132.52/34.29	YES
132.95/34.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
132.95/34.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
132.95/34.39	
132.95/34.39	
132.95/34.39	Termination of the given RelTRS could be proven:
132.95/34.39	
132.95/34.39	(0) RelTRS
132.95/34.39	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
132.95/34.39	(2) RelTRS
132.95/34.39	(3) FlatCCProof [EQUIVALENT, 0 ms]
132.95/34.39	(4) RelTRS
132.95/34.39	(5) RootLabelingProof [EQUIVALENT, 0 ms]
132.95/34.39	(6) RelTRS
132.95/34.39	(7) RelTRSRRRProof [EQUIVALENT, 520 ms]
132.95/34.39	(8) RelTRS
132.95/34.39	(9) SIsEmptyProof [EQUIVALENT, 0 ms]
132.95/34.39	(10) QTRS
132.95/34.39	(11) QTRSRRRProof [EQUIVALENT, 11 ms]
132.95/34.39	(12) QTRS
132.95/34.39	(13) DependencyPairsProof [EQUIVALENT, 0 ms]
132.95/34.39	(14) QDP
132.95/34.39	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
132.95/34.39	(16) QDP
132.95/34.39	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
132.95/34.39	(18) YES
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(0)
132.95/34.39	Obligation:
132.95/34.39	Relative term rewrite system:
132.95/34.39	The relative TRS consists of the following R rules:
132.95/34.39	
132.95/34.39	   c(c(b(x1))) -> b(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> b(a(a(x1)))
132.95/34.39	   c(c(c(x1))) -> a(a(b(x1)))
132.95/34.39	   c(b(c(x1))) -> c(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> b(b(c(x1)))
132.95/34.39	   c(b(c(x1))) -> c(c(b(x1)))
132.95/34.39	
132.95/34.39	The relative TRS consists of the following S rules:
132.95/34.39	
132.95/34.39	   a(b(b(x1))) -> b(a(c(x1)))
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(1) RelTRS Reverse (EQUIVALENT)
132.95/34.39	We have reversed the following relative TRS [REVERSE]:
132.95/34.39	The set of rules R is 
132.95/34.39	   c(c(b(x1))) -> b(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> b(a(a(x1)))
132.95/34.39	   c(c(c(x1))) -> a(a(b(x1)))
132.95/34.39	   c(b(c(x1))) -> c(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> b(b(c(x1)))
132.95/34.39	   c(b(c(x1))) -> c(c(b(x1)))
132.95/34.39	
132.95/34.39	The set of rules S is 
132.95/34.39	   a(b(b(x1))) -> b(a(c(x1)))
132.95/34.39	
132.95/34.39	We have obtained the following relative TRS:
132.95/34.39	The set of rules R is 
132.95/34.39	   b(c(c(x1))) -> b(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> a(a(b(x1)))
132.95/34.39	   c(c(c(x1))) -> b(a(a(x1)))
132.95/34.39	   c(b(c(x1))) -> b(b(c(x1)))
132.95/34.39	   c(a(c(x1))) -> c(b(b(x1)))
132.95/34.39	   c(b(c(x1))) -> b(c(c(x1)))
132.95/34.39	
132.95/34.39	The set of rules S is 
132.95/34.39	   b(b(a(x1))) -> c(a(b(x1)))
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(2)
132.95/34.39	Obligation:
132.95/34.39	Relative term rewrite system:
132.95/34.39	The relative TRS consists of the following R rules:
132.95/34.39	
132.95/34.39	   b(c(c(x1))) -> b(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> a(a(b(x1)))
132.95/34.39	   c(c(c(x1))) -> b(a(a(x1)))
132.95/34.39	   c(b(c(x1))) -> b(b(c(x1)))
132.95/34.39	   c(a(c(x1))) -> c(b(b(x1)))
132.95/34.39	   c(b(c(x1))) -> b(c(c(x1)))
132.95/34.39	
132.95/34.39	The relative TRS consists of the following S rules:
132.95/34.39	
132.95/34.39	   b(b(a(x1))) -> c(a(b(x1)))
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(3) FlatCCProof (EQUIVALENT)
132.95/34.39	We used flat context closure [ROOTLAB]
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(4)
132.95/34.39	Obligation:
132.95/34.39	Relative term rewrite system:
132.95/34.39	The relative TRS consists of the following R rules:
132.95/34.39	
132.95/34.39	   b(c(c(x1))) -> b(b(b(x1)))
132.95/34.39	   c(a(c(x1))) -> c(b(b(x1)))
132.95/34.39	   b(c(a(c(x1)))) -> b(a(a(b(x1))))
132.95/34.39	   c(c(a(c(x1)))) -> c(a(a(b(x1))))
132.95/34.39	   a(c(a(c(x1)))) -> a(a(a(b(x1))))
132.95/34.39	   b(c(c(c(x1)))) -> b(b(a(a(x1))))
132.95/34.39	   c(c(c(c(x1)))) -> c(b(a(a(x1))))
132.95/34.39	   a(c(c(c(x1)))) -> a(b(a(a(x1))))
132.95/34.39	   b(c(b(c(x1)))) -> b(b(b(c(x1))))
132.95/34.39	   c(c(b(c(x1)))) -> c(b(b(c(x1))))
132.95/34.39	   a(c(b(c(x1)))) -> a(b(b(c(x1))))
132.95/34.39	   b(c(b(c(x1)))) -> b(b(c(c(x1))))
132.95/34.39	   c(c(b(c(x1)))) -> c(b(c(c(x1))))
132.95/34.39	   a(c(b(c(x1)))) -> a(b(c(c(x1))))
132.95/34.39	
132.95/34.39	The relative TRS consists of the following S rules:
132.95/34.39	
132.95/34.39	   b(b(b(a(x1)))) -> b(c(a(b(x1))))
132.95/34.39	   c(b(b(a(x1)))) -> c(c(a(b(x1))))
132.95/34.39	   a(b(b(a(x1)))) -> a(c(a(b(x1))))
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(5) RootLabelingProof (EQUIVALENT)
132.95/34.39	We used plain root labeling [ROOTLAB] with the following heuristic:
132.95/34.39	LabelAll: All function symbols get labeled
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(6)
132.95/34.39	Obligation:
132.95/34.39	Relative term rewrite system:
132.95/34.39	The relative TRS consists of the following R rules:
132.95/34.39	
132.95/34.39	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	The relative TRS consists of the following S rules:
132.95/34.39	
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(7) RelTRSRRRProof (EQUIVALENT)
132.95/34.39	We used the following monotonic ordering for rule removal:
132.95/34.39	Polynomial interpretation [POLO]:
132.95/34.39	
132.95/34.39	   POL(a_{a_1}(x_1)) = 1 + 2*x_1
132.95/34.39	   POL(a_{b_1}(x_1)) = 2*x_1
132.95/34.39	   POL(a_{c_1}(x_1)) = 3 + x_1
132.95/34.39	   POL(b_{a_1}(x_1)) = 1 + 2*x_1
132.95/34.39	   POL(b_{b_1}(x_1)) = 2*x_1
132.95/34.39	   POL(b_{c_1}(x_1)) = 3 + x_1
132.95/34.39	   POL(c_{a_1}(x_1)) = 4*x_1
132.95/34.39	   POL(c_{b_1}(x_1)) = 4*x_1
132.95/34.39	   POL(c_{c_1}(x_1)) = 4 + 2*x_1
132.95/34.39	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
132.95/34.39	Rules from R:
132.95/34.39	
132.95/34.39	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
132.95/34.39	   c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	Rules from S:
132.95/34.39	
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
132.95/34.39	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
132.95/34.39	
132.95/34.39	
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(8)
132.95/34.39	Obligation:
132.95/34.39	Relative term rewrite system:
132.95/34.39	The relative TRS consists of the following R rules:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	S is empty.
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(9) SIsEmptyProof (EQUIVALENT)
132.95/34.39	The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(10)
132.95/34.39	Obligation:
132.95/34.39	Q restricted rewrite system:
132.95/34.39	The TRS R consists of the following rules:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	Q is empty.
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(11) QTRSRRRProof (EQUIVALENT)
132.95/34.39	Used ordering:
132.95/34.39	Polynomial interpretation [POLO]:
132.95/34.39	
132.95/34.39	   POL(a_{a_1}(x_1)) = x_1
132.95/34.39	   POL(a_{c_1}(x_1)) = x_1
132.95/34.39	   POL(b_{a_1}(x_1)) = x_1
132.95/34.39	   POL(b_{c_1}(x_1)) = x_1
132.95/34.39	   POL(c_{a_1}(x_1)) = x_1
132.95/34.39	   POL(c_{b_1}(x_1)) = x_1
132.95/34.39	   POL(c_{c_1}(x_1)) = 1 + x_1
132.95/34.39	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
132.95/34.39	
132.95/34.39	
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(12)
132.95/34.39	Obligation:
132.95/34.39	Q restricted rewrite system:
132.95/34.39	The TRS R consists of the following rules:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	Q is empty.
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(13) DependencyPairsProof (EQUIVALENT)
132.95/34.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(14)
132.95/34.39	Obligation:
132.95/34.39	Q DP problem:
132.95/34.39	The TRS P consists of the following rules:
132.95/34.39	
132.95/34.39	   C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1))
132.95/34.39	   C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1))
132.95/34.39	   C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1))
132.95/34.39	
132.95/34.39	The TRS R consists of the following rules:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	Q is empty.
132.95/34.39	We have to consider all minimal (P,Q,R)-chains.
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(15) DependencyGraphProof (EQUIVALENT)
132.95/34.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(16)
132.95/34.39	Obligation:
132.95/34.39	Q DP problem:
132.95/34.39	The TRS P consists of the following rules:
132.95/34.39	
132.95/34.39	   C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1))
132.95/34.39	   C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1))
132.95/34.39	
132.95/34.39	The TRS R consists of the following rules:
132.95/34.39	
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
132.95/34.39	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
132.95/34.39	
132.95/34.39	Q is empty.
132.95/34.39	We have to consider all minimal (P,Q,R)-chains.
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(17) QDPSizeChangeProof (EQUIVALENT)
132.95/34.39	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
132.95/34.39	
132.95/34.39	From the DPs we obtained the following set of size-change graphs:
132.95/34.39	*C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1))
132.95/34.39	The graph contains the following edges 1 > 1
132.95/34.39	
132.95/34.39	
132.95/34.39	*C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1))
132.95/34.39	The graph contains the following edges 1 > 1
132.95/34.39	
132.95/34.39	
132.95/34.39	----------------------------------------
132.95/34.39	
132.95/34.39	(18)
132.95/34.39	YES
133.11/36.98	EOF