{d_1}(d_{c_1}(c_{d_1}(x1))) -> a_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{f_1}(x1))) -> b_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   b_{d_1}(d_{c_1}(c_{d_1}(x1))) -> b_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{f_1}(x1))) -> c_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{f_1}(x1))) -> d_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{b_1}(x1))) -> d_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{a_1}(x1))) -> d_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{c_1}(x1))) -> d_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   d_{d_1}(d_{c_1}(c_{d_1}(x1))) -> d_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{f_1}(x1))) -> f_{b_1}(b_{f_1}(f_{f_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{a_1}(x1))) -> f_{b_1}(b_{f_1}(f_{a_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{c_1}(x1))) -> f_{b_1}(b_{f_1}(f_{c_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   a_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   b_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   c_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   d_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> d_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{f_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{f_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{b_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{a_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{a_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{c_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{c_1}(x1)))
7.83/2.88	   f_{d_1}(d_{d_1}(d_{d_1}(d_{d_1}(x1)))) -> f_{a_1}(a_{c_1}(c_{d_1}(x1)))
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(6)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	The TRS R consists of the following rules:
7.83/2.88	
7.83/2.88	   c_{d_1}(d_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{b_1}(x1))) -> f_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{d_1}(x1))) -> f_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(7) QTRSRRRProof (EQUIVALENT)
7.83/2.88	Used ordering:
7.83/2.88	Polynomial interpretation [POLO]:
7.83/2.88	
7.83/2.88	   POL(b_{f_1}(x_1)) = x_1
7.83/2.88	   POL(c_{b_1}(x_1)) = x_1
7.83/2.88	   POL(c_{d_1}(x_1)) = 1 + x_1
7.83/2.88	   POL(d_{c_1}(x_1)) = x_1
7.83/2.88	   POL(f_{b_1}(x_1)) = x_1
7.83/2.88	   POL(f_{d_1}(x_1)) = 2 + x_1
7.83/2.88	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.83/2.88	
7.83/2.88	   c_{d_1}(d_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{b_1}(x1))) -> f_{b_1}(b_{f_1}(f_{b_1}(x1)))
7.83/2.88	   f_{d_1}(d_{c_1}(c_{d_1}(x1))) -> f_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(8)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	The TRS R consists of the following rules:
7.83/2.88	
7.83/2.88	   c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(9) QTRSRRRProof (EQUIVALENT)
7.83/2.88	Used ordering:
7.83/2.88	Polynomial interpretation [POLO]:
7.83/2.88	
7.83/2.88	   POL(b_{f_1}(x_1)) = x_1
7.83/2.88	   POL(c_{b_1}(x_1)) = x_1
7.83/2.88	   POL(c_{d_1}(x_1)) = x_1
7.83/2.88	   POL(d_{c_1}(x_1)) = 1 + x_1
7.83/2.88	   POL(f_{d_1}(x_1)) = x_1
7.83/2.88	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.83/2.88	
7.83/2.88	   c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{f_1}(f_{d_1}(x1)))
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(10)
7.83/2.88	Obligation:
7.83/2.88	Q restricted rewrite system:
7.83/2.88	R is empty.
7.83/2.88	Q is empty.
7.83/2.88	
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(11) RisEmptyProof (EQUIVALENT)
7.83/2.88	The TRS R is empty. Hence, termination is trivially proven.
7.83/2.88	----------------------------------------
7.83/2.88	
7.83/2.88	(12)
7.83/2.88	YES
8.06/2.93	EOF