5.66/2.28	YES
5.66/2.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.66/2.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.66/2.30	
5.66/2.30	
5.66/2.30	Termination of the given RelTRS could be proven:
5.66/2.30	
5.66/2.30	(0) RelTRS
5.66/2.30	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
5.66/2.30	(2) RelTRS
5.66/2.30	(3) FlatCCProof [EQUIVALENT, 0 ms]
5.66/2.30	(4) RelTRS
5.66/2.30	(5) RootLabelingProof [EQUIVALENT, 0 ms]
5.66/2.30	(6) RelTRS
5.66/2.30	(7) RelTRSRRRProof [EQUIVALENT, 44 ms]
5.66/2.30	(8) RelTRS
5.66/2.30	(9) RelTRSRRRProof [EQUIVALENT, 17 ms]
5.66/2.30	(10) RelTRS
5.66/2.30	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
5.66/2.30	(12) YES
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(0)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	The relative TRS consists of the following R rules:
5.66/2.30	
5.66/2.30	   b(c(c(x1))) -> a(c(b(x1)))
5.66/2.30	   c(a(a(x1))) -> c(b(b(x1)))
5.66/2.30	   a(b(a(x1))) -> c(b(a(x1)))
5.66/2.30	
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   a(b(c(x1))) -> a(a(c(x1)))
5.66/2.30	   c(a(a(x1))) -> c(a(b(x1)))
5.66/2.30	   a(b(c(x1))) -> c(c(c(x1)))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(1) RelTRS Reverse (EQUIVALENT)
5.66/2.30	We have reversed the following relative TRS [REVERSE]:
5.66/2.30	The set of rules R is 
5.66/2.30	   b(c(c(x1))) -> a(c(b(x1)))
5.66/2.30	   c(a(a(x1))) -> c(b(b(x1)))
5.66/2.30	   a(b(a(x1))) -> c(b(a(x1)))
5.66/2.30	
5.66/2.30	The set of rules S is 
5.66/2.30	   a(b(c(x1))) -> a(a(c(x1)))
5.66/2.30	   c(a(a(x1))) -> c(a(b(x1)))
5.66/2.30	   a(b(c(x1))) -> c(c(c(x1)))
5.66/2.30	
5.66/2.30	We have obtained the following relative TRS:
5.66/2.30	The set of rules R is 
5.66/2.30	   c(c(b(x1))) -> b(c(a(x1)))
5.66/2.30	   a(a(c(x1))) -> b(b(c(x1)))
5.66/2.30	   a(b(a(x1))) -> a(b(c(x1)))
5.66/2.30	
5.66/2.30	The set of rules S is 
5.66/2.30	   c(b(a(x1))) -> c(a(a(x1)))
5.66/2.30	   a(a(c(x1))) -> b(a(c(x1)))
5.66/2.30	   c(b(a(x1))) -> c(c(c(x1)))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(2)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	The relative TRS consists of the following R rules:
5.66/2.30	
5.66/2.30	   c(c(b(x1))) -> b(c(a(x1)))
5.66/2.30	   a(a(c(x1))) -> b(b(c(x1)))
5.66/2.30	   a(b(a(x1))) -> a(b(c(x1)))
5.66/2.30	
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   c(b(a(x1))) -> c(a(a(x1)))
5.66/2.30	   a(a(c(x1))) -> b(a(c(x1)))
5.66/2.30	   c(b(a(x1))) -> c(c(c(x1)))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(3) FlatCCProof (EQUIVALENT)
5.66/2.30	We used flat context closure [ROOTLAB]
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(4)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	The relative TRS consists of the following R rules:
5.66/2.30	
5.66/2.30	   a(b(a(x1))) -> a(b(c(x1)))
5.66/2.30	   c(c(c(b(x1)))) -> c(b(c(a(x1))))
5.66/2.30	   b(c(c(b(x1)))) -> b(b(c(a(x1))))
5.66/2.30	   a(c(c(b(x1)))) -> a(b(c(a(x1))))
5.66/2.30	   c(a(a(c(x1)))) -> c(b(b(c(x1))))
5.66/2.30	   b(a(a(c(x1)))) -> b(b(b(c(x1))))
5.66/2.30	   a(a(a(c(x1)))) -> a(b(b(c(x1))))
5.66/2.30	
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   c(b(a(x1))) -> c(a(a(x1)))
5.66/2.30	   c(b(a(x1))) -> c(c(c(x1)))
5.66/2.30	   c(a(a(c(x1)))) -> c(b(a(c(x1))))
5.66/2.30	   b(a(a(c(x1)))) -> b(b(a(c(x1))))
5.66/2.30	   a(a(a(c(x1)))) -> a(b(a(c(x1))))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(5) RootLabelingProof (EQUIVALENT)
5.66/2.30	We used plain root labeling [ROOTLAB] with the following heuristic:
5.66/2.30	LabelAll: All function symbols get labeled
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(6)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	The relative TRS consists of the following R rules:
5.66/2.30	
5.66/2.30	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
5.66/2.30	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
5.66/2.30	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{a_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(x1)))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(7) RelTRSRRRProof (EQUIVALENT)
5.66/2.30	We used the following monotonic ordering for rule removal:
5.66/2.30	Polynomial interpretation [POLO]:
5.66/2.30	
5.66/2.30	   POL(a_{a_1}(x_1)) = 1 + x_1
5.66/2.30	   POL(a_{b_1}(x_1)) = x_1
5.66/2.30	   POL(a_{c_1}(x_1)) = x_1
5.66/2.30	   POL(b_{a_1}(x_1)) = 1 + x_1
5.66/2.30	   POL(b_{b_1}(x_1)) = x_1
5.66/2.30	   POL(b_{c_1}(x_1)) = x_1
5.66/2.30	   POL(c_{a_1}(x_1)) = x_1
5.66/2.30	   POL(c_{b_1}(x_1)) = x_1
5.66/2.30	   POL(c_{c_1}(x_1)) = x_1
5.66/2.30	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.66/2.30	Rules from R:
5.66/2.30	
5.66/2.30	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
5.66/2.30	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
5.66/2.30	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
5.66/2.30	Rules from S:
5.66/2.30	
5.66/2.30	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{c_1}(c_{a_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{c_1}(c_{b_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{c_1}(c_{c_1}(x1)))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	
5.66/2.30	
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(8)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	The relative TRS consists of the following R rules:
5.66/2.30	
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(9) RelTRSRRRProof (EQUIVALENT)
5.66/2.30	We used the following monotonic ordering for rule removal:
5.66/2.30	Polynomial interpretation [POLO]:
5.66/2.30	
5.66/2.30	   POL(a_{a_1}(x_1)) = x_1
5.66/2.30	   POL(a_{b_1}(x_1)) = x_1
5.66/2.30	   POL(a_{c_1}(x_1)) = x_1
5.66/2.30	   POL(b_{a_1}(x_1)) = x_1
5.66/2.30	   POL(b_{b_1}(x_1)) = x_1
5.66/2.30	   POL(b_{c_1}(x_1)) = x_1
5.66/2.30	   POL(c_{a_1}(x_1)) = x_1
5.66/2.30	   POL(c_{b_1}(x_1)) = x_1
5.66/2.30	   POL(c_{c_1}(x_1)) = 1 + x_1
5.66/2.30	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.66/2.30	Rules from R:
5.66/2.30	
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.66/2.30	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.66/2.30	Rules from S:
5.66/2.30	none
5.66/2.30	
5.66/2.30	
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(10)
5.66/2.30	Obligation:
5.66/2.30	Relative term rewrite system:
5.66/2.30	R is empty.
5.66/2.30	The relative TRS consists of the following S rules:
5.66/2.30	
5.66/2.30	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.66/2.30	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.66/2.30	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.66/2.30	
5.66/2.30	
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(11) RIsEmptyProof (EQUIVALENT)
5.66/2.30	The TRS R is empty. Hence, termination is trivially proven.
5.66/2.30	----------------------------------------
5.66/2.30	
5.66/2.30	(12)
5.66/2.30	YES
6.04/2.39	EOF