129.25/33.55	YES
129.25/33.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
129.25/33.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
129.25/33.56	
129.25/33.56	
129.25/33.56	Termination of the given RelTRS could be proven:
129.25/33.56	
129.25/33.56	(0) RelTRS
129.25/33.56	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
129.25/33.56	(2) RelTRS
129.25/33.56	(3) FlatCCProof [EQUIVALENT, 0 ms]
129.25/33.56	(4) RelTRS
129.25/33.56	(5) RootLabelingProof [EQUIVALENT, 0 ms]
129.25/33.56	(6) RelTRS
129.25/33.56	(7) RelTRSRRRProof [EQUIVALENT, 652 ms]
129.25/33.56	(8) RelTRS
129.25/33.56	(9) RelTRSRRRProof [EQUIVALENT, 4 ms]
129.25/33.56	(10) RelTRS
129.25/33.56	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
129.25/33.56	(12) YES
129.25/33.56	
129.25/33.56	
129.25/33.56	----------------------------------------
129.25/33.56	
129.25/33.56	(0)
129.25/33.56	Obligation:
129.25/33.56	Relative term rewrite system:
129.25/33.56	The relative TRS consists of the following R rules:
129.25/33.56	
129.25/33.56	   a(b(b(x1))) -> c(b(a(x1)))
129.25/33.56	   b(a(a(x1))) -> a(c(c(x1)))
129.25/33.56	   c(a(c(x1))) -> c(c(b(x1)))
129.25/33.56	   b(c(a(x1))) -> b(b(a(x1)))
129.25/33.56	
129.25/33.56	The relative TRS consists of the following S rules:
129.25/33.56	
129.25/33.56	   b(c(c(x1))) -> a(a(a(x1)))
129.25/33.56	   a(b(c(x1))) -> a(a(b(x1)))
129.25/33.56	   b(c(b(x1))) -> c(a(b(x1)))
129.25/33.56	
129.25/33.56	
129.25/33.56	----------------------------------------
129.25/33.56	
129.25/33.56	(1) RelTRS Reverse (EQUIVALENT)
129.25/33.56	We have reversed the following relative TRS [REVERSE]:
129.25/33.57	The set of rules R is 
129.25/33.57	   a(b(b(x1))) -> c(b(a(x1)))
129.25/33.57	   b(a(a(x1))) -> a(c(c(x1)))
129.25/33.57	   c(a(c(x1))) -> c(c(b(x1)))
129.25/33.57	   b(c(a(x1))) -> b(b(a(x1)))
129.25/33.57	
129.25/33.57	The set of rules S is 
129.25/33.57	   b(c(c(x1))) -> a(a(a(x1)))
129.25/33.57	   a(b(c(x1))) -> a(a(b(x1)))
129.25/33.57	   b(c(b(x1))) -> c(a(b(x1)))
129.25/33.57	
129.25/33.57	We have obtained the following relative TRS:
129.25/33.57	The set of rules R is 
129.25/33.57	   b(b(a(x1))) -> a(b(c(x1)))
129.25/33.57	   a(a(b(x1))) -> c(c(a(x1)))
129.25/33.57	   c(a(c(x1))) -> b(c(c(x1)))
129.25/33.57	   a(c(b(x1))) -> a(b(b(x1)))
129.25/33.57	
129.25/33.57	The set of rules S is 
129.25/33.57	   c(c(b(x1))) -> a(a(a(x1)))
129.25/33.57	   c(b(a(x1))) -> b(a(a(x1)))
129.25/33.57	   b(c(b(x1))) -> b(a(c(x1)))
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(2)
129.25/33.57	Obligation:
129.25/33.57	Relative term rewrite system:
129.25/33.57	The relative TRS consists of the following R rules:
129.25/33.57	
129.25/33.57	   b(b(a(x1))) -> a(b(c(x1)))
129.25/33.57	   a(a(b(x1))) -> c(c(a(x1)))
129.25/33.57	   c(a(c(x1))) -> b(c(c(x1)))
129.25/33.57	   a(c(b(x1))) -> a(b(b(x1)))
129.25/33.57	
129.25/33.57	The relative TRS consists of the following S rules:
129.25/33.57	
129.25/33.57	   c(c(b(x1))) -> a(a(a(x1)))
129.25/33.57	   c(b(a(x1))) -> b(a(a(x1)))
129.25/33.57	   b(c(b(x1))) -> b(a(c(x1)))
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(3) FlatCCProof (EQUIVALENT)
129.25/33.57	We used flat context closure [ROOTLAB]
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(4)
129.25/33.57	Obligation:
129.25/33.57	Relative term rewrite system:
129.25/33.57	The relative TRS consists of the following R rules:
129.25/33.57	
129.25/33.57	   a(c(b(x1))) -> a(b(b(x1)))
129.25/33.57	   b(b(b(a(x1)))) -> b(a(b(c(x1))))
129.25/33.57	   a(b(b(a(x1)))) -> a(a(b(c(x1))))
129.25/33.57	   c(b(b(a(x1)))) -> c(a(b(c(x1))))
129.25/33.57	   b(a(a(b(x1)))) -> b(c(c(a(x1))))
129.25/33.57	   a(a(a(b(x1)))) -> a(c(c(a(x1))))
129.25/33.57	   c(a(a(b(x1)))) -> c(c(c(a(x1))))
129.25/33.57	   b(c(a(c(x1)))) -> b(b(c(c(x1))))
129.25/33.57	   a(c(a(c(x1)))) -> a(b(c(c(x1))))
129.25/33.57	   c(c(a(c(x1)))) -> c(b(c(c(x1))))
129.25/33.57	
129.25/33.57	The relative TRS consists of the following S rules:
129.25/33.57	
129.25/33.57	   b(c(b(x1))) -> b(a(c(x1)))
129.25/33.57	   b(c(c(b(x1)))) -> b(a(a(a(x1))))
129.25/33.57	   a(c(c(b(x1)))) -> a(a(a(a(x1))))
129.25/33.57	   c(c(c(b(x1)))) -> c(a(a(a(x1))))
129.25/33.57	   b(c(b(a(x1)))) -> b(b(a(a(x1))))
129.25/33.57	   a(c(b(a(x1)))) -> a(b(a(a(x1))))
129.25/33.57	   c(c(b(a(x1)))) -> c(b(a(a(x1))))
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(5) RootLabelingProof (EQUIVALENT)
129.25/33.57	We used plain root labeling [ROOTLAB] with the following heuristic:
129.25/33.57	LabelAll: All function symbols get labeled
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(6)
129.25/33.57	Obligation:
129.25/33.57	Relative term rewrite system:
129.25/33.57	The relative TRS consists of the following R rules:
129.25/33.57	
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	
129.25/33.57	The relative TRS consists of the following S rules:
129.25/33.57	
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(7) RelTRSRRRProof (EQUIVALENT)
129.25/33.57	We used the following monotonic ordering for rule removal:
129.25/33.57	Polynomial interpretation [POLO]:
129.25/33.57	
129.25/33.57	   POL(a_{a_1}(x_1)) = 2*x_1
129.25/33.57	   POL(a_{b_1}(x_1)) = 1 + 2*x_1
129.25/33.57	   POL(a_{c_1}(x_1)) = 1 + 4*x_1
129.25/33.57	   POL(b_{a_1}(x_1)) = 3 + 2*x_1
129.25/33.57	   POL(b_{b_1}(x_1)) = 2 + 2*x_1
129.25/33.57	   POL(b_{c_1}(x_1)) = 1 + 4*x_1
129.25/33.57	   POL(c_{a_1}(x_1)) = x_1
129.25/33.57	   POL(c_{b_1}(x_1)) = 1 + x_1
129.25/33.57	   POL(c_{c_1}(x_1)) = 2*x_1
129.25/33.57	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
129.25/33.57	Rules from R:
129.25/33.57	
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
129.25/33.57	   c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	Rules from S:
129.25/33.57	
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
129.25/33.57	
129.25/33.57	
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(8)
129.25/33.57	Obligation:
129.25/33.57	Relative term rewrite system:
129.25/33.57	The relative TRS consists of the following R rules:
129.25/33.57	
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	
129.25/33.57	The relative TRS consists of the following S rules:
129.25/33.57	
129.25/33.57	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(9) RelTRSRRRProof (EQUIVALENT)
129.25/33.57	We used the following monotonic ordering for rule removal:
129.25/33.57	Knuth-Bendix order [KBO] with precedence:c_{a_1}_1 > b_{c_1}_1 > b_{b_1}_1 > c_{c_1}_1 > a_{c_1}_1 > a_{b_1}_1 > b_{a_1}_1 > c_{b_1}_1
129.25/33.57	
129.25/33.57	and weight map:
129.25/33.57	
129.25/33.57	   a_{c_1}_1=2
129.25/33.57	   c_{b_1}_1=2
129.25/33.57	   b_{a_1}_1=1
129.25/33.57	   a_{b_1}_1=3
129.25/33.57	   b_{b_1}_1=1
129.25/33.57	   b_{c_1}_1=2
129.25/33.57	   c_{c_1}_1=3
129.25/33.57	   c_{a_1}_1=2
129.25/33.57	
129.25/33.57	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
129.25/33.57	Rules from R:
129.25/33.57	
129.25/33.57	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1)))
129.25/33.57	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
129.25/33.57	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
129.25/33.57	Rules from S:
129.25/33.57	
129.25/33.57	   b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
129.25/33.57	
129.25/33.57	
129.25/33.57	
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(10)
129.25/33.57	Obligation:
129.25/33.57	Relative term rewrite system:
129.25/33.57	R is empty.
129.25/33.57	S is empty.
129.25/33.57	
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(11) RIsEmptyProof (EQUIVALENT)
129.25/33.57	The TRS R is empty. Hence, termination is trivially proven.
129.25/33.57	----------------------------------------
129.25/33.57	
129.25/33.57	(12)
129.25/33.57	YES
129.25/33.61	EOF